The Chinese University of Hong Kong Michigan State University

DIMENSION THEORY OF ITERATED FUNCTION SYSTEMS

DE-JUN FENG AND HUYI HU

Abstract. Let {S_{i}}_{i=1}^{ℓ} be an iterated function system (IFS) on ℝ^{d} with attractor K. Let (Σ, σ) denote the one-sided full shift over the alphabet {1, …, ℓ}. We define the projection entropy function h_{π} on the space of invariant measures on Σ associated with the coding map π : Σ → K, and develop some basic ergodic properties about it. This concept turns out to be crucial in the study of dimensional properties of invariant measures on K. We show that for any conformal IFS (resp., the direct product of finitely many conformal IFS), without any separation condition, the projection of an ergodic measure under π is always exactly dimensional and, its Hausdorff dimension can be represented as the ratio of its projection entropy to its Lyapunov exponent (resp., the linear combination of projection entropies associated with several coding maps). Furthermore, for any conformal IFS and certain affine IFS, we prove a variational principle between the Hausdorff dimension of the attractor and that of projections of ergodic measures.

Contents

1. Introduction 2 2. Statement of main results 4 3. Density results about conditional measures 10 4. Projection measure-theoretic entropies associated with IFS 17 4.1. Some basic properties 17 4.2. A version of Shannon-McMillan-Breiman Theorem associated with

IFS 22

4.3. Projection entropy under the ergodic decomposition 26 4.4. The projection entropy for certain affine IFS and the proof of

Theorem 2.3 28

4.5. Upper semi-continuity of h_{π}(σ, ·) under the AWSC 31 5. Some geometric properties of C^{1} IFS 34 6. Estimates for local dimensions of invariant measures for C^{1} IFS 39 7. Proofs of Theorem 2.11 and Theorem 2.12 44 7.1. Proof of Theorem 2.11 44 7.2. Proof of Theorem 2.12 48 8. A variational principle about dimensions of self-conformal sets 49 9. Proof of Theorem 2.15 51 10. A final remark about infinite non-contractive IFS 57 References 58

1. Introduction

Let {S_{i} : X → X}_{i=1}^{ℓ} be a family of contractive maps on a nonempty closed set X ⊂ ℝ^{d}. Following Barnsley [2], we say that Φ = {S_{i}}_{i=1}^{ℓ} is an iterated function system (IFS) on X. Hutchinson [28] showed that there is a unique nonempty compact set K ⊂ X, called the attractor of {S_{i}}_{i=1}^{ℓ}, such that K = ⋃_{i=1}^{ℓ} S_{i}(K). A probability measure μ on ℝ^{d} is said to be exactly dimensional if there is a constant C such that the local dimension

d(μ, x) = lim_{r→0} \frac{log μ(B(x, r))}{log r}

exists and equals C for μ-a.e. x ∈ ℝ^{d}, where B(x, r) denotes the closed ball of radius r centered at x. It was shown by Young [65] that in such case, the Hausdorff dimension of μ is equal to C. (See also [14, 43, 51].)

The motivation of the paper is to study the Hausdorff dimension of an invariant measure μ (see Section 2 for precise meaning) for conformal and affine IFS with overlaps. To deal with overlaps, we regard such a system as the image of a natural projection π from the one-sided full shift space over ℓ symbols. Hence we obtain a dynamical system. We introduce a notion projection entropy, which plays the similar role as the classical entropy for IFS satisfying the open set condition, and it becomes the classical entropy if the projection is finite to one. The concept of projection entropy turns out to be crucial in the study of dimensional properties of invariant measures on attractors of either conformal IFS with overlaps or affine IFS.

We develop some basic properties about projection entropy (Theorem 2.2, 2.3). We prove that for conformal IFS with overlaps, every ergodic measure μ is exactly dimensional and d(μ, x) is equal to the projection entropy divided by the Lyapunov exponent (Theorem 2.8). Furthermore, if Φ is a direct product of conformal IFS (see Definition 2.10 for precise meaning), then for every ergodic measure on K the local dimension can be expressed by a Ledrappier-Young type formula in terms of projection entropies and Lyapunov exponents (Theorem 2.11). We also prove variational results about Hausdorff dimension for conformal IFS and certain affine IFS (Theorem 2.13 and 2.15), which says that the Hausdorff dimension of the attractor K is equal to the supremum of Hausdorff dimension of μ taking over all ergodic measures. The results we obtain cover some interesting cases such as S_{i}(x) = diag(ρ_{1}, …, ρ_{d})x + a_{i}, where i = 1, …, ℓ and ρ_{i}^{−1} are Pisot or Salem numbers and a_{i} ∈ ℤ^{d}.

The problem whether a given measure is exactly dimensional, and whether the Hausdorff dimension of an attractor can be assumed or approximated by that of an invariant measure have been well studied in the literature for C^{1+α} conformal IFS which satisfy the open set condition (cf. [6, 23, 47]). It is well known that in such case, any ergodic measure μ is exactly dimensional with the Hausdorff dimension given by the classic entropy divided by the Lyapunov exponent. Furthermore there is a unique invariant measure μ with dim_{H}(μ) = dim_{H}(K), the Hausdorff dimension of K. However the problems become much complicated and intractable without the assumption of the open set condition. Partial results have only been obtained for conformal IFS that satisfy the finite type condition (see [45] for the definition). In that case, a Bernoulli measure is exactly dimensional and its Hausdorff dimension may be expressed as the upper Lyapunov exponent of certain random matrices (see e.g. [16, 17, 36, 39, 37]), and furthermore the Hausdorff dimension of K can be computed (see e.g. [35, 54, 45]).

There are some results for certain special non-overlapping affine IFS. McMullen [44] and Bedford [5] independently computed the Hausdorff dimension and the box dimension of the attractor of the following planar affine IFS

S_{i}(x) = [\begin{matrix}n^{−1} & 0 \\ 0 & k^{−1} \end{matrix}] x + [\begin{matrix}a_{i}/n \\ b_{i}/k, \end{matrix}] , i = 1, …, ℓ,

where all a_{i}, b_{i} are integers, 0 ≤ a_{i} < n and 0 ≤ b_{i} < k. Furthermore they showed that there is a Bernoulli measure of full Hausdorff dimension. This result was extended by Kenyon and Peres [33] to higher dimensional self-affine Sierpinski sponges, for which ergodic measures are proved to be exactly dimensional with Hausdorff dimension given by a Ledrappier-Young type formula. Another extension of McMullen and Bedford’s result to a boarder class of planar affine IFS {S_{i}}_{i=1}^{ℓ} was given by Gatzouras and Lalley [20], in which S_{i} map the unit square (0, 1)^{2} into disjoint rectangles with sides parallel to the axes (where the longer sides are parallel to the x-axis, furthermore once projected onto the x-axis these rectangles are either identical, or disjoint). Further extensions were given recently by Barański [1], Feng and Wang [19], Luzia [41] and Olivier [46]. For other related results, see e.g. [52, 38, 34, 21, 25, 27, 17, 60, 3, 31].

Along another direction, in [11] Falconer gave a variational formula for the Hausdorff and box dimensions for “almost all” self-affine sets under some assumptions. This formula remains true under some weaker conditions [61, 29]. Käenmäki [30] proved that for “almost all” self-affine sets there exists an ergodic measure m so that m ∘ π^{−1} is of full Hausdorff dimension.

Our arguments use ergodic theory and Rohlin’s theory about conditional measures. The proofs of Theorem 2.6 and Theorem 2.11 are based on some ideas from the work of Ledrappier and Young [40] and techniques in analyzing the densities of conditional measures associated with overlapping IFS.

So far we have restricted ourselves on the study of finite contractive IFS. However we point out that part of our results remain valid for certain non-contractive infinite IFS (see Section 10 for details).

The paper is organized as follows. The main results are given in Section 2. In Section 3, we prove some density results about conditional measures. In Section 4, we investigate the properties of projection entropy and prove Theorem 2.2 and 2.3. In Section 5, we give some local geometric properties of a C^{1} IFS. In Section 6, we prove a generalized version of Theorem 2.6, which is based on a key proposition (Proposition 6.1) about the densities of conditional measures. In Section 7, we prove Theorem 2.11 and 2.12. In Section 8, we prove Theorem 2.13 and in Section 9, we prove Theorem 2.15. In Section 10 we give a remark regarding certain non-contractive infinite IFS.

2. Statement of main results

Let {S_{i}}_{i=1}^{ℓ} be an IFS on a closed set X ⊂ ℝ^{d}. Denote by K its attractor. Let Σ = {1, …, ℓ}^{ℕ} associated with the left shift σ (cf. [9]). Let M_{σ}(Σ) denote the space of σ-invariant measures on Σ, endowed with the weak-star topology. Let π : Σ → K be the canonical projection defined by

(1)  {π(x)} = ⋂_{n=1}^{∞} S_{x_{1}} ∘ S_{x_{2}} ∘ ⋯ ∘ S_{x_{n}}(K),  where x = (x_{i})_{i=1}^{∞}.

A measure μ on K is called invariant (resp., ergodic) for the IFS if there is an invariant (resp. ergodic) measure ν on Σ such that μ = ν ∘ π^{−1}.

Let (Ω, F, ν) be a probability space. For a sub-σ-algebra A of F and f ∈ L^{1}(Ω, F, ν), we denote by E_{ν}(f|A) the the conditional expectation of f given A. For countable F-measurable partition ξ of Ω, we denote by I_{ν}(ξ|A) the conditional information of ξ given A, which is given by the formula

(2)  I_{ν}(ξ|A) = − ∑_{A∈ξ} χ_{A} log E_{ν}(χ_{A}|A),

where χ_{A} denotes the characteristic function on A. The conditional entropy of ξ given A, written H_{ν}(ξ|A), is defined by the formula

H_{ν}(ξ|A) = ∫ I_{ν}(ξ|A) dν.

(See e.g. [48] for more details.) The above information and entropy are unconditional when A = N, the trivial σ-algebra consisting of sets of measure zero and one, and in this case we write

I_{ν}(ξ|N) =: I_{ν}(ξ) and H_{ν}(ξ|N) =: H_{ν}(ξ).

Now we consider the space (Σ, B(Σ), m), where B(Σ) is the Borel σ-algebra on Σ and m ∈ M_{σ}(Σ). Let P denote the Borel partition

(3)  P = {[j] : 1 ≤ j ≤ ℓ}

of Σ, where [j] = {(x_{i})_{i=1}^{∞} ∈ Σ : x_{1} = j}. Let I denote the σ-algebra

I = {B ∈ B(Σ) : σ^{−1}B = B}.

For convenience, we use γ to denote the Borel σ-algebra B(ℝ^{d}) on ℝ^{d}.

Definition 2.1. For any m ∈ M_{σ}(Σ), we call

h_{π}(σ, m) := H_{m}(P|σ^{−1}π^{−1}γ) − H_{m}(P|π^{−1}γ)

the projection entropy of m under π w.r.t. {S_{i}}_{i=1}^{ℓ}, and we call

h_{π}(σ, m, x) := E_{m} (f|I) (x)

the local projection entropy of m at x under π w.r.t. {S_{i}}_{i=1}^{ℓ}, where f denotes the function I_{m}(P|σ^{−1}π^{−1}γ) − I_{m}(P|π^{−1}γ).

It is clear that h_{π}(σ, m) = ∫ h_{π}(σ, m, x) dm(x). Our first result is the following theorem.

Theorem 2.2. Let {S_{i}}_{i=1}^{ℓ} be an IFS. Then

(i) For any m ∈ M_{σ}(Σ), we have 0 ≤ h_{π}(σ, m) ≤ h(σ, m), where h(σ, m) denotes the classical measure-theoretic entropy of m associated with σ.

(ii) The map m ↦ h_{π}(σ, m) is affine on M_{σ}(Σ). Furthermore if m = ∫ ν dℙ(ν) is the ergodic decomposition of m, we have

h_{π}(σ, m) = ∫ h_{π}(σ, ν) dℙ(ν).

(iii) For any m ∈ M_{σ}(Σ), we have

lim_{n→∞} \frac{1}{n}I_{m}(P_{0}^{n−1}|π^{−1}γ)(x) = h(σ, m, x) − h_{π}(σ, m, x)

for m-a.e. x ∈ Σ, where h(σ, m, x) denotes the local entropy of m at x, that is, h(σ, m, x) = I_{m}(P|σ^{−1}B(Σ))(x).

Part (iii) of the theorem is an analogue of the classical relativized Shannon-McMillan-Breiman theorem (see, e.g. [8, Lemma 4.1]). However, we should notice that the sub σ-algebra π^{−1}γ in our consideration is not σ-invariant in general (see Remark 4.11).

Part (iii) also implies that if the map π: Σ → K is finite-to-one, then

h_{π}(σ, m) = h(σ, m)

for any m ∈ M_{σ}(Σ). In Section 4, we will present a sufficient and necessary condition for the equality (see Corollary 4.16). However for general overlapping IFS, the projection entropy can be strictly less than the classical entropy.

In our next theorem, we give a geometric characterization of the projection entropy for certain affine IFS, which will be used later in the proof of our variational results about the Hausdorff and box dimensions of self-affine sets.

Theorem 2.3. Assume that Φ = {S_{i}}_{i=1}^{ℓ} is an IFS on ℝ^{d} of the form

S_{i}(x) = Ax + c_{i} (i = 1, …, ℓ),

where A is a d×d non-singular contractive real matrix and c_{i} ∈ ℝ^{d}. Let K denote the attractor of Φ. Let Q denote the partition {[0, 1)^{d} + α : α ∈ ℤ^{d}} of ℝ^{d}. For n = 0, 1, …, and x ∈ ℝ^{d}, we set Q_{n} = {A^{n}Q : Q ∈ Q}. Then

(i) For any m ∈ M_{σ}(Σ), we have

h_{π}(σ, m) = lim_{n→∞} \frac{H_{m}(π^{−1}Q_{n})}{n}.

(ii) Moreover,

lim_{n→∞} \frac{log #\{Q ∈ Q : A^{n}Q ∩ K ≠ ∅\}}{n} = sup\{h_{π}(σ, m) : m ∈ M_{σ}(Σ)\}.

To give the applications of projection entropy in dimension theory of IFS, we need some more notation and definitions.

Definition 2.4. {S_{i} : X → X}_{i=1}^{ℓ} is called a C^{1} IFS on a compact set X ⊂ ℝ^{d} if each S_{i} extends to a contracting C^{1}-diffeomorphism S_{i} : U → S_{i}(U) ⊂ U on an open set U ⊃ X.

For any d × d real matrix M, we use ∥M∥ to denote the usual norm of M, and []M[] the smallest singular value of M, i.e.,

∥M∥ = max{|Mv| : v ∈ ℝ^{d}, |v| = 1} and
[]M[] = min{|Mv| : v ∈ ℝ^{d}, |v| = 1}.

(4)

Definition 2.5. Let {S_{i}}_{i=1}^{ℓ} be a C^{1} IFS. For x = (x_{j})_{j=1}^{∞} ∈ Σ, the upper and lower Lyapunov exponents of {S_{i}}_{i=1}^{ℓ} at x are defined respectively by

\overline{λ}(x) = − lim inf_{n→∞} \frac{1}{n} log[]S′_{x_{1}...x_{n}}(πσ^{n}x)[],

\underline{λ}(x) = − lim sup_{n→∞} \frac{1}{n} log ∥S′_{x_{1}...x_{n}}(πσ^{n}x)∥,

where S′_{x_{1}...x_{n}}(πσ^{n}x) denotes the differential of S_{x_{1}...x_{n}} := S_{x_{1}} ∘ S_{x_{2}} ∘ … ∘ S_{x_{n}} at πσ^{n}x. When \overline{λ}(x) = \underline{λ}(x), the common value, denoted as λ(x), is called the Lyapunov exponent of {S_{i}}_{i=1}^{ℓ} at x.

It is easy to check that both \overline{λ} and \underline{λ} are positive-valued σ-invariant functions on Σ (i.e. \overline{λ} = \overline{λ} ∘ σ and \underline{λ} = \underline{λ} ∘ σ). Recall that for a probability measure μ on ℝ^{d}, the local upper and lower dimensions are defined respectively by

\overline{d}(μ, x) = lim sup_{r→0} \frac{log μ(B(x, r))}{log r}, \underline{d}(μ, x) = lim inf_{r→0} \frac{log μ(B(x, r))}{log r},

where B(x, r) denotes the closed ball of radius r centered at x. If \overline{d}(μ, x) = \underline{d}(μ, x), the common value is denoted as d(μ, x) and is called the local dimension of m at x.

The following theorem gives an estimate of local dimensions of invariant measures on the attractor of an arbitrary C^{1} IFS, without any separation condition.

Theorem 2.6. Let {S_{i}}_{i=1}^{ℓ} be a C^{1} IFS with attractor K. Then for μ = m ∘ π^{−1}, where m ∈ M_{σ}(Σ), we have the following estimates:

\overline{d}(μ, πx) ≤ \frac{h_{π}(σ, m, x)}{\underline{λ}(x)} and \underline{d}(μ, πx) ≥ \frac{h_{π}(σ, m, x)}{\overline{λ}(x)} for m-a.e. x ∈ Σ,

where h_{π}(σ, m, x) denotes the local projection entropy of m at x under π (see Definition 2.1). In particular, if m is ergodic, we have

\frac{h_{π}(σ, m)}{∫ \overline{λ} dm} ≤ \underline{d}(μ, z) ≤ \overline{d}(μ, z) ≤ \frac{h_{π}(σ, m)}{∫ \underline{λ} dm} for μ-a.e. z ∈ K.

Definition 2.7. Let {S_{i}}_{i=1}^{ℓ} be a C^{1} IFS and m ∈ M_{σ}(Σ). We say that {S_{i}}_{i=1}^{ℓ} is m-conformal if λ(x) exists (i.e., \overline{λ}(x) = \underline{λ}(x)) for m-a.e. x ∈ Σ.

As a direct application of Theorem 2.6, we have Theorem 2.8. Assume that {S_{i}}_{i=1}^{ℓ} is m-conformal for some m ∈ M_{σ}(Σ). Let μ = m ∘ π^{−1}. Then we have

(5)  d(μ, πx) = \frac{h_{π}(σ, m, x)}{λ(x)} for m-a.e. x ∈ Σ.

In particular, if m is ergodic, we have

(6)  d(μ, z) = \frac{h_{π}(σ, m)}{∫ λ dm} for μ-a.e. z ∈ K.

Recall that S : U → S(U) is a conformal map if S′(x) : ℝ^{d} → ℝ^{d} satisfies ∥S′(x)∥ ≠ 0 and |S′(x)y| = ∥S′(x)∥|y| for all x ∈ U and y ∈ ℝ^{d}.

Definition 2.9. A C^{1} IFS {S_{i}}_{i=1}^{ℓ} is said to be weakly conformal if

\frac{1}{n}(log[]S′_{x_{1}...x_{n}}(πσ^{n}x)[] − log ∥S′_{x_{1}...x_{n}}(πσ^{n}x)∥)

converges to 0 uniformly on Σ as n tends to ∞. We say that {S_{i}}_{i=1}^{ℓ} is conformal if each S_{i} extends to a conformal map S_{i} : U → S_{i}(U) ⊂ U on an open set U ⊃ K, where K is the attractor of {S_{i}}_{i=1}^{ℓ}.

By definition, a conformal IFS is always weakly conformal. Furthermore, a weakly conformal IFS is m-conformal for each m ∈ M_{σ}(Σ) (see Proposition 5.6(ii)). There are some natural examples of weakly conformal IFS which are not conformal. For instance, let S_{i}(x) = A_{i}x + a_{i} (i = 1, …, ℓ) such that, for each i, A_{i} is a contracting linear map with eigenvalues equal to each other in modulus, and A_{i}A_{j} = A_{j}A_{i} for different i, j. Then such an IFS is always weakly conformal but not necessary to be conformal. The first conclusion follows from the asymptotic behavior

lim_{n→∞}[]A_{i}^{n}[]^{1/n} = lim_{n→∞} ∥A_{i}^{n}∥^{1/n} = ρ(A_{i}) (i = 1, …, ℓ),

where ρ(A_{i}) denotes the spectral radius of A_{i} (cf. [64]).

Theorem 2.8 verifies the existence of local dimensions for invariant measures on the attractor of an arbitrary weakly conformal IFS attractors, without any separation assumption. We point out that the exact dimensionality for overlapping self-similar measures was first claimed by Ledrappier, nevertheless no proof has been written out (cf. [50, p. 1619]). We remark that this property was also conjectured later by Fan, Lau and Rao in [15].

We can extend the above result to a class of non-conformal IFS.

Definition 2.10. Assume for j = 1, …, k, Φ_{j} := {S_{i,j}}_{i=1}^{ℓ} is a C^{1} IFS defined on a compact set X_{j} ⊂ ℝ^{q_{j}}. Let Φ := {S_{i}}_{i=1}^{ℓ} be the IFS on X_{1} × ⋯ × X_{k} ⊂ ℝ^{q_{1}} × ⋯ × ℝ^{q_{k}} given by

S_{i}(z_{1}, …, z_{k}) = (S_{i,1}(z_{1}), …, S_{i,k}(z_{k})) (i = 1, …, ℓ, j = 1, …, k, z_{j} ∈ X_{j}).

We say that Φ is the direct product of Φ_{1}, …, Φ_{k}, and write Φ = Φ_{1} × ⋯ × Φ_{k}.

Theorem 2.11. Let Φ = {S_{i}}_{i=1}^{ℓ} be the direct product of k C^{1} IFS Φ_{1}, …, Φ_{k}. Let μ = m ∘ π^{−1}, where m ∈ M_{σ}(Σ). Assume that Φ_{1}, …, Φ_{k} are m-conformal. Then

(i) d(μ, z) exists for μ-a.e. z.

(ii) Assume furthermore that m is ergodic. Then μ is exactly dimensional. Let τ be a permutation on {1, …, k} such that

λ_{τ(1)} ≤ λ_{τ(2)} ≤ ⋯ ≤ λ_{τ(k)},

where λ_{j} = ∫ λ_{j}(x) dm(x), and λ_{j}(x) denotes the Lyapunov exponent of Φ_{j} at x ∈ Σ. Then we have

(7) d(μ, z) = \frac{h_{π_{1}}(σ, m)}{λ_{τ(1)}} + ∑_{j=2}^{k} \frac{h_{π_{j}}(σ, m) − h_{π_{j−1}}(σ, m)}{λ_{τ(j)}} for μ-a.e. z,

where π_{j} denotes the canonical projection w.r.t. the IFS Φ_{τ(1)}×⋯×Φ_{τ(j)}, and h_{π_{j}}(σ, m) denotes the projection entropy of m under π_{j}.

We mention that fractals satisfy the conditions of the theorem include many interesting examples such as those studied in [44, 5, 20, 33], etc.

As an application of Theorem 2.11, we have

Theorem 2.12. Let {S_{i}}_{i=1}^{ℓ} be an IFS on ℝ^{d} of the form

S_{i}(x) = A_{i}x + a_{i}, i = 1, …, ℓ,

such that each A_{i} is a nonsingular contracting linear map on ℝ^{d}, and A_{i}A_{j} = A_{j}A_{i} for any 1 ≤ i, j ≤ ℓ. Then for any ergodic measure m on Σ, μ = m ∘ π^{−1} is exactly dimensional.

Indeed, under the assumption of Theorem 2.12, we can show that there is a nonsingular linear transformation T on ℝ^{d} such that the IFS {T ∘ S_{i} ∘ T^{−1}}_{i=1}^{ℓ} is the direct product of some weakly conformal IFS. Hence we can apply Theorem 2.11 in this situation.

We remark that formula (7) provides an analogue of that for the Hausdorff dimension of C^{1+α} hyperbolic measures along the unstable (resp. stable) manifold established by Ledrappier and Young [40].

The problem of the existence of local dimensions has also a long history in smooth dynamical systems. In [65], Young proved that an ergodic hyperbolic measure invariant under a C^{1+α} surface diffeomorphism is always exact dimensional. For a measures μ in high-dimensional C^{1+α} systems, Ledrappier and Young [40] proved the existence of δ^{u} and δ^{s}, the local dimensions along stable and unstable local manifolds, respectively, and the upper local dimension of μ is bounded by the sum of δ^{u}, δ^{s}, and the multiplicity of 0 as an exponent. Eckmann and Ruelle [10] indicated that it is unknown whether the local dimension of μ is the sum of δ^{u} and δ^{s} if μ is a hyperbolic measure. Then the problem was referred as Eckmann-Ruelle conjecture, and affirmatively answered by Barreira, Pesin and Schmeling in [4] seventeen years later. Some partial dimensional results were obtained for measures invariant under hyperbolic endomorphism [58, 59]. Recently, Qian and Xie [53] proved the exact dimensionality of ergodic measures invariant under C^{2} expanding endomorphism on smooth Riemannian manifolds.

In the remaining part of this section, we present some variational results about the Hausdorff dimension and the box dimension of attractors of IFS and that of invariant measures. First we consider conformal IFS.

Theorem 2.13. Let K be the attractor of a weakly conformal IFS {S_{i}}_{i=1}^{ℓ}. Then we have

(8) dim_{H} K = dim_{B} K
(9)  = sup {dim_{H} μ : μ = m ∘ π^{−1}, m ∈ M_{σ}(Σ), m is ergodic}
= max {dim_{H} μ : μ = m ∘ π^{−1}, m ∈ M_{σ}(Σ)}
(10)  = sup \{\frac{h_{π}(σ, m)}{∫ λ dm} : m ∈ M_{σ}(Σ)\} ,

where dim_{B} K denotes the box dimension of K.

Equality (8) was first proved by Falconer [12] for C^{1+α} conformal IFS. It is not known whether the supremum in (9) and (10) can be attained in the general setting of Theorem 2.13. However, this is true if the IFS {S_{i}}_{i=1}^{ℓ} satisfies an additional separation condition defined as follows.

Definition 2.14. An IFS {S_{i}}_{i=1}^{ℓ} on a compact set X ⊂ ℝ^{d} is said to satisfy the asymptotically weak separation condition (AWSC), if

lim_{n→∞} \frac{1}{n} log t_{n} = 0,

where t_{n} is given by

(11)  t_{n} = sup_{x∈ℝ^{d}} #{S_{u} : u ∈ {1, …, ℓ}^{n}, x ∈ S_{u}(K)},

here K is the attractor of {S_{i}}_{i=1}^{ℓ}.

The above definition was first introduced in [18] under a slightly different setting. For example, if 1/ρ is a Pisot or Salem number, then the IFS {ρx + a_{i}}_{i=1}^{ℓ} on ℝ, with a_{i} ∈ ℤ, satisfies the AWSC (see Proposition 5.3 and Remark 5.5 in [18]). Recall that a real number β > 1 is said to be a Salem number if it is an algebraic integer whose algebraic conjugates all have modulus not greater than 1, with at least one of which on the unit circle. Whilst β > 1 is called a Pisot number if it is an algebraic integer whose algebraic conjugates all have modulus less than 1. For instance, the largest root (≈ 1.72208) of x^{4} − x^{3} − x^{2} − x + 1 is a Salem number, and the golden ratio (\sqrt{5} + 1)/2 is a Pisot number. One is referred to [57] for more examples and properties about Pisot and Salem numbers. Under the AWSC assumption, we can show that the projection entropy map m ↦ h_{π}(σ, m) is upper semi-continuous on M_{σ}(Σ) (see Proposition 4.20) and, as a consequence, the supremum (9) and (10) can be attained at ergodic measures (see Remark 8.2).

Next we consider some affine IFS.

Theorem 2.15. Let Φ = {S_{i}}_{i=1}^{ℓ} be an affine IFS on ℝ^{d} given by

S_{i}(x_{1}, …, x_{d}) = (ρ_{1}x_{1}, ⋯, ρ_{d}x_{d}) + (a_{i,1}, …, a_{i,d}),

where ρ_{1} > ρ_{2} > ⋯ > ρ_{d} > 0 and a_{i,j} ∈ ℝ. Let K denote the attractor of Φ, and write λ_{j} = log(1/ρ_{j}) for j = 1, …, d and λ_{d+1} = ∞. View Φ as the direct product of Φ_{1}, …, Φ_{d}, where Φ_{j} = {S_{i,j}(x_{j}) = ρ_{j}x_{j} + a_{i,j}}_{i=1}^{ℓ}. Assume that Φ_{1} × ⋯ × Φ_{j} satisfies the AWSC for j = 1, …, d. Then we have

dim_{H} K = max {dim_{H} μ : μ = m ∘ π^{−1}, m is ergodic}
= max \{∑_{j=1}^{d} (\frac{1}{λ_{j}} − \frac{1}{λ_{j+1}} ) h_{π_{j}}(σ, m) : m is ergodic\} ,

where π_{j} is the canonical projection w.r.t. the IFS Φ_{1} × ⋯ × Φ_{j}. Furthermore

dim_{B} K = ∑_{j=1}^{d} (\frac{1}{λ_{j}} − \frac{1}{λ_{j+1}} ) H_{j},

where H_{j} := max{h_{π_{j}}(σ, m) : m is ergodic}.

It is direct to check that if Φ_{j} satisfies the AWSC for each 1 ≤ j ≤ d, then so does Φ_{1}×⋯×Φ_{j}. Hence for instance, the condition of Theorem 2.15 fulfills when 1/ρ_{j} are Pisot numbers or Salem numbers and (a_{i,1}, …, a_{i,d}) ∈ ℤ^{d}. Different from the earlier works on the Hausdorff dimension of deterministic self-affine sets and self-affine measures (see e.g. [44, 5, 33, 20, 27, 1, 46]), our model in Theorem 2.15 admits certain overlaps. The two variational results in Theorem 2.15 provide some new insights in the study of overlapping self-affine IFS. An interesting question is whether the results of Theorem 2.15 remain true without the AWSC assumption. It is related to the open problem whether a non-conformal repeller carries an ergodic measure of full dimension (see [22] for a survey). We remark that in the general case, we do have the following inequality(see Lemma 9.2):

\underline{dim}_{B}K ≥ ∑_{j=1}^{d} (\frac{1}{λ_{j}} − \frac{1}{λ_{j+1}} ) sup\{h_{π_{j}}(σ, m) : m is ergodic\}.

Furthermore Theorem 2.15 can be extended somewhat (see Remark 9.3 and Theorem 9.4).

3. Density results about conditional measures

We prove some density results about conditional measures in this section. To begin with, we give a brief introduction to Rohlin’s theory of Lebesgue spaces, measurable partitions and conditional measures. The reader is referred to [55, 49] for more details.

A probability space (X, B, m) is called a Lebesgue space if it is isomorphic to a probability space which is the union of [0, s] (0 ≤ s ≤ 1) with Lebesgue measure and a countable number of atoms. Now let (X, B, m) be a Lebesgue space. A measurable partition η of X is a partition of X such that, up to a set of measure zero, the quotient space X/η is separated by a countable number of measurable sets {B_{i}}. The quotient space X/η with its inherit probability space structure, written as (X_{η}, B_{η}, m_{η}), is again a Lebesgue space. Also, any measurable partition η determine a sub-σ-algebra of B, denoted by η̂, whose elements are unions of elements of η. Conversely, any sub-σ-algebra of B′ of B is also countably generated, say by {B′_{i}}, and therefore all the sets of the form ∩A_{i}, where A_{i} = B′_{i} or its complement, form a measurable partition. In particular, B itself is corresponding to a partition into single points. An important property of Lebesgue space and measurable partitions is the following.

Theorem 3.1 (Rohlin [55]). Let η be a measurable partition of a Lebesgue space (X, B, m). Then, for every x in a set of full m-measure, there is a probability measure m_{x}^{η} defined on η(x), the element of η containing x. These measures are uniquely characterized (up to sets of m-measure 0) by the following properties: if A ⊂ X is a measurable set, then x ↦ m_{x}^{η}(A) is η̂-measurable and m(A) = ∫ m_{x}^{η}(A)dm(x). These properties imply that for any f ∈ L^{1}(X, B, m), m_{x}^{η}(f) = E_{m}(f|η̂)(x) for m-a.e. x, and m(f) = ∫ E_{m}(f|η̂)dm.

The family of measures {m_{x}^{η}} in the above theorem is called the canonical system of conditional measures associated with η.

Throughout the remaining part of this section, we assume that (X, B, m) is a Lebesgue space. Let η be a measurable partition of X, and let {m_{x}^{η}} denote the corresponding canonical system of conditional measures. Suppose that π : X → ℝ^{d} is a B-measurable map. Denote γ := B(ℝ^{d}), the Borel-σ-algebra on ℝ^{d}. For y ∈ ℝ^{d}, we use B(y, r) to denote the closed ball in ℝ^{d} of radius r centered at y. Also, we denote for x ∈ X,

(12)  B^{π}(x, r) = π^{−1}B(πx, r).

Lemma 3.2. Let A ∈ B.

(i) The map x ↦ m_{x}^{η}(B^{π}(x, r) ∩ A) is η̂ ∨ π^{−1}γ-measurable for each r > 0, where η̂ ∨ π^{−1}γ denotes the smallest sub-σ-algebra of B containing η̂ and π^{−1}γ.

(ii) The following functions

lim inf_{r→0} \frac{m_{x}^{η}(B^{π}(x, r) ∩ A)}{m_{x}^{η}(B^{π}(x, r))}, lim sup_{r→0} \frac{m_{x}^{η}(B^{π}(x, r) ∩ A)}{m_{x}^{η}(B^{π}(x, r))}

and

inf_{r>0} \frac{m_{x}^{η}(B^{π}(x, r) ∩ A)}{m_{x}^{η}(B^{π}(x, r))}

are η̂ ∨ π^{−1}γ-measurable, where we interpret 0/0 = 0.

Proof. We first prove (i). Let A ∈ B and r > 0. For n ∈ ℕ, let D_{n} denote the collection

D_{n} = {[0, 2^{−n})^{d} + α : α ∈ 2^{−n}ℤ^{d}}.

For y ∈ ℝ^{d}, denote

W_{n}(y) = ⋃_{Q∈D_{n}: Q∩B(y,r)≠∅} Q.

Write W_{n} := {W_{n}(y) : y ∈ ℝ^{d}}. It is clear that W_{n} is countable for each n ∈ ℕ. Furthermore, we have W_{n}(y) ↓ B(y, r) for each y ∈ ℝ^{d} as n → ∞, that is, W_{n+1}(y) ⊂ W_{n}(y) and ⋂_{n=1}^{∞} W_{n}(y) = B(y, r). As a consequence, we have π^{−1}W_{n}(πx) ↓ B^{π}(x, r) and hence

m_{x}^{η}(B^{π}(x, r) ∩ A) = lim_{n→∞} m_{x}^{η}(π^{−1}W_{n}(πx) ∩ A) (x ∈ X).

Therefore to show that x ↦ m_{x}^{η}(B^{π}(x, r) ∩ A) is η̂∨ π^{−1}γ-measurable, it suffices to show that x ↦ m_{x}^{η}(π^{−1}W_{n}(πx) ∩ A) is η̂ ∨ π^{−1}γ-measurable for each n ∈ ℕ.

Fix n ∈ ℕ. For F ∈ W_{n}, let Γ_{n}(F) = {x ∈ X : W_{n}(πx) = F }. Then Γ_{n}(F) ∈ π^{−1}γ. By Theorem 3.1, m_{x}^{η}(π^{−1}F ∩ A) is an η̂-measurable function of x for each F ∈ W_{n}. However

m_{x}^{η}(π^{−1}W_{n}(πx) ∩ A) = ∑_{F ∈W_{n}} χ_{Γ_{n}(F)}(x)m_{x}^{η}(π^{−1}F ∩ A).

Hence m_{x}^{η}(π^{−1}W_{n}(πx) ∩ A) is η̂ ∨ π^{−1}γ-measurable, so is m_{x}^{η}(B^{π}(x, r) ∩ A).

To see (ii), note that for x ∈ Σ and r > 0 satisfying m_{x}^{η}(B^{π}(x, r)) > 0, we have

\frac{m_{x}^{η}(B^{π}(x, r) ∩ A)}{m_{x}^{η}(B^{π}(x, r))} = lim_{q↓r: q∈ℚ^{+}} \frac{m_{x}^{η}(B^{π}(x, q) ∩ A)}{m_{x}^{η}(B^{π}(x, q))}.

Hence for the three limits in (ii), we can restrict r to be positive rationals. It together with (i) yields the desired measurability. □

Lemma 3.3. Let A ∈ B. Then for m-a.e. x ∈ X,

(13)  lim_{r→0} \frac{m_{x}^{η}(B^{π}(x, r) ∩ A)}{m_{x}^{η}(B^{π}(x, r))} = E_{m}(χ_{A}|η̂ ∨ π^{−1}γ)(x).

Proof. Let \overline{f}(x) and \underline{f}(x) be the values obtained by taking the upper and lower limits in the left hand side of (13). By Lemma 3.2, both \overline{f} and \underline{f} are η̂ ∨ π^{−1}γmeasurable. In the following we only show that \overline{f}(x) = E_{m}(χ_{A}|η̂ ∨ π^{−1}γ)(x) for m-a.e. x. The proof for \underline{f}(x) = E_{m}(χ_{A}|η̂ ∨ π^{−1}γ)(x) is similar.

We first prove that

(14) ∫_{B∩π^{−1}D} \overline{f} dm = ∫_{B∩π^{−1}D} E_{m}(χ_{A}|η̂ ∨ π^{−1}γ) dm  (B ∈ η̂, D ∈ γ).

By Theorem 3.1, for any given C ∈ η, m_{x}^{η} (x ∈ C) represent the same measure supported on C, which we rewrite as m_{C}. Fix C ∈ η. We define measures μ_{C} and ν_{C} on ℝ^{d} by μ_{C}(E) = m_{C}(π^{−1}E ∩ A) and ν_{C}(E) = m_{C}(π^{−1}E) for all E ∈ γ. It is clear that μ_{C} ≪ ν_{C}. Define

g_{C}(z) = lim sup_{r→0} \frac{μ_{C}(B(z, r))}{ν_{C}(B(z, r))} (z ∈ ℝ^{d}).

Then \overline{f}(x) = g_{η(x)}(πx) for all x ∈ Σ. According to the differentiation theory of measures on ℝ^{d} (see, e.g., [43, Theorem 2.12]), g_{C} = \frac{dμ_{C}}{dν_{C}} ν_{C}-a.e. Hence for each D ∈ γ, we have ∫_{D} g_{C}(z) dν_{C}(z) = μ_{C}(D), i.e., ∫_{π^{−1}D} g_{C}(πy) dm_{C}(y) = μ_{C}(D) = m_{C}(π^{−1}D ∩ A). That is,

(15)  ∫_{π^{−1}D} \overline{f} dm_{x}^{η} = m_{x}^{η}(π^{−1}D ∩ A)  (x ∈ X).

To see (14), let B ∈ η̂. Then

∫_{B∩π^{−1}D} \overline{f} dm = ∫ χ_{B}χ_{π^{−1}D}\overline{f} dm = ∫ E_{m} (χ_{B}χ_{π^{−1}D}\overline{f}|η̂) dm
= ∫ χ_{B}E_{m} (χ_{π^{−1}D}\overline{f}|η̂) dm
= ∫_{B} (∫_{π^{−1}D} \overline{f} dm_{x}^{η}) dm(x)  (by Theorem 3.1)
= ∫_{B} m_{x}^{η}(π^{−1}D ∩ A)dm(x)  (by (15))
= ∫ χ_{B}(x)E_{m} (χ_{π^{−1}D∩A}|η̂) (x) dm(x)  (by Theorem 3.1).

Thus we have

∫_{B∩π^{−1}D} \overline{f} dm = ∫ E_{m} (χ_{B}χ_{π^{−1}D∩A}|η̂) (x) dm(x)
= ∫ χ_{B}χ_{π^{−1}D∩A}dm = m(B ∩ π^{−1}D ∩ A)
= ∫ E_{m}(χ_{B∩π^{−1}D}χ_{A}|η̂ ∨ π^{−1}γ) dm
= ∫ χ_{B∩π^{−1}D}E_{m}(χ_{A}|η̂ ∨ π^{−1}γ) dm
= ∫_{B∩π^{−1}D} E_{m}(χ_{A}|η̂ ∨ π^{−1}γ) dm.

This establishes (14).

Let R = \overline{f} − E_{m}(χ_{A}|η̂ ∨ π^{−1}γ). Then R is η̂ ∨ π^{−1}γ-measurable and

∫_{B∩π^{−1}(D)} R dm = 0 (B ∈ η̂, D ∈ π^{−1}γ).

Denote F = {B ∩ π^{−1}(D) : B ∈ η̂, D ∈ π^{−1}γ} and let

F′ = {⋃_{i=1}^{k} F_{i} : k ∈ ℕ, F_{1}, …, F_{k} ∈ F are disjoint} .

It is clear that ∫_{F} R dm = 0 for all F ∈ F′. Moreover it is a routine to check that F′ is an algebra which contains η̂ and π^{−1}γ, and hence F′ generates the σ-algebra η̂ ∨ π^{−1}γ.

We claim that R = 0 m-a.e. Assume this is not true. Then there exists ϵ > 0 such that the set {R > ϵ}, or {R < −ϵ}, has positive m-measure. Without loss of generality, we assume that m{R > ϵ} > 0. Since F′ is an algebra which generates η̂ ∨ π^{−1}γ, there exists a sequence F_{i} ∈ F′ such that m(F_{i}△{R > ϵ}) tends to 0 as i → ∞ (cf. [63, Theorem 0.7]). We conclude that ∫_{F_{i}} R dm tends to ∫_{{R>ϵ}} R dm > 0 as i → ∞, which contradicts the fact ∫_{F_{i}} R dm = 0. □

Remark 3.4. (i) Letting η = N be the trivial partition of X in the above

lemma, we obtain lim_{r→0} \frac{m(B^{π}(x, r) ∩ A)}{m(B^{π}(x, r))} = E_{m}(χ_{A}|π^{−1}γ)(x) m-a.e.

(ii) In general, E_{m_{x}^{η}}(χ_{A}|π^{−1}γ)(x) = E_{m}(χ_{A}|η̂ ∨ π^{−1}γ)(x) m-a.e., both of

them equal lim_{r→0} \frac{m_{x}^{η}(B^{π}(x, r) ∩ A)}{m_{x}^{η}(B^{π}(x, r))} m-a.e. by (i).

Proposition 3.5. Let ξ be a countable measurable partition of X. Then for m-a.e. x ∈ X,

(16)  lim_{r→0} log \frac{m_{x}^{η} (B^{π}(x, r) ∩ ξ(x))}{m_{x}^{η} (B^{π}(x, r))} = −I_{m} (ξ|η̂ ∨ π^{−1}γ) (x),

where I_{m}(·|·) denotes the conditional information (see (2) for the definition). Furthermore, set

(17)  g(x) = − inf_{r>0} log \frac{m_{x}^{η} (B^{π}(x, r) ∩ ξ(x))}{m_{x}^{η} (B^{π}(x, r))}

and assume H_{m}(ξ) < ∞. Then g ≥ 0 and g ∈ L^{1}(X, B, m).

Proof. (16) follows directly from Lemma 3.3 and the following equality

lim_{r→0} log \frac{m_{x}^{η} (B^{π}(x, r) ∩ ξ(x))}{m_{x}^{η} (B^{π}(x, r))} = ∑_{A∈ξ} χ_{A}(x) lim_{r→0} log \frac{m_{x}^{η} (B^{π}(x, r) ∩ A)}{m_{x}^{η} (B^{π}(x, r))}.

Now we turn to the proof of (17). It is clear that g is non-negative. By Lemma 3.2, g is measurable. In the following we show that g ∈ L^{1}(X, B, m).

Let C ∈ η and A ∈ ξ be given. As in the proof of Lemma 3.3, we define measures μ_{C} and ν_{C} on ℝ^{d} by μ_{C}(E) = m_{C}(π^{−1}E ∩ A) and ν_{C}(E) = m_{C}(π^{−1}E) for all E ∈ γ. By Theorem 7.4 in [56], we have

μ_{C} \{z ∈ ℝ^{d} : inf_{r>0} \frac{μ_{C}(B(z, r))}{ν_{C}(B(z, r))} < λ\} ≤ 3^{d}λ (λ > 0).

Hence for any λ > 0,

m_{C} ( \{x ∈ X : inf_{r>0} \frac{m_{C} (B^{π}(x, r) ∩ A)}{m_{C} (B^{π}(x, r))} < λ\} ∩ A) ≤ 3^{d}λ.

Integrating C with respect to m_{η}, we obtain

m ( \{x ∈ X : inf_{r>0} \frac{m_{x}^{η} (B^{π}(x, r) ∩ A)}{m_{x}^{η} (B^{π}(x, r))} < λ\} ∩ A) ≤ 3^{d}λ.

Denote g^{A}(x) = inf_{r>0} \frac{m_{x}^{η} (B^{π}(x, r) ∩ A)}{m_{x}^{η} (B^{π}(x, r))}. Then the above inequality can be rewrit-

ten as

m(A ∩ {g^{A} < λ}) ≤ 3^{d}λ.

Note that by (17), g(x) = − ∑_{A∈ξ} χ_{A}(x) log g^{A}(x). Since g is non-negative, we have

∫ g dm = ∫_{0}^{∞} m{g > t} dt = ∫_{0}^{∞} ∑_{A∈ξ} m(A ∩ {g^{A} < e^{−t}}) dt

≤ ∑_{A∈ξ} ∫_{0}^{∞} min{m(A), 3^{d}e^{−t}} dt

≤ ∑_{A∈ξ} (−m(A) log m(A) + m(A) + m(A) log 3^{d})
= H_{m}(ξ) + 1 + log 3^{d}.

This finishes the proof of the proposition. □

Remark 3.6. Consider the case X = Σ and ξ = P, where P is defined as in (3). Suppose that {S_{i}}_{i=1}^{ℓ} is a family of mappings such that S_{i}: π(Σ) → S_{i}(π(Σ)) ⊂ ℝ^{d} is homeomorphic for each i. Then in (16) and (17), we can change the terms B^{π}(x, r) to π^{−1}R_{r,x}(πx), where R_{r,x}(z) := S_{x_{1}}^{−1}B(S_{x_{1}}(z), r). To see it, fix i and define π′ = S_{i} ∘ π. Then we have

lim_{r→0} \frac{m_{x}^{η} (π^{−1}R_{r,x}(πx) ∩ [i])}{m_{x}^{η} (π^{−1}R_{r,x}(πx))} = lim_{r→0} \frac{m_{x}^{η} (B^{π′}(x, r) ∩ [i])}{m_{x}^{η} (B^{π′}(x, r))}
= E_{m}(χ_{[i]}|η̂ ∨ (π′)^{−1}γ)(x).

However, (π′)^{−1}γ = π^{−1}γ due to the assumption on S_{i}. Hence the last term in the above formula equals E_{m}(χ_{[i]}|η̂ ∨ π^{−1}γ)(x). Thus we can replace the terms B^{π}(x, r) by π^{−1}R_{r,x}(πx) in (16). For the change in (17), we may use a similar argument.

Lemma 3.7. Let π: X → ℝ^{d} and ϕ: X → ℝ^{k} be two B-measurable maps. Let η be the partition of X given by η = {π^{−1}(z): z ∈ ℝ^{d}}. Let A ∈ B and t > 0. Then for m-a.e. x ∈ X, we have

(18)  m_{x}^{η}(B^{ϕ}(x, t) ∩ A) ≥ lim sup_{r→0} \frac{m (B^{ϕ}(x, t) ∩ A ∩ B^{π}(x, r))}{m (B^{π}(x, r))}

and

(19)  m_{x}^{η}(U^{ϕ}(x, t) ∩ A) ≤ lim inf_{r→0} \frac{m (U^{ϕ}(x, t) ∩ A ∩ B^{π}(x, r))}{m (B^{π}(x, r))},

where B^{ϕ}(x, t) := ϕ^{−1}B(ϕx, t), U^{ϕ}(x, t) := ϕ^{−1}U(ϕx, t), here U(z, t) denotes the open ball in ℝ^{d} centered at z of radius t.

Proof. Fix A ∈ B and t > 0. Similar to the proof of Lemma 3.2, for n ∈ ℕ, let D_{n} denote the collection

D_{n} = {[0, 2^{−n})^{k} + α : α ∈ 2^{−n}ℤ^{k}}.

For y ∈ ℝ^{k}, denote

W_{n}(y) = ⋃_{Q∈D_{n}: Q∩B(y,t)≠∅} Q,  Ŵ_{n}(y) = ⋃_{Q∈D_{n}: Q⊂U(y,t)} Q.

Write W_{n} := {W_{n}(y) : y ∈ ℝ^{k}} and Ŵ_{n} := {Ŵ_{n}(y) : y ∈ ℝ^{k}}. It is clear that both W_{n} and Ŵ_{n} are countable for each n ∈ ℕ. Furthermore, we have W_{n}(y) ↓ B(y, t) and Ŵ_{n}(y) ↑ U(y, t) for each y ∈ ℝ^{k} as n → ∞. As a consequence, we have ϕ^{−1}W_{n}(ϕx) ↓ B^{ϕ}(x, t) and ϕ^{−1}Ŵ_{n}(ϕx) ↑ U^{ϕ}(x, t) for x ∈ X. Therefore

m_{x}^{η}(B^{ϕ}(x, t) ∩ A) = lim_{n→∞} m_{x}^{η}(ϕ^{−1}W_{n}(ϕx) ∩ A)

and

m_{x}^{η}(U^{ϕ}(x, t) ∩ A) = lim_{n→∞} m_{x}^{η}(ϕ^{−1}Ŵ_{n}(ϕx) ∩ A)

for each x ∈ X.

In the following we only prove (18). The proof of (19) is essentially identical. For n ∈ ℕ and F ∈ W_{n}, let Γ_{n}(F) = {x ∈ X : W_{n}(ϕx) = F }. Then for m-a.e. x and all n ∈ ℕ, we have

m_{x}^{η}(π^{−1}W_{n}(ϕx) ∩ A) = ∑_{F ∈W_{n}} χ_{Γ_{n}(F)}(x)m_{x}^{η}(ϕ^{−1}F ∩ A)
= ∑_{F ∈W_{n}} χ_{Γ_{n}(F)}(x)E_{m}(χ_{ϕ^{−1}F ∩A}|η̂)(x)
= ∑_{F ∈W_{n}} χ_{Γ_{n}(F)}(x)E_{m}(χ_{ϕ^{−1}F ∩A}|π^{−1}γ)(x)

= ∑_{F ∈W_{n}} χ_{Γ_{n}(F)}(x) lim_{r→0} \frac{m (ϕ^{−1}F ∩ A ∩ B^{π}(x, r))}{m (B^{π}(x, r))}

( by Lemma 3.3)

= lim_{r→0} \frac{m (ϕ^{−1}W_{n}(ϕx) ∩ A ∩ B^{π}(x, r))}{m (B^{π}(x, r))}
≥ lim sup_{r→0} \frac{m (B^{ϕ}(x, t) ∩ A ∩ B^{π}(x, r))}{m (B^{π}(x, r))}.

Letting n → ∞, we obtain (18). □

Remark 3.8. Under the condition of Lemma 3.7, assume that

g: π(X) → g(π(X)) ⊂ ℝ^{d}

is a homeomorphism. Then we may replace the terms B^{π}(x, r) in (18) and (19) by B^{gπ}(x, r). To see it, let π′ = g ∘ π. It is easy to see the partition η is just the same as {(π′)^{−1}(z): z ∈ ℝ^{d}}.

Proposition 3.9. Let T: X → X be a measure-preserving transformation on (X, B, m), and let η be a measurable partition of X. Suppose that π: X → ℝ^{d} is a bounded B-measurable function. Then for any r > 0,

lim_{n→∞} \frac{1}{n} log m_{T^{n}x}^{η} (B^{π}(T^{n}x, r)) = 0 for m-a.e. x ∈ X.

Proof. Fix r > 0 and t > 0. Since π(X) is a bounded subset of ℝ^{d}, we can cover it by ℓ balls B(πx_{i}, r/2) of radius r/2, where x_{i} ∈ X and i = 1, …, ℓ. Define

A_{n} = {x ∈ X: m_{x}^{η}(B^{π}(x, r)) ≤ e^{−nt}}, n ∈ ℕ.

If a ball B^{π}(x_{i}, r/2) intersects A_{n}, then for any y ∈ A_{n} ∩ B^{π}(x_{i}, r/2), we have B^{π}(x_{i}, r/2) ⊂ B^{π}(y, r) because B(πx_{i}, r/2) ⊂ B(πy, r) by the triangle inequality. So the definition of A_{n} gives m_{y}^{η}(A_{n} ∩ B^{π}(x_{i}, r/2)) ≤ m_{y}^{η}(B^{π}(y, r)) ≤ e^{−nt}. Hence

m(A_{n} ∩ B^{π}(x_{i}, r/2)) = ∫ m_{y}^{η}(A_{n} ∩ B^{π}(x_{i}, r/2)) dm(y) ≤ e^{−nt}

and m(A_{n}) ≤ ℓe^{−nt}.

This estimate gives directly that g(x) := log m_{x}^{η}(B^{π}(x, r)) ∈ L^{1}(X, B, m). Note that g(T^{n}x) = ∑_{i=1}^{n} g(T^{i}x) − ∑_{i=1}^{n−1} g(T^{i}x). By the Birkhoff ergodic theorem we can get lim_{n→∞} \frac{1}{n}g(T^{n}x) = 0 for m-a.e. x ∈ X, which is the desired result. □

Lemma 3.10. Let A be a sub-σ-algebra of B. Let A ∈ B with m(A) > 0. Then

E_{m}(χ_{A}|A)(x) > 0

for m-a.e. x ∈ A.

Proof. Let W := {E_{m}(χ_{A}|A) ≤ 0}. Then W ∈ A. Hence

0 ≥ ∫_{W} E_{m}(χ_{A}|A) dm = ∫_{W} χ_{A} dm(x) = m(A ∩ W),

which implies m(A ∩ W) = 0. This finishes the proof. □

4. Projection measure-theoretic entropies associated with IFS

Throughout this section, let {S_{i}}_{i=1}^{ℓ} be an IFS on a closed set X ⊂ ℝ^{d}, and (Σ, σ) the one-sided full shift over {1, …, ℓ}. Let M_{σ}(Σ) denote the collection of all σ-invariant Borel probability measures on Σ. Let π: Σ → ℝ^{d} be defined as in (1), and h_{π}(σ, ·) as in Definition 2.1.

4.1. Some basic properties. In this subsection, we present some basic properties of projection measure-theoretic entropy. Our first result is the following.

Proposition 4.1. (i) 0 ≤ h_{π}(σ, m) ≤ h(σ, m) for every m ∈ M_{σ}(Σ), where

h(σ, m) denotes the classical measure-theoretic entropy of m. (ii) The projection entropy function is affine on M_{σ}(Σ), i.e., for any m_{1}, m_{2} ∈

M_{σ}(Σ) and any 0 ≤ p ≤ 1, we have

(20)  h_{π}(σ, pm_{1} + (1 − p)m_{2}) = ph_{π}(σ, m_{1}) + (1 − p)h_{π}(σ, m_{2}).

The proof of the above proposition will be given later. Now let us recall some notation. If ξ is a partition of Σ, then ξ̂ denotes the σ-algebra generated by ξ. If ξ_{1}, …, ξ_{n} are countable partitions of Σ, then ⋁_{i=1}^{n} ξ_{i} denotes the partition consisting of sets A_{1} ∩ ⋯ ∩ A_{n} with A_{i} ∈ ξ_{i}. Similarly for σ-algebras A_{1}, A_{2}, …, ⋁_{n} A_{n} denotes the σ-algebra generated by ⋃_{n} A_{n}.

Let P be the partition of Σ defined as in (3). Write P_{0}^{n} = ⋁_{i=0}^{n} σ^{−i}P for n ≥ 0. Let γ denote the Borel σ-algebra B(ℝ^{d}) on ℝ^{d}. Similar to Definition 2.1, we give the following definition.

Definition 4.2. Let k ∈ ℕ and ν ∈ M_{σ^{k}}(Σ). Define

h_{π}(σ^{k}, ν) := H_{ν} (P_{0}^{k−1}|σ^{−k}π^{−1}γ) − H_{ν} (P_{0}^{k−1}|π^{−1}γ) .

The term h_{π}(σ^{k}, ν) can be viewed as the projection measure-theoretic entropy of ν w.r.t. the IFS {S_{i_{1}} ∘ ⋯ ∘ S_{i_{k}} : 1 ≤ i_{j} ≤ ℓ for 1 ≤ j ≤ k}. The following proposition exploits the connection between h_{π}(σ^{k}, ν) and h_{π}(σ, m), where m = \frac{1}{k} ∑_{i=0}^{k−1} ν ∘ σ^{−i}.

Proposition 4.3. Let k ∈ ℕ and ν ∈ M_{σ^{k}}(Σ). Set m = \frac{1}{k} ∑_{i=0}^{k−1} ν ∘ σ^{−i}. Then m is σ-invariant, and h_{π}(σ, m) = \frac{1}{k}h_{π}(σ^{k}, ν).

To prove Propositions 4.1 and 4.3, we first give some lemmas about the (conditional) information and entropy (see §2 for the definitions).

Lemma 4.4 (cf. [48]). Let m be a Borel probability measure on Σ. Let ξ, η be two countable Borel partitions of Σ with H_{m}(ξ) < ∞, H_{m}(η) < ∞, and A a sub-σ-algebra of B(Σ). Then we have

(i) I_{m∘σ^{−1}}(ξ|A) ∘ σ = I_{m}(σ^{−1}ξ|σ^{−1}A).

(ii) I_{m}(ξ ∨ η|A) = I_{m}(ξ|A) + I_{m}(η|ξ̂∨ A).

(iii) H_{m}(ξ ∨ η|A) = H_{m}(ξ|A) + H(η|ξ̂∨ A).

(iv) If A_{1} ⊂ A_{2} ⊂ ⋯ is an increasing sequence of sub-σ-algebras with A_{n} ↑ A, then I_{m}(ξ|A_{n}) converges almost everywhere and in L^{1} to I_{m}(ξ|A). In particular, lim_{n→∞} H_{m}(ξ|A_{n}) = H_{m}(ξ|A).

Lemma 4.5. Denote g(x) = −x log x for x ≥ 0. For any integer k ≥ 2 and x_{1}, …, x_{k} ≥ 0, we have \frac{1}{k} ∑_{i=1}^{k} g(x_{i}) ≤ g (\frac{1}{k} ∑_{i=1}^{k} x_{i}) ≤ ∑_{i=1}^{k} g(x_{i}/k) and

(21) ∑_{i=1}^{k} g(x_{i}) − (x_{1} + … + x_{k}) log k ≤ g(x_{1} + … + x_{k}) ≤ ∑_{i=1}^{k} g(x_{i}).

Moreover for any p_{1}, p_{2} ≥ 0 with p_{1} + p_{2} = 1,

(22)  ∑_{j=1}^{2} p_{j}g(x_{j}) ≤ g (∑_{j=1}^{2} p_{j}x_{j}) ≤ ∑_{j=1}^{2} p_{j}g(x_{j}) + g(p_{j})x_{j}.

Proof. Standard. □

Lemma 4.6. Let m be a Borel probability measure on Σ. Assume ξ and η are two countable Borel partitions of Σ such that each member in ξ intersects at most k members of η. Then H_{m}(ξ) ≥ H_{m}(ξ ∨ η) − log k.

Proof. Although the result is standard, we give a short proof for the convenience of the reader. Denote g(x) = −x log x for x ∈ [0, 1]. Then

H_{m}(ξ) = ∑_{A∈ξ} g(m(A)) = ∑_{A∈ξ} g ( ∑_{B∈η, B∩A≠∅} m(A ∩ B))

≥ ∑_{A∈ξ} [( ∑_{B∈η, B∩A≠∅} g(m(A ∩ B))) − m(A) log k] (by (21))

≥ (∑_{A∈ξ} ∑_{B∈η} g(m(A ∩ B))) − log k
= H_{m}(ξ ∨ η) − log k.

This finishes the proof. □

The following simple lemma plays an important role in our analysis.

Lemma 4.7. P̂ ∨ σ^{−1}π^{−1}γ = P̂ ∨ π^{−1}γ.

Proof. We only prove P̂∨σ^{−1}π^{−1}γ ⊆ P̂∨π^{−1}γ. The other direction can be proved by an essentially identical argument. Note that each member in P̂ ∨ σ^{−1}π^{−1}γ can be written as

⋃_{j=1}^{ℓ}[j] ∩ σ^{−1}π^{−1}A_{j}

with A_{j} ∈ γ. However, it is direct to check that

[j] ∩ σ^{−1}π^{−1}A_{j} = [j] ∩ π^{−1}(S_{j}(A_{j})).

Since S_{j} is injective and contractive (thus continuous), we have S_{j}(A_{j}) ∈ γ. Therefore ⋃_{j=1}^{ℓ}[j] ∩ σ^{−1}π^{−1}A_{j} ∈ P̂ ∨ π^{−1}γ. □

Lemma 4.8. Let m be a Borel probability measure on Σ and k ∈ ℕ. We have

H_{m} (P_{0}^{k−1}|σ^{−k}π^{−1}γ) − H_{m} (P_{0}^{k−1}|π^{−1}γ)

= ∑_{j=0}^{k−1} H_{m∘σ^{−j}}(P|σ^{−1}π^{−1}γ) − H_{m∘σ^{−j}}(P|π^{−1}γ).

Moreover if m ∈ M_{σ}(Σ), then

H_{m} (P_{0}^{k−1}|σ^{−k}π^{−1}γ) − H_{m} (P_{0}^{k−1}|π^{−1}γ) = kh_{π}(σ, m).

20 DE-JUN FENG AND HUYI HU

Proof. For j = 0, 1, …, k − 1, we have I_{m} (P_{0}^{k−1}|σ^{−j}π^{−1}γ) − I_{m} (P_{0}^{k−1}|σ^{−(j+1)}π^{−1}γ)

= I_{m} (σ^{−j}P|σ^{−j}π^{−1}γ) + I_{m} ( ⋁_{0≤i≤k−1, i≠j} σ^{−i}P|σ^{−j}P̂ ∨ σ^{−j}π^{−1}γ)
− I_{m} (P_{0}^{k−1}|σ^{−(j+1)}π^{−1}γ) (by Lemma 4.4(ii))
= I_{m} (σ^{−j}P|σ^{−j}π^{−1}γ) + I_{m} ( ⋁_{0≤i≤k−1, i≠j} σ^{−i}P|σ^{−j}P̂ ∨ σ^{−(j+1)}π^{−1}γ)
− I_{m} (P_{0}^{k−1}|σ^{−(j+1)}π^{−1}γ) (by Lemma 4.7)
= I_{m} (σ^{−j}P|σ^{−j}π^{−1}γ) − I_{m} (σ^{−j}P|σ^{−(j+1)}π^{−1}γ) (by Lemma 4.4(ii))
= I_{m∘σ^{−j}} (P|π^{−1}γ) ∘ σ^{j} − I_{m∘σ^{−j}} (P|σ^{−1}π^{−1}γ) ∘ σ^{j} (by Lemma 4.4(i)).

Summing j over {0, …, k − 1} yields

\begin{matrix}I_{m} (P_{0}^{k−1}|π^{−1}γ) − I_{m} (P_{0}^{k−1}|σ^{−k}π^{−1}γ) \\ = ∑_{j=0}^{k−1} (I_{m∘σ^{−j}} (P|π^{−1}γ) ∘ σ^{j} − I_{m∘σ^{−j}} (P|σ^{−1}π^{−1}γ) ∘ σ^{j}) . \end{matrix} ; (23)

Taking integration, we obtain the desired formula. □

For any n ∈ ℕ, let D_{n} be the partition of ℝ^{d} given by

(24)  D_{n} = {[0, 2^{−n})^{d} + α : α ∈ 2^{−n}ℤ^{d}}.

Lemma 4.9. Let m ∈ M_{σ}(Σ). For each n ∈ ℕ, we have

H_{m}(P|σ^{−1}π^{−1}D̂_{n}) − H_{m}(P|π^{−1}D̂_{n}) ≥ −d log(\sqrt{d} + 1).

Proof. Since m is σ-invariant, by Lemma 4.4(iii), we have

\begin{matrix}H_{m}(P|σ^{−1}π^{−1}D̂_{n}) − H_{m}(P|π^{−1}D̂_{n}) \\ = H_{m}(P ∨ σ^{−1}π^{−1}D_{n}) − H_{m}(σ^{−1}π^{−1}D_{n}) \\ − H_{m}(P ∨ π^{−1}D_{n}) + H_{m}(π^{−1}D_{n}) \\ = H_{m}(P ∨ σ^{−1}π^{−1}D_{n}) − H_{m}(P ∨ π^{−1}D_{n}). \end{matrix} ; (25)

Observe that for each 1 ≤ j ≤ ℓ and Q ∈ D_{n},

[j] ∩ σ^{−1}π^{−1}(Q) = [j] ∩ π^{−1}(S_{j}(Q)).

Since S_{j} is contractive, diam(S_{j}(Q)) ≤ 2^{−n}\sqrt{d} and thus S_{j}(Q) intersects at most (\sqrt{d} + 1)^{d} members in D_{n}. It deduces that [j] ∩ σ^{−1}π^{−1}(Q) intersects at most (\sqrt{d} + 1)^{d} members in P ∨ π^{−1}D_{n}. By Lemma 4.6, we have

\begin{aligned}H_{m}(P ∨ σ^{−1}π^{−1}D_{n}) & ≥ H_{m}(P ∨ σ^{−1}π^{−1}D_{n} ∨ π^{−1}D_{n}) \\ & − d log(\sqrt{d} + 1) \\ & ≥ H_{m}(P ∨ π^{−1}D_{n}) − d log(\sqrt{d} + 1). \end{aligned} ; (26)

Combining it with (25) yields the desired inequality. □

Proof of Proposition 4.1. We first prove part (i) of the proposition, i.e.,

0 ≤ h_{π}(σ, m) ≤ h(σ, m).

Since D̂_{n} ↑ γ as n tends to ∞, by Lemma 4.4(iv), we have

lim_{n→∞} H_{m}(P|σ^{−1}π^{−1}D_{n})−H_{m}(P|π^{−1}D_{n}) = H_{m}(P|σ^{−1}π^{−1}γ)−H_{m}(P|π^{−1}γ).

It together with Lemma 4.9 yields

H_{m}(P|σ^{−1}π^{−1}γ) − H_{m}(P|π^{−1}γ) ≥ −d log(\sqrt{d} + 1).

Using the same argument to the IFS {S_{i_{1}...i_{k}} : 1 ≤ i_{j} ≤ ℓ, 1 ≤ j ≤ k}, we have

H_{m} (P_{0}^{k−1}|σ^{−k}π^{−1}γ) − H_{m} (P_{0}^{k−1}|π^{−1}γ) ≥ −d log(\sqrt{d} + 1).

It together with Lemma 4.8 yields h_{π}(σ, m) ≥ −d log(\sqrt{d} + 1)/k. Since k is arbitrary, we have h_{π}(σ, m) ≥ 0. To see h_{π}(σ, m) ≤ h(σ, m), it suffices to observe that

kh_{π}(σ, m) = H_{m} (P_{0}^{k−1}|σ^{−k}π^{−1}γ) − H_{m} (P_{0}^{k−1}|π^{−1}γ)
≤ H_{m} (P_{0}^{k−1}|σ^{−k}π^{−1}γ) ≤ H_{m} (P_{0}^{k−1}) .

Now we turn to the proof of part (ii). Let m_{1}, m_{2} ∈ M_{σ}(Σ) and m = pm_{1} + (1−p)m_{2} for some p ∈ [0, 1]. Using (22), for any finite or countable Borel partition ξ we have

(27)  |H_{m}(ξ) − pH_{m_{1}}(ξ) − (1 − p)H_{m_{2}}(ξ)| ≤ g(p) + g(1 − p) ≤ log 2.

Let k ∈ ℕ. By Lemma 4.8, Lemma 4.4(iv), and (25), we have

\begin{aligned}h_{π}(σ, m) & = \frac{1}{k} (H_{m} (P_{0}^{k−1}|σ^{−k}π^{−1}γ) − H_{m} (P_{0}^{k−1}|π^{−1}γ)) \\ & = \frac{1}{k} lim_{n→∞} (H_{m} (P_{0}^{k−1}|σ^{−k}π^{−1}D̂_{n}) − H_{m} (P_{0}^{k−1}|π^{−1}D̂_{n})) \\ & = \frac{1}{k} lim_{n→∞} (H_{m} (P_{0}^{k−1} ∨ σ^{−k}π^{−1}D_{n}) − H_{m} (P_{0}^{k−1} ∨ π^{−1}D_{n})) . \end{aligned} (28)

The above statement is true when m is replaced by m_{1} and m_{2}. However by (27),

H_{m} (P_{0}^{k−1} ∨ σ^{−k}π^{−1}D_{n}) − H_{m} (P_{0}^{k−1} ∨ π^{−1}D_{n})

differs from

∑_{j=1}^{2} p_{j} [H_{m_{j}} (P_{0}^{k−1} ∨ σ^{−k}π^{−1}D_{n}) − H_{m_{j}} (P_{0}^{k−1} ∨ π^{−1}D_{n})]

at most 2 log 2, where p_{1} = p and p_{2} = 1 − p. This together with (28) yields (20). □

Proof of Proposition 4.3. Let k ≥ 2 and ν ∈ M_{σ^{k}}(Σ). We claim that h_{π}(σ^{k}, ν ∘ σ^{−j}) = h_{π}(σ^{k}, ν) for any 1 ≤ j ≤ k − 1. To prove the claim, it suffices to prove h_{π}(σ^{k}, ν ∘ σ^{−1}) = h_{π}(σ^{k}, ν). Note that both ν and ν ∘ σ^{−1} are σ^{k}-invariant. By Lemma 4.8, we have

h_{π}(σ^{k}, ν) = H_{ν} (P_{0}^{k−1}|σ^{−k}π^{−1}γ) − H_{ν} (P_{0}^{k−1}|π^{−1}γ)
= ∑_{j=0}^{k−1} (H_{ν∘σ^{−j}} (P|σ^{−1}π^{−1}γ) − H_{ν∘σ^{−j}} (P|π^{−1}γ)) ,

whilst

h_{π}(σ^{k}, ν ∘ σ^{−1}) = H_{ν∘σ^{−1}} (P_{0}^{k−1}|σ^{−k}π^{−1}γ) − H_{ν∘σ^{−1}} (P_{0}^{k−1}|π^{−1}γ)
= ∑_{j=0}^{k−1} (H_{ν∘σ^{−j−1}} (P|σ^{−1}π^{−1}γ) − H_{ν∘σ^{−j−1}} (P|π^{−1}γ)) .

Since ν is σ^{k}-invariant, we obtain h_{π}(σ^{k}, ν ∘ σ^{−1}) = h_{π}(σ^{k}, ν). This finishes the proof of the claim. To complete the proof of the proposition, let m = \frac{1}{k} ∑_{i=0}^{k−1} ν ∘ σ^{−i}. It is clear that m is σ-invariant. By Proposition 4.1(ii), h_{π}(σ^{k}, ·) is affine on M_{σ^{k}}(Σ). Hence

h_{π}(σ^{k}, m) = \frac{1}{k} ∑_{i=0}^{k−1} h_{π}(σ^{k}, ν ∘ σ^{−i}) = h_{π}(σ^{k}, ν).

Combining it with Lemma 4.8 yields the equality h_{π}(σ, m) = \frac{1}{k}h_{π}(σ^{k}, ν). □

4.2. A version of Shannon-McMillan-Breiman Theorem associated with IFS. In this subsection, we prove the following Shannon-McMillan-Breiman type theorem associated with IFS, which is needed in the proof of Theorem 2.11. It is also of independent interest.

Proposition 4.10. Let {S_{i}}_{i=1}^{ℓ} be an IFS and m ∈ M_{σ}(Σ). Then

(29) lim_{k→∞} \frac{1}{k} I_{m} (P_{0}^{k−1}|π^{−1}γ) (x) = E_{m}(f|I)(x) = h(σ, m, x) − h_{π}(σ, m, x).

almost everywhere and in L^{1}, where

f := I_{m}(P|σ^{−1}B(Σ)) + I_{m}(P|π^{−1}γ) − I_{m}(P|σ^{−1}π^{−1}γ),

I = {B ∈ B(Σ) : σ^{−1}B = B}, and h(σ, m, x), h_{π}(σ, m, x) denote the classical local entropy and the local projection entropy of m at x (see Definition 2.1), respectively. Moreover if m is ergodic, then the limit in (29) equals h(σ, m) − h_{π}(σ, m) for m-a.e. x ∈ Σ.

Remark 4.11. If ξ is a countable Borel partition of Σ, and A ⊂ B(Σ) is a sub-σ-algebra with σ^{−1}A = A, then the relativized Shannon-McMillan-Breiman Theorem states that

lim_{k→∞} \frac{1}{k} I_{m} (ξ_{0}^{k−1}|A) (x) = E_{m}(g|I)(x) for m-a.e. x ∈ Σ,

where g = I_{m} (ξ|A ∨ ξ_{1}^{∞}) (see, e.g., [8, Lemma 4.1]). However under the setting of Proposition 4.10, the sub-σ-algebra π^{−1}γ is not invariant in general.

In the following we present a generalized version of Proposition 4.10.

Proposition 4.12. Let ξ be a countable Borel partition of Σ with H_{m}(ξ) < ∞, and let A ⊂ B(Σ) be a sub-σ-algebra so that ξ̂∨ σ^{−1}A = ξ̂∨ A. Let m ∈ M_{σ}(Σ). Then

(30)  lim_{k→∞} \frac{1}{k} I_{m} (ξ_{0}^{k−1}|A) (x) = E_{m}(f|I)(x)

almost everywhere and in L^{1}, where

f := I_{m} (ξ|σ^{−1}A ∨ ⋁_{i=1}^{∞} σ^{−i}ξ̂) + I_{m}(ξ|A) − I_{m}(ξ|σ^{−1}A),

and I = {B ∈ B(Σ) : σ^{−1}B = B}.

To prove Proposition 4.12, we need the following lemma.

Lemma 4.13 ([42], Corollary 1.6, p. 96). Let m ∈ M_{σ}(Σ). Let F_{k} ∈ L^{1}(Σ, m) be a sequence that converges almost everywhere and in L^{1} to F ∈ L^{1}(Σ, m). Then

lim_{k→∞} \frac{1}{k} ∑_{j=0}^{k−1} F_{k−j}(σ^{j}(x)) = E_{m}(F |I)(x)

almost everywhere and in L^{1}.

Proof of Proposition 4.12. For k ≥ 2 and x ∈ Σ, we write

g_{k}(x) = I_{m} (ξ_{0}^{k−1}|A) (x) − I_{m} (ξ_{0}^{k−2}|A) (σx).

Then

(31)  I_{m} (ξ_{0}^{k−1}|A) (x) = I_{m}(ξ|A)(σ^{k−1}x) + ∑_{j=0}^{k−2} g_{k−j}(σ^{j}x).

We claim that (32)

g_{k}(x) = I_{m} (ξ|σ^{−1}A ∨ ⋁_{i=1}^{k−1} σ^{−i}ξ̂) (x) + I_{m} (ξ|A) (x) − I_{m} (ξ|σ^{−1}A) (x).

By the claim and Lemma 4.4(iv), g_{k} converges almost everywhere and in L^{1} to f. It together with (31) and Lemma 4.13 yields (30).

Now we turn to the proof of (32). Let k ≥ 2. We have

\begin{aligned}I_{m} (ξ_{0}^{k−1}|σ^{−1}A) (x) & = I_{m} (ξ|σ^{−1}A) (x) + I_{m} ( ⋁_{i=1}^{k−1} σ^{−i}ξ|σ^{−1}A ∨ ξ̂) (x) \\ & = I_{m} (ξ|σ^{−1}A) (x) + I_{m} ( ⋁_{i=1}^{k−1} σ^{−i}ξ|A ∨ ξ̂) (x), \end{aligned} (33)

using the property σ^{−1}A ∨ ξ̂ = A ∨ ξ̂. Meanwhile, we have

\begin{matrix}I_{m} (ξ_{0}^{k−1}|σ^{−1}A) (x) \\ = I_{m} ( ⋁_{i=1}^{k−1} σ^{−i}ξ|σ^{−1}A) (x) + I_{m} (ξ|σ^{−1}A ∨ ⋁_{i=1}^{k−1} σ^{−i}ξ̂) (x) \\ = I_{m} (ξ_{0}^{k−2}|A) (σx) + I_{m} (ξ|σ^{−1}A ∨ ⋁_{i=1}^{k−1} σ^{−i}ξ̂) (x). \end{matrix} ; (34)

Combining (33) with (34) yields

\begin{matrix}I_{m} (ξ|σ^{−1}A) (x) + I_{m} ( ⋁_{i=1}^{k−1} σ^{−i}ξ|A ∨ ξ̂) (x) \\ = I_{m} (ξ_{0}^{k−2}|A) (σx) + I_{m} (ξ|σ^{−1}A ∨ ⋁_{i=1}^{k−1} σ^{−i}ξ̂) (x). \end{matrix} ; (35)

However

(36) I_{m} (ξ_{0}^{k−1}|A) (x) = I_{m} (ξ|A) (x) + I_{m} ( ⋁_{i=1}^{k−1} σ^{−i}ξ|A ∨ ξ̂) (x).

Combining (35) with (36) yields (32). This finishes the proof of Proposition 4.12. □

We remark that Proposition 4.10 can be stated in terms of conditional measures. To see it, let

η = {π^{−1}(z) : z ∈ ℝ^{d}}

be the measurable partition of Σ generated by the canonical projection π associated with {S_{i}}_{i=1}^{ℓ}. For m ∈ M_{σ}(Σ), let {m_{x}^{η}}_{x∈Σ} denote the canonical system of conditional measures w.r.t. η. For x ∈ Σ and k ∈ ℕ, let P_{0}^{k}(x) denote the element in the partition P_{0}^{k} containing x. Then Proposition 4.10 can be restated as the following.

DIMENSION THEORY OF ITERATED FUNCTION SYSTEMS 25

Proposition 4.14. For m ∈ M_{σ}(Σ), we have

(37) − lim_{k→∞} \frac{1}{k} log m_{x}^{η}(P_{0}^{k}(x)) = E_{m}(f|I)(x)  for m-a.e. x ∈ Σ,

where f := I_{m}(P|σ^{−1}B(Σ)) + I_{m}(P|π^{−1}γ) − I_{m}(P|σ^{−1}π^{−1}γ). Moreover if m is ergodic, then the limit in (37) equals h(σ, m) − h_{π}(σ, m) for m-a.e. x ∈ Σ. Proof. It suffices to show that for each k ∈ ℕ,

log m_{x}^{η}(P_{0}^{k}(x)) = −I_{m}(P_{0}^{k}|π^{−1}γ)(x) almost everywhere.

To see this, by Theorem 3.1 we have

∑_{A∈P_{0}^{k}} χ_{A}(x)m_{x}^{η}(A) = ∑_{A∈P_{0}^{k}} χ_{A}(x)E_{m}(χ_{A}|π^{−1}γ)(x) for m-a.e. x ∈ Σ.

Taking logarithm yields the desired result. □ Remark 4.15. In Proposition 4.14, for m-a.e. x ∈ Σ, we have

lim_{k→∞} −\frac{1}{k} log m_{x}^{η}(P_{0}^{k}(y)) = E_{m}(f|I)(y) for m_{x}^{η}-a.e. y ∈ η(x).

To see this, denote

R = \{y ∈ Σ : − lim_{k→∞} \frac{1}{k} log m_{y}^{η}(P_{0}^{k}(y)) = E_{m}(f|I)(y)\} .

Then 1 = m(R) = ∫ m_{x}^{η}(R ∩ η(x)) dm(x). Hence m_{x}^{η}(R ∩ η(x)) = 1 m-a.e. For y ∈ R ∩ η(x), we have

lim_{k→∞} −\frac{1}{k} log m_{x}^{η}(P_{0}^{k}(y)) = lim_{k→∞} −\frac{1}{k} log m_{y}^{η}(P_{0}^{k}(y)) = E_{m}(f|I)(y).

As a corollary of Proposition 4.14, we have Corollary 4.16. Let m ∈ M_{σ}(Σ). Then

h_{π}(σ, m) = h(σ, m) ⟺ lim_{k→∞} \frac{1}{k} log m_{x}^{η}(P_{0}^{k}(x)) = 0 m-a.e.
⟺ dim_{H} m_{x}^{η} = 0 m-a.e.

In particular, if dim_{H} π^{−1}(z) = 0 for each z ∈ ℝ^{d}, then h_{π}(σ, m) = h(σ, m). Here dim_{H} denotes the Hausdorff dimension. Proof. Let f be defined as in Proposition 4.14. Then

∫ E_{m}(f|I) dm = ∫ f dm = h(σ, m) − h_{π}(σ, m).

By (37), E_{m}(f|I)(x) ≥ 0 for m-a.e. x ∈ Σ. Hence we have

h(σ, m) = h_{π}(σ, m) ⟺ E_{m}(f|I) = 0 m-a.e.
⟺ lim_{k→∞} \frac{1}{k} log m_{x}^{η}(P_{0}^{k}(x)) = 0 m-a.e.

Using dimension theory of measures (see, e.g., [14]), we have

dim_{H} m_{x}^{η} = esssup_{y∈η(x)} lim inf_{k→∞} \frac{log m_{x}^{η}(P_{0}^{k}(y))}{log ℓ^{−k}}.

It together with Remark 4.15 yields

E_{m}(f|I) = 0 m-a.e. ⟺ dim_{H} m_{x}^{η} = 0 m-a.e.

This finishes the proof of the first part of the corollary.

To complete the proof, assume that dim_{H} π^{−1}(z) = 0 for each z ∈ ℝ^{d}. Then for each x ∈ Σ, dim_{H} η(x) = 0 and hence dim_{H} m_{x}^{η} = 0. Thus h_{π}(σ, m) = h(σ, m). □

4.3. Projection entropy under the ergodic decomposition. In this subsection, we first prove the following result.

Proposition 4.17. Let {S_{i}}_{i=1}^{ℓ} be an IFS and m ∈ M_{σ}(Σ). Assume that m = ∫ ν dℙ(ν) is the ergodic decomposition of m. Then

h_{π}(σ, m) = ∫ h_{π}(σ, ν) dℙ(ν).

Proof. Let I denote the σ-algebra {B ∈ B(Σ) : σ^{−1}B = B}, and let m ∈ M_{σ}(Σ). Then there exists an m-measurable partition ε of Σ such that ε̂ = I modulo sets of zero m-measure (see [49, pp. 37-38]). Let {m_{x}^{ε}} denote the conditional measures of m associated with the partition ε. Then m = ∫ m_{x}^{ε} dm(x) is just the ergodic decomposition of m (see e.g., [32, Theorem 2.3.3]). Hence to prove the proposition, we need to show that

(38)  h_{π}(σ, m) = ∫ h_{π}(σ, m_{x}^{ε}) dm(x).

We first show the direction “≤” in (38). Note that I is σ-invariant and P̂ ∨ σ^{−1}π^{−1}γ = P̂ ∨ π^{−1}γ. Hence we have P̂ ∨ σ^{−1}π^{−1}γ ∨ I = P̂ ∨ π^{−1}γ ∨ I. Taking ξ = P and A = π^{−1}γ ∨ I in Proposition 4.12 yields

(39)  lim_{k→∞} \frac{1}{k} I_{m} (P_{0}^{k−1}|π^{−1}γ ∨ I) (x) = E_{m}(f|I)(x)

almost everywhere and in L^{1}, where

f := I_{m} (P|σ^{−1}B(Σ)) + I_{m}(P|π^{−1}γ ∨ I) − I_{m}(P|σ^{−1}π^{−1}γ ∨ I).

By Remark 3.4(ii), we have

I_{m_{x}^{ε}} (P_{0}^{k−1}|π^{−1}γ) (x) = I_{m} (P_{0}^{k−1}|π^{−1}γ ∨ I) (x).

Hence according to the ergodicity of m_{x}^{ε} and Proposition 4.10, we have

h(σ, m_{x}^{ε}) − h_{π}(σ, m_{x}^{ε}) = lim_{k→∞} \frac{1}{k} I_{m_{x}^{ε}} (P_{0}^{k−1}|π^{−1}γ) (x)
= lim_{k→∞} \frac{1}{k} I_{m} (P_{0}^{k−1}|π^{−1}γ ∨ I) (x)

almost everywhere and

(40) ∫ h(σ, m_{x}^{ε}) − h_{π}(σ, m_{x}^{ε}) dm(x) = lim_{k→∞} \frac{1}{k} H_{m} (P_{0}^{k−1}|π^{−1}γ ∨ I) .

Using Proposition 4.10 again we have

(41)  h(σ, m) − h_{π}(σ, m) = lim_{k→∞} \frac{1}{k} H_{m} (P_{0}^{k−1}|π^{−1}γ) .

However, H_{m} (P_{0}^{k−1}|π^{−1}γ ∨ I) ≤ H_{m} (P_{0}^{k−1}|π^{−1}γ) (see e.g. [63, Theorem 4.3 (v)]). By (40), (41) and the above inequality, we have

∫ h(σ, m_{x}^{ε}) − h_{π}(σ, m_{x}^{ε}) dm(x) ≤ h(σ, m) − h_{π}(σ, m).

It is well known (see [63, Theorem 8.4]) that ∫ h(σ, m_{x}^{ε}) dm(x) = h(σ, m). Hence we obtain the inequality h_{π}(σ, m) ≤ ∫ h_{π}(σ, m_{x}^{ε}) dm(x).

Now we prove the direction “≥ ” in (38). For any n ∈ ℕ, let D_{n} be defined as in (24). Since D̂_{n} ↑ γ, we have

(42)  h_{π}(σ, m) = lim_{n→∞} H_{m}(P|σ^{−1}π^{−1}D̂_{n}) − H_{m}(P|π^{−1}D̂_{n}).

Now fix n ∈ ℕ and denote A(m) = H_{m}(P|σ^{−1}π^{−1}D̂_{n}) − H_{m}(P|π^{−1}D̂_{n}) and

B(m) = H_{m}(σ^{−1}π^{−1}D_{n}|P ∨ π^{−1}D̂_{n})
= H_{m}(P ∨ σ^{−1}π^{−1}D̂_{n} ∨ π^{−1}D̂_{n}) − H_{m}(P ∨ π^{−1}D̂_{n}).

Then by (25) and (26), we have

(43)  B(m) − c ≤ A(m) ≤ B(m),

where c = d log(\sqrt{d} + 1). As a conditional entropy function, B(m) is concave on M_{σ}(Σ) (see, e.g., [26, Lemma 3.3 (1)]). Hence by Jensen’s inequality and (43), we have

A(m) ≥ B(m) − c ≥ ∫ B(m_{x}^{ε}) dm(x) − c ≥ ∫ A(m_{x}^{ε}) dm(x) − c.

That is,

H_{m}(P|σ^{−1}π^{−1}D̂_{n}) − H_{m}(P|π^{−1}D̂_{n})
≥ ∫ H_{m_{x}^{ε}}(P|σ^{−1}π^{−1}D̂_{n}) − H_{m_{x}^{ε}}(P|π^{−1}D̂_{n}) dm(x) − c.

Letting n → ∞, using (42) and Lebesgue dominated convergence theorem, we have

h_{π}(σ, m) ≥ ∫ h_{π}(σ, m_{x}^{ε}) dm(x) − c.

Replacing σ by σ^{k} we have

(44)  h_{π}(σ^{k}, m) ≥ ∫ h_{π}(σ^{k}, m_{x}^{ε_{k}}) dm(x) − c,

where ε_{k} denotes a measurable partition of Σ such that

ε̂_{k} = {B ∈ B(Σ) : σ^{−k}B = B}

modulo sets of zero m-measure. Note that m = ∫ m_{x}^{ε_{k}} dm(x) is the ergodic decomposition of m with respect to σ^{k}. Hence m = ∫ (1/k) ∑_{i=0}^{k−1} m_{x}^{ε_{k}} ∘ σ^{−i} dm(x) is the ergodic decomposition of m with respect to σ. It follows that

(45)  \frac{1}{k} ∑_{i=0}^{k−1} m_{x}^{ε_{k}} ∘ σ^{−i} = m_{x}^{ε} m-a.e.

By (44), Proposition 4.3 and (45), we have

h_{π}(σ^{k}, m) = \frac{1}{k} ∑_{i=0}^{k−1} h_{π}(σ^{k}, m ∘ σ^{−i})

≥ \frac{1}{k} ∑_{i=0}^{k−1} ∫ h_{π}(σ^{k}, m_{x}^{ε_{k}} ∘ σ^{−i}) dm(x) − c

= ∫ h_{π}(σ^{k}, \frac{1}{k} ∑_{i=0}^{k−1} m_{x}^{ε_{k}} ∘ σ^{−i}) dm(x) − c

= ∫ h_{π}(σ^{k}, m_{x}^{ε}) dm(x) − c.

Using Proposition 4.3 again yields

h_{π}(σ, m) ≥ ∫ h_{π}(σ, m_{x}^{ε}) dm(x) − c/k for any k ∈ ℕ.

Hence we have h_{π}(σ, m) ≥ ∫ h_{π}(σ, m_{x}^{ε}) dm(x), as desired. □

Proof of Theorem 2.2. It follows directly from Propositions 4.1, 4.10 and 4.17. □

4.4. The projection entropy for certain affine IFS and the proof of Theorem 2.3. In this subsection, we assume that Φ = {S_{i}}_{i=1}^{ℓ} is an IFS on ℝ^{d} of the form

S_{i}(x) = Ax + c_{i} (i = 1, …, ℓ),

where A is a d × d non-singular real matrix with ∥A∥ < 1 and c_{i} ∈ ℝ^{d}. Let K denote the attractor of Φ.

Let Q denote the partition {[0, 1)^{d} + α : α ∈ ℤ^{d}} of ℝ^{d}. For n = 0, 1, …, and x ∈ ℝ^{d}, we set

Q_{n} = {A^{n}Q : Q ∈ Q}, Q_{n} + x = {A^{n}Q + x : Q ∈ Q}.

We have the following geometric characterization of h_{π} for the IFS Φ (i.e., Theorem 2.3).

Proposition 4.18. (i) Let m ∈ M_{σ}(Σ). Then

(46)  h_{π}(σ, m) = lim_{n→∞} \frac{H_{m}(π^{−1}Q_{n})}{n}.

(ii)

lim_{n→∞} \frac{log #\{Q ∈ Q : A^{n}Q ∩ K ≠ ∅\}}{n} = sup\{h_{π}(σ, m) : m ∈ M_{σ}(Σ)\}.

To prove the above proposition, we need the following lemma.

Lemma 4.19. Assume that Ω is a subset of {1, …, ℓ} such that S_{i}(K)∩S_{j}(K) = ∅ for all i, j ∈ Ω with i ≠ j. Suppose that ν is an invariant measure on Σ supported on Ω^{ℕ}, i.e., ν([j]) = 0 for all j ∈ {1, …, ℓ}\Ω. Then h_{π}(σ, ν) = h(σ, ν).

Proof. It suffices to prove that h_{π}(σ, ν) ≥ h(σ, ν). Recall that

h_{π}(σ, ν) = H_{ν}(P|σ^{−1}π^{−1}γ) − H_{ν}(P|π^{−1}γ)

and H_{ν}(P|σ^{−1}π^{−1}γ) ≥ H_{ν}(P|σ^{−1}B(Σ)) = h(σ, ν). Hence we only need to show H_{ν}(P|π^{−1}γ) = 0. To do this, denote

δ = min{d(S_{i}(K), S_{j}(K)) : i, j ∈ Ω, i ≠ j}.

Then δ > 0. Let ξ be an arbitrary finite Borel partition of K so that diam(A) < δ/2 for A ∈ ξ. Set W = {[i] : i ∈ Ω}. Since ν is supported on Ω^{ℕ}, we have

H_{ν}(P|π^{−1}ξ̂) = H_{ν}(P ∨ π^{−1}ξ) − H_{ν}(π^{−1}ξ) = H_{ν}(W ∨ π^{−1}ξ) − H_{ν}(π^{−1}ξ).

However for each A ∈ ξ, there is at most one i ∈ Ω such that S_{i}(K) ∩ A ≠ ∅, i.e.,

[i] ∩ π^{−1}A ≠ ∅. This forces that H_{ν}(W ∨ π^{−1}ξ) = H_{ν}(π^{−1}ξ). Hence

H_{ν}(P|π^{−1}ξ̂) = 0.

By the arbitrariness of ξ and Lemma 4.4(iv), we have H_{ν}(P|π^{−1}γ) = 0. □

Proof of Proposition 4.18. We first prove (i). Let m ∈ M_{σ}(Σ). Denote γ = B(ℝ^{d}). According to Proposition 4.3, we have

H_{m}(P_{0}^{p−1}|σ^{−p}π^{−1}γ) − H_{m}(P_{0}^{p−1}|π^{−1}γ) = ph_{π}(σ, m) (p ∈ ℕ).

Now fix p. Since Q̂_{n} ↑ γ, by Lemma 4.4(iv), there exists k_{0} such that for k ≥ k_{0},

|H_{m}(P_{0}^{p−1}|σ^{−p}π^{−1}γ) − H_{m}(P_{0}^{p−1}|σ^{−p}π^{−1}Q̂_{kp})| ≤ 1, and

|H_{m}(P_{0}^{p−1}|π^{−1}γ) − H_{m}(P_{0}^{p−1}|π^{−1}{ \atop Q_{(k+1)p}})| ≤ 1.

It follows that for k ≥ k_{0},

\begin{aligned}ph_{π}(σ, m) − 2 & ≤ H_{m}(P_{0}^{p−1}|σ^{−p}π^{−1}Q̂_{kp}) − H_{m}(P_{0}^{p−1}|π^{−1}{ \atop Q_{(k+1)p}}) \\ & ≤ ph_{π}(σ, m) + 2. \end{aligned} ; (47)

Now we estimate the difference of conditional entropies in (47). Note that

H_{m}(P_{0}^{p−1}|σ^{−p}π^{−1}Q̂_{kp}) = H_{m}(P_{0}^{p−1} ∨ σ^{−p}π^{−1}Q_{kp}) − H_{m}(σ^{−p}π^{−1}Q_{kp})
= H_{m}(P_{0}^{p−1} ∨ σ^{−p}π^{−1}Q_{kp}) − H_{m}(π^{−1}Q_{kp})

and

H_{m}(P_{0}^{p−1}|π^{−1}{ \atop Q_{(k+1)p}}) = H_{m}(P_{0}^{p−1} ∨ π^{−1}Q_{(k+1)p}) − H_{m}(π^{−1}Q_{(k+1)p}).

Hence we have

\begin{matrix}H_{m}(P_{0}^{p−1}|σ^{−p}π^{−1}Q̂_{kp}) − H_{m}(P_{0}^{p−1}|π^{−1}{ \atop Q_{(k+1)p}}) \\ = H_{m}(P_{0}^{p−1} ∨ σ^{−p}π^{−1}Q_{kp}) − H_{m}(P_{0}^{p−1} ∨ π^{−1}Q_{(k+1)p}) \\ + H_{m}(π^{−1}Q_{(k+1)p}) − H_{m}(π^{−1}Q_{kp}). \end{matrix} ; (48)

Observe that for each [u] ∈ P_{0}^{p−1} and any Q ∈ Q,

[u] ∩ σ^{−p}π^{−1}A^{kp}Q = [u] ∩ π^{−1}S_{u}A^{kp}Q.

Since the linear part of S_{u} is A^{p}, the set S_{u}A^{kp}Q intersects at most 2^{d} elements of Q_{(k+1)p}. Therefore each element of P_{0}^{p−1} ∨ σ^{−p}π^{−1}Q_{kp} intersects at most 2^{d} elements of P_{0}^{p−1} ∨ π^{−1}Q_{(k+1)p}. Similarly, the statement is also true if the two partitions are interchanged. Therefore by Lemma 4.6, we have

|H_{m}(P_{0}^{p−1} ∨ σ^{−p}π^{−1}Q_{kp}) − H_{m}(P_{0}^{p−1} ∨ π^{−1}Q_{(k+1)p})| ≤ d log 2.

It together with (47) and (48) yields

ph_{π}(σ, m) − 2 − d log 2 ≤ H_{m}(π^{−1}Q_{(k+1)p}) − H_{m}(π^{−1}Q_{kp})
≤ ph_{π}(σ, m) + 2 + d log 2

for k ≥ k_{0}. Hence we have

lim sup_{k→∞} \frac{H_{m}(π^{−1}Q_{kp})}{kp} ≤ h_{π}(σ, m) + \frac{2 + d log 2}{p} and

lim inf_{k→∞} \frac{H_{m}(π^{−1}Q_{kp})}{kp} ≥ h_{π}(σ, m) − \frac{2 + d log 2}{p}.

By a volume argument, there is a large integer N (N depends on A, d, p; and it is independent of k) such that for any i = 0, …, p − 1, each element of Q_{kp+i} intersects at most N elements of Q_{kp}, and vice versa. Hence by Lemma 4.6, |H_{m}(π^{−1}Q_{kp}) − H_{m}(π^{−1}Q_{kp+i})| < log N for 0 ≤ i ≤ p − 1. It follows that

lim sup_{k→∞} H_{m}(π^{−1}Q_{kp})/(kp) = lim sup_{n→∞} H_{m}(π^{−1}Q_{n})/n and

lim inf_{k→∞} H_{m}(π^{−1}Q_{kp})/(kp) = lim inf_{n→∞} H_{m}(π^{−1}Q_{n})/n.

Thus we have

h_{π}(σ, m) − \frac{2 + d log 2}{p} ≤ lim inf_{n→∞} \frac{H_{m}(π^{−1}Q_{n})}{n} ≤ lim sup_{n→∞} \frac{H_{m}(π^{−1}Q_{n})}{n}
≤ h_{π}(σ, m) + \frac{2 + d log 2}{p}.

Letting p tend to infinity, we obtain (46).

To show (ii), we assume K ⊂ [0, 1)^{d}, without loss of generality. Note that the number of (non-empty) elements in the partition π^{−1}Q_{n} is just equal to

N_{n} := #{Q ∈ Q : A^{n}Q ∩ K ≠ ∅}.

Hence by (21), we have

H_{m}(π^{−1}Q_{n}) ≤ log N_{n}, ∀ m ∈ M_{σ}(Σ).

This together with (i) proves

lim inf_{n→∞} \frac{log N_{n}}{n} ≥ sup\{h_{π}(σ, m) : m ∈ M_{σ}(Σ)\}.

To prove (ii), we still need to show

(49)  lim sup_{n→∞} log N_{n}/n ≤ sup{h_{π}(σ, m) : m ∈ M_{σ}(Σ)}.

We may assume that lim sup_{n→∞} log N_{n}/n > 0, otherwise there is nothing to prove. Let n be a large integer so that N_{n} > 7^{d}. Choose a subset Γ of

{Q : A^{n}Q ∩ K ≠ ∅, Q ∈ Q}

such that #Γ > 7^{−d}N_{n}, and

(50)  2Q ∩ 2Q̃ = ∅  for different Q, Q̃ ∈ Γ,

where 2Q := ⋃_{P ∈Q: \overline{P} ∩\overline{Q}≠∅} P, and \overline{P} denotes the closure of P. For each Q ∈ Γ, since A^{n}Q∩K ≠ ∅, we can pick a word u = u(Q) ∈ Σ_{n} such that S_{u}K ∩A^{n}Q ≠ ∅. Consider the collection W = {u(Q) : Q ∈ Γ}. The separation condition (50) for elements in Γ guarantees that

S_{u(Q)}(K) ∩ S_{u(Q̃)}(K) = ∅ for all Q, Q̃ ∈ Γ with Q ≠ Q̃.

Define a Bernoulli measure ν on W^{ℕ} by

ν([w_{1} … w_{k}]) = (#Γ)^{−k} (k ∈ ℕ, w_{1}, …, w_{k} ∈ W).

Then ν can be viewed as a σ^{n}-invariant measure on Σ (by viewing W^{ℕ} as a subset of Σ). By Lemma 4.19, we have h_{π}(σ^{n}, ν) = h(σ^{n}, ν) = log #Γ. Define μ = \frac{1}{n} ∑_{i=0}^{n−1} ν ∘ σ^{−i}. Then μ ∈ M_{σ}(Σ), and by Proposition 4.3,

h_{π}(σ, μ) = \frac{h_{π}(σ^{n}, ν)}{n} = \frac{log #Γ}{n} ≥ \frac{log(7^{−d}N_{n})}{n},

from which (49) follows. □

4.5. Upper semi-continuity of h_{π}(σ, ·) under the AWSC. In this subsection, we prove the following proposition.

Proposition 4.20. Assume that {S_{i}}_{i=1}^{ℓ} is an IFS which satisfies the AWSC (see Definition 2.14). Then the map m ↦ h_{π}(σ, m) is upper semi-continuous on M_{σ}(Σ).

We first prove a lemma.

Lemma 4.21. Let {S_{i}}_{i=1}^{ℓ} be an IFS with attractor K ⊂ ℝ^{d}. Assume that

#{1 ≤ i ≤ ℓ : x ∈ S_{i}(K)} ≤ k

for some k ∈ ℕ and each x ∈ ℝ^{d}. Then H_{ν}(P|π^{−1}γ) ≤ log k for any Borel probability measure ν on Σ.

Proof. A compactness argument shows that there is r_{0} > 0 such that

#{1 ≤ i ≤ ℓ : B(x, r_{0}) ∩ S_{i}(K) ≠ ∅} ≤ k

for each x ∈ ℝ^{d}. Let n ∈ ℕ so that 2^{−n}\sqrt{d} < r_{0}. Then for each Q ∈ D_{n}, where D_{n} is defined as in (24), there are at most k different i ∈ {1, …, ℓ} such that S_{i}(K) ∩ Q ≠ ∅. It follows that each member in π^{−1}D_{n} intersects at most k members of P ∨ π^{−1}D_{n}. By Lemma 4.6, we have

H_{ν}(P|π^{−1}D̂_{n}) = H_{ν}(P ∨ π^{−1}D_{n}) − H_{ν}(π^{−1}D_{n}) ≤ log k.

Note that π^{−1}D̂_{n} ↑ π^{−1}γ. Applying Lemma 4.4(iv), we obtain

H_{ν}(P|π^{−1}γ) = lim_{n→∞} H_{ν}(P|π^{−1}D̂_{n}) ≤ log k.

□

As a corollary, we have

Corollary 4.22. Under the condition of Lemma 4.21, we have

h_{π}(σ, m) ≥ h(σ, m) − log k

for any m ∈ M_{σ}(Σ).

Proof. By the definition of h_{π}(σ, m) and Lemma 4.21, we have

h_{π}(σ, m) = H_{m}(P|σ^{−1}π^{−1}γ) − H_{m}(P|π^{−1}γ) ≥ H_{m}(P|σ^{−1}π^{−1}γ) − log k.

However, H_{m}(P|σ^{−1}π^{−1}γ) ≥ H_{m}(P|σ^{−1}B(Σ)) = h(σ, m). This implies the desired result. □

To prove Proposition 4.20, we need the following lemma.

Lemma 4.23. Let {S_{i}}_{i=1}^{ℓ} be an IFS with attractor K. Suppose that Ω is a subset of {1, …, ℓ} such that there is a map g: {1, …, ℓ} → Ω so that

S_{i} = S_{g(i)} (i = 1, …, ℓ).

Let (Ω^{ℕ}, σ̃) denote the one-sided full shift over Ω. Define G : Σ → Ω^{ℕ} by (x_{j})_{j=1}^{∞} ↦ (g(x_{j}))_{j=1}^{∞}. Then

(i) K is also the attractor of {S_{i}}_{i∈Ω}. Moreover if we let π̃: Ω^{ℕ} → K denote the canonical projection w.r.t. {S_{i}}_{i∈Ω}, then we have π = π̃ ∘ G.

(ii) Let m ∈ M_{σ}(Σ). Then ν = m ∘ G^{−1} ∈ Mσ̃(Ω^{ℕ}). Furthermore, h_{π}(σ, m) = hπ̃(σ̃, ν). In particular, h_{π}(σ, m) ≤ log(#Ω).

Proof. (i) is obvious. To see (ii), let m ∈ M_{σ}(Σ). It is easily seen that the following diagram commutes:

Σ ⟶^{σ} Σ
G↓  ↓^{G}
Ω^{ℕ} \underrightarrow{σ̃} Ω^{ℕ}.

That is, σ̃ ∘ G = G ∘ σ. Hence ν = m ∘ G^{−1} ∈ Mσ̃(Ω^{ℕ}). To show that h_{π}(σ, m) = hπ̃(σ̃, ν), let Q = {[i] : i ∈ Ω} be the canonical partition of Ω^{ℕ}. Then

hπ̃(σ̃, ν) = H_{m∘G^{−1}}(Q|σ^{−1}π^{−1}γ) − H_{m∘G^{−1}}(Q|π^{−1}γ)
= H_{m}(G^{−1}(Q)|G^{−1}σ^{−1}π^{−1}γ) − H_{m}(G^{−1}(Q)|G^{−1}π^{−1}γ)
= H_{m}(G^{−1}(Q)|σ^{−1}π^{−1}γ) − H_{ν}(G^{−1}(Q)|π^{−1}γ),

using the facts G ∘ σ = σ̃ ∘ G and π̃ ∘ G = π. Since P ∨ G^{−1}(Q) = P, we have

h_{π}(σ, m) − hπ̃(σ̃, m ∘ G^{−1})
= (H_{m} (P|σ^{−1}π^{−1}γ) − H_{m} (P|π^{−1}γ))
− (H_{m}(G^{−1}(Q)|σ^{−1}π^{−1}γ) − H_{m}(G^{−1}(Q)|π^{−1}γ))
= (H_{m} (P|σ^{−1}π^{−1}γ) − H_{m}(G^{−1}(Q)|σ^{−1}π^{−1}γ))
− (H_{m} (P|π^{−1}γ) − H_{m}(G^{−1}(Q)|π^{−1}γ))
= H_{m} (P|σ^{−1}π^{−1}γ ∨ G^{−1}(Q̂)) − H_{m} (P|π^{−1}γ ∨ G^{−1}(Q̂)) .

An argument similar to the proof of Lemma 4.7 shows that

σ^{−1}π^{−1}γ ∨ G^{−1}(Q̂) = π^{−1}γ ∨ G^{−1}(Q̂).

Hence we have h_{π}(σ, m) = hπ̃(σ̃, m ∘ G^{−1}). □

Proof of Proposition 4.20. Let (ν_{n}) be a sequence in M_{σ}(Σ) converging to m in the weak-star topology. We need to show that lim sup_{n→∞} h_{π}(σ, ν_{n}) ≤ h_{π}(σ, m). To see this, it suffices to show that

(51)  lim sup_{n→∞} h_{π}(σ, ν_{n}) ≤ h_{π}(σ, m) + \frac{1}{k} log t_{k}

for each k ∈ ℕ, where t_{k} is given as in Definition 2.14.

To prove (51), we fix k ∈ ℕ. Define an equivalence relation ∼ on {1, …, ℓ}^{k} by u ∼ v if S_{u} = S_{v}. Let \underline{u} denotes the equivalence class containing u. Denote S\underline{u} = S_{u}. Set J = \{\underline{u} : u ∈ \{1, …, ℓ\}^{k}\}. Let (J^{ℕ}, T) denote the one-sided full shift space over the alphabet J. Let G : Σ → J^{N} be defined by

(x_{i})_{i=1}^{∞} ↦ (\underline{x_{jk+1} ⋯ x_{(j+1)k}})_{j=0}^{∞} .

It is clear that the following diagram commutes:

Σ ⟶^{σ^{k}} Σ
G↓  ↓^{G}
J^{N} ⟶^{T} J^{N}

That is, T ∘ G = G ∘ σ^{k}. It implies that ν_{n} ∘ G^{−1}, m ∘ G^{−1} ∈ M_{T} (J^{ℕ}) and

lim_{n→∞} ν_{n} ∘ G^{−1} = m ∘ G^{−1}.

Hence we have

(52)  h(T, m ∘ G^{−1}) ≥ lim sup_{n→∞} h(T, ν_{n} ∘ G^{−1}),

where we use the upper semi-continuity of the classical measure-theoretic entropy map on (J^{ℕ}, T). Define π̃ : J^{ℕ} → K by

π̃ ((\underline{u_{i}})_{i=1}^{∞}) = lim_{n→∞} S\underline{u_{1}} ∘ ⋯ ∘ S\underline{u_{n}}(K).

Then π̃∘G = π. By the assumption of AWSC (11) and Corollary 4.22 (considering the IFS \{S\underline{u} : \underline{u} ∈ J \}), we have

hπ̃(T, m ∘ G^{−1}) ≥ h(T, m ∘ G^{−1}) − log t_{k}
≥ lim sup_{n→∞} h(T, ν_{n} ∘ G^{−1}) − log t_{k}  ( by (52))
≥ lim sup_{n→∞} hπ̃(T, ν_{n} ∘ G^{−1}) − log t_{k},

where the last inequality follows from Proposition 4.1(i). Then (51) follows from the above inequality, together with Proposition 4.3 and the following claim:

(53)  hπ̃(T, ν ∘ G^{−1}) = h_{π}(σ^{k}, ν)  (ν ∈ M_{σ}(Σ)).

However, (53) just comes from Lemma 4.23, where we consider the IFS {S_{u} : u ∈ {1, …, ℓ}^{k}} rather than {S_{i}}_{i=1}^{ℓ}. □

5. Some geometric properties of C^{1} IFS

In this section we give some geometric properties of C^{1} IFS.

Lemma 5.1. Let S : U → S(U) ⊂ ℝ^{d} be a C^{1} diffeomorphism on an open set U ⊂ ℝ^{d}, and X a compact subset of U. Let c > 1. Then there exists r_{0} > 0 such that

(54)  c^{−1}[]S′(x)[] · |x − y| ≤ |S(x) − S(y)| ≤ c∥S′(x)∥ · |x − y|

for all x ∈ X, y ∈ U with |x − y| ≤ r_{0}, where S′(x) denotes the differential of S at x, [] · [] and ∥ · ∥ are defined as in (4). As a consequence,

(55)  B(S(x), c^{−1}[]S′(x)[]r) ⊂ S (B(x, r)) ⊂ B(S(x), c∥S′(x)∥r)

for all x ∈ X and 0 < r ≤ r_{0}.

Proof. Let c > 1. We only prove (54), for it is not hard to derive (55) from (54). Assume on the contrary that (54) is not true. Then there exist two sequences (x_{n}) ⊂ X, (y_{n}) ⊂ U such that x_{n} ≠ y_{n}, lim_{n→∞} |x_{n} − y_{n}| = 0 and for each n ≥ 1,

either |S(x_{n}) − S(y_{n})| ≥ c∥S′(x_{n})∥ · |x_{n} − y_{n}|,
or |S(x_{n}) − S(y_{n})| ≤ c^{−1}[]S′(x_{n})[] · |x_{n} − y_{n}|. (56)

Since X is compact, without lost of generality, we assume that

lim_{n→∞} x_{n} = x = lim_{n→∞} y_{n}.

Write S = (f_{1}, f_{2}, …, f_{d})^{t}. Then each component f_{j} of S is a C^{1} real-valued function defined on U. Choose a small ϵ > 0 such that

{z ∈ ℝ^{d} : |z − x| ≤ ϵ for some x ∈ X} ⊂ U.

Take N ∈ ℕ such that |x_{n} − y_{n}| < ϵ for n ≥ N. By the mean value theorem, for each n ≥ N and 1 ≤ j ≤ d, there exists z_{n,j} on the segment L_{x_{n},y_{n}} connecting x_{n} and y_{n} such that

f_{j}(x_{n}) − f_{j}(y_{n}) = ∇f_{j}(z_{n,j}) · (x_{n} − y_{n}),

where ∇f_{j} denote the gradient of f_{j}. Therefore |S(x_{n})−S(y_{n})| = |M_{n}(x_{n} −y_{n})| with M_{n} := (∇f_{1}(z_{n,1}), …, ∇f_{d}(z_{n,d}))^{t}. It follows

(57)  []M_{n}[] · |x_{n} − y_{n}| ≤ |S(x_{n}) − S(y_{n})| ≤ ∥M_{n}∥ · |x_{n} − y_{n}|.

Since S is C^{1}, M_{n} tends to S′(x) as n → ∞. Thus we have []M_{n}[] → []S′(x)[] and ∥M_{n}∥ → ∥S′(x)∥. Meanwhile, []S′(x_{n})[] → []S′(x)[] and ∥S′(x_{n})∥ → ∥S′(x)∥. These limits together (57) lead to a contradiction with (56). □

Let {S_{1}, …, S_{ℓ}} be a C^{1} IFS on a compact set X ⊂ ℝ^{d}. Let π : Σ → ℝ^{d} be defined as in (1). By Lemma 5.1, we have directly

Lemma 5.2. Let c > 1. Then there exists r_{0} > 0 such that for any 1 ≤ i ≤ ℓ, x ∈ Σ and 0 < r < r_{0},

B(S_{i}(πx), c^{−1}[]S′_{i}(πx)[]r) ⊂ S_{i} (B(πx, r)) ⊂ B(S_{i}(πx), c∥S′_{i}(πx)∥r).

Let \overline{ρ}, \underline{ρ} : Σ → ℝ be defined by

(58) \overline{ρ}(x) = ∥S′_{x_{1}}(πσx)∥, \underline{ρ}(x) = []S′_{x_{1}}(πσx)[]  (x = (x_{i})_{i=1}^{∞} ∈ Σ).

Let P be the partition of Σ defined as in (3). For x ∈ Σ, let P(x) denote the element in P which contains x. Then we have

Lemma 5.3. Let c > 1. Then there exists r_{0} > 0 such that for any z ∈ Σ and 0 < r < r_{0},

B^{π}(z, c^{−1}\underline{ρ}(z)r) ∩ P(z) ⊂ B^{πσ}(z, r) ∩ P(z) ⊂ B^{π}(z, c\overline{ρ}(z)r) ∩ P(z),

where B^{π}(z, r) is defined as in (12).

Proof. Let z = (z_{j})_{j=1}^{∞} ∈ Σ. Taking i = z_{1} and x = σz in Lemma 5.2 we obtain

B(S_{z_{1}}(πσz), c^{−1}[]S′_{z_{1}}(πσz)[]r) ⊂ S_{z_{1}}(B(πσz, r)) ⊂ B(S_{z_{1}}(πσz), c∥S′_{z_{1}}(πσz)∥r).

That is,

B(πz, c^{−1}\underline{ρ}(z)r) ⊂ S_{z_{1}}B(πσz, r) ⊂ B(πz, c\overline{ρ}(z)r),

where we use the fact S_{z_{1}}(πσz) = πz, which can be checked directly from the definition of π. Thus we have

B^{π}(πz, c^{−1}\underline{ρ}(z)r) ∩ P(z) ⊂ π^{−1} (S_{z_{1}} (B(πσz, r))) ∩ P(z) ⊂ B^{π}(z, c\overline{ρ}(z)r) ∩ P(z).

At last we show that π^{−1} (S_{z_{1}} (B(πσz, r))) ∩ P(z) = B^{πσ}(z, r) ∩ P(z). To see this, let y = (y_{j})_{j=1}^{∞} ∈ Σ. Then we have the following equivalent implications.

y ∈ π^{−1} (S_{z_{1}} (B(πσz, r))) ∩ P(z)
⟺ y_{1} = z_{1}, πy ∈ S_{z_{1}} (B(πσz, r))
⟺ y_{1} = z_{1}, S_{y_{1}}(πσy) ∈ S_{z_{1}} (B(πσz, r))
⟺ y_{1} = z_{1}, πσy ∈ B(πσz, r)
⟺ y_{1} = z_{1}, y ∈ B^{πσ}(z, r)
⟺ y ∈ B^{πσ}(z, r) ∩ P(z).

This finishes the proof of the lemma. □

Lemma 5.4. Assume that {S_{i}}_{i=1}^{ℓ} is a weakly conformal IFS with attractor K. Then for any c > 1, there exists D > 0 such that for any n ∈ ℕ, u ∈ {1, …, ℓ}^{n}, and x, y ∈ K we have

D^{−1}c^{−n}∥S′_{u}(x)∥ · |x − y| ≤ |S_{u}(x) − S_{u}(y)| ≤ Dc^{n}∥S′_{u}(x)∥ · |x − y|.

and

(59)  D^{−1}c^{−n}∥S′_{u}(x)∥ ≤ diam(S_{u}(K)) ≤ Dc^{n}∥S′_{u}(x)∥.

Proof. The results were proved in the conformal case in [18, Lemma 3.5 and Corollary 3.6]. A slight modification of that proof works for the weakly conformal case. □

As a corollary, we have

Corollary 5.5. Under the assumption of Lemma 5.4, for α > 0, there is r_{0} > 0 such that for any 0 < r < r_{0} and z ∈ K, there exist n ∈ ℕ and u ∈ {1, …, ℓ}^{n} such that S_{u}(K) ⊂ B(z, r) and

(60)  |S_{u}(x) − S_{u}(y)| ≥ r^{1+α}|x − y|  (x, y ∈ K).

Proof. Denote a = inf{[]S′_{i}(x)[] : x ∈ K, 1 ≤ i ≤ ℓ} and b = sup{∥S′_{i}(x)∥ : x ∈ K, 1 ≤ i ≤ ℓ}. Then 0 < a ≤ b < 1. Choose c so that

(61)  1 < c < b^{\frac{−α}{3(2+α)}} .

Let D be the constant in Lemma 5.4 corresponding to c. Take n_{0} ∈ ℕ and r_{0} > 0 such that

(62)  (c^{3}b^{α/(2+α)})^{n_{0}} < D^{−3}ab^{α/(2+α)}, (1 + α/2) · \frac{log r_{0}}{log a} = n_{0}.

Now fix z ∈ K and 0 < r < r_{0}. We shall show that there exist n ∈ ℕ and u ∈ {1, …, ℓ}^{n} such that S_{u}(K) ⊂ B(z, r) and (60) holds. To see this, take ω = (ω_{i})_{i=1}^{∞} ∈ Σ such that z = πω, where π is defined as in (1). Let n be the unique integer such that

(63)  ∥S′_{ω_{1}···ω_{n}}(πσ^{n}ω)∥ < r^{1+α/2} ≤ ∥S′_{ω_{1}···ω_{n−1}}(πσ^{n−1}ω)∥.

It follows a^{n} < r^{1+α/2} ≤ b^{n−1}, which together with (62) forces that

(64)  n > n_{0} and c^{3n} < D^{−3}ar^{−α/2}.

To see (64), we first assume on the contrary that n ≤ n_{0}. Then

a^{n} ≥ a^{n_{0}} = a^{(1+α/2) log r_{0}/ log α} = r_{0}^{1+α/2} > r^{1+α/2},

which contradicts the fact a^{n} < r^{1+α/2}. Hence n > n_{0}. To see c^{3n} < D^{−3}ar^{−α/2}, note that

c^{3n}r^{α/2} ≤ c^{3n}b^{(n−1)α/(2+α)}  ( using r^{1+α/2} ≤ b^{n−1} )
≤ (c^{3}b^{α/(2+α)})^{n} b^{−α/(2+α)}
≤ (c^{3}b^{α/(2+α)})^{n_{0}} b^{−α/(2+α)}  ( using n > n_{0} and (61) )
≤ D^{−3}a  (by (62)).

This completes the proof of (64). By (59), we have

diamS_{ω_{1}···ω_{n}}(K) ≤ Dc^{n}∥S′_{ω_{1}···ω_{n}}(πσ^{n}ω)∥ ≤ Dc^{n}r^{1+α/2} < r.

Since z ∈ S_{ω_{1}···ω_{n}}(K), the above inequality implies S_{ω_{1}···ω_{n}}(K) ⊂ B(z, r). By (59) again, we have

(65)  ∥S′_{u}(x)∥ ≥ D^{−2}c^{−2n}∥S′_{u}(y)∥, ∀ u ∈ {1, …, ℓ}^{n}, ∀ x, y ∈ K.

By Lemma 5.4, we have for x, y ∈ K,

|S_{ω_{1}···ω_{n}}(x) − S_{ω_{1}···ω_{n}}(y)|
≥ D^{−1}c^{−n}∥S′_{ω_{1}···ω_{n}}(x)∥ · |x − y|
≥ D^{−3}c^{−3n}∥S′_{ω_{1}···ω_{n}}(πσ^{n}ω)∥ · |x − y| (by (65))
≥ D^{−3}c^{−3n}∥S′_{ω_{1}···ω_{n−1}}(πσ^{n−1}ω)∥[]S′_{ω_{n}}(πσ^{n}ω)[] · |x − y|
≥ D^{−3}c^{−3n}ar^{1+α/2}|x − y| (by (63))
≥ r^{1+α}|x − y| (by (64)).

Hence the corollary follows by taking u = ω_{1} ⋯ ω_{n}. □

Proposition 5.6. Let {S_{i}}_{i=1}^{ℓ} be a C^{1} IFS with attractor K. Assume that K is not a singleton. Then

(i) for any m ∈ M_{σ}(Σ), we have for m-a.e. x = (x_{i})_{i=1}^{∞} ∈ Σ,

lim inf_{n→∞} \frac{log diamS_{x_{1}...x_{n}}(K)}{n} ≥ −\overline{λ}(x),

lim sup_{n→∞} \frac{log diamS_{x_{1}...x_{n}}(K)}{n} ≤ −\underline{λ}(x),

where \underline{λ}, \overline{λ} are defined as in Definition 2.5. In particular, if \{S_{i}\}_{i=1}^{ℓ} is

m-conformal, then for m-a.e. x = (x_{i})_{i=1}^{∞} ∈ Σ,

lim_{n→∞} \frac{log diamS_{x_{1}...x_{n}}(K)}{n} = −λ(x).

(ii) If {S_{i}}_{i=1}^{ℓ} is weakly conformal, then it is m-conformal for each m ∈ M_{σ}(Σ).

Proof. We first prove (i). Take c > 1 small enough so that c sup_{x∈Σ} \overline{ρ}(x) < 1. Let r_{0} > 0 be given as in Lemma 5.2. Let x = (x_{i})_{i=1}^{∞} ∈ Σ. Applying Lemma 5.2 repeatedly, we have

(66)  S_{x_{1}···x_{n}}(B(πσ^{n}x, r_{0})) ⊂ B(πx, c^{n}\overline{ρ}(x) ⋯ \overline{ρ}(σ^{n−1}x)r_{0}).

Since {S_{i}}_{i=1}^{ℓ} is contractive, there is a constant k such that

S_{x_{n+1}···x_{n+k}}(K) ⊂ B(πσ^{n}x, r_{0}).

This together with (66) yields (67)

diamS_{x_{1}...x_{n+k}}(K) ≤ diamS_{x_{1}···x_{n}}(B(πσ^{n}x, r_{0})) ≤ c^{n}\overline{ρ}(x) … \overline{ρ}(σ^{n−1}x)r_{0}.

Since K is not a singleton, there exists 0 < r_{1} < r_{0} such that for each z ∈ K, there exists w ∈ K such that r_{1} ≤ |z − w| ≤ r_{0}. Indeed, to obtain r_{1}, one chooses an integer n_{0} large enough such that sup_{u∈Σ_{n_{0}}} diamS_{u}(K) ≤ r_{0}, then set

r_{1} = (1/2) inf_{u∈Σ_{n_{0}}} diamS_{u}(K).

For each such pair (z, w), applying (54) repeatedly yields

diamS_{x_{1}...x_{n}}(K) ≥ |S_{x_{1}...x_{n}}(z) − S_{x_{1}...x_{n}}(w)| ≥ r_{1}c^{−n} ∏_{j=1}^{n}[]S′_{x_{j}}(S_{x_{j+1}...x_{n}}(z)[].

Hence by taking z = πσ^{n}x, we have

(68)  diamS_{x_{1}...x_{n}}(K) ≥ r_{1}c^{−n}\underline{ρ}(x) … \underline{ρ}(σ^{n−1}x).

Denote

g_{∗}(x) = lim inf_{n→∞} \frac{log diamS_{x_{1}...x_{n}}(K)}{n} and

g^{∗}(x) = lim sup_{n→∞} \frac{log diamS_{x_{1}...x_{n}}(K)}{n}.

It is clear that g_{∗}(x) = g_{∗}(σx) and g^{∗}(x) = g^{∗}(σx). Let I denote the σ-algebra {B ∈ B(Σ) : σ^{−1}B = B}. Then by (68), the Birkhoff ergodic theorem, and Theorem 34.2 in [7], we have for m-a.e. x ∈ Σ,

g_{∗}(x) = E_{m}(g_{∗}|I)(x) ≥ E_{m} ( lim_{n→∞} \frac{−n log c + ∑_{i=0}^{n−1} log \underline{ρ} ∘ σ^{−i}}{n} |I) (x)

= − log c + lim_{n→∞} \frac{1}{n} ∑_{i=0}^{n−1} E_{m}(log \underline{ρ} ∘ σ^{−i}|I)(x)

= − log c + E_{m}(log \underline{ρ}|I)(x)

(69)

and similarly by (67),

(70)  g^{∗}(x) ≤ log c + E_{m}(log \overline{ρ}|I)(x).

For p ∈ ℕ, write A_{p}(x) = log[]S′_{x_{1}···x_{p}}(πσ^{p}x)[] and A_{p}^{∗}(x) = log ∥S′_{x_{1}···x_{p}}(πσ^{p}x)∥. Consider the IFS {S_{i_{1}...i_{p}} : 1 ≤ i_{j} ≤ ℓ, 1 ≤ j ≤ p} rather than {S_{i}}_{i=1}^{ℓ}. Then (69) and (70) can be replaced by

g_{∗}(x) ≥ − log c + \frac{1}{p}E_{m}(A_{p}|I_{p})(x), g^{∗}(x) ≤ log c + \frac{1}{p}E_{m}(A_{p}^{∗}|I_{p})(x),

where I_{p} := {B ∈ B(Σ) : σ^{−p}B = B}. Taking the conditional expectation with respect to I in the above inequalities and noting that g_{∗}, g^{∗} are σ-invariant, we obtain

(71) g_{∗}(x) ≥ − log c + \frac{1}{p}E_{m}(A_{p}|I)(x),  g^{∗}(x) ≤ log c + \frac{1}{p}E_{m}(A_{p}^{∗}|I)(x).

Since A_{p}(x) is sup-additive (i.e., A_{p+q}(x) ≥ A_{p}(x) + A_{q}(σ^{p}x)) and A_{p}^{∗}(x) is subadditive (i.e., A_{p+q}^{∗}(x) ≤ A_{p}^{∗}(x) + A_{q}^{∗}(σ^{p}x)), by Kingman’s sub-additive ergodic theorem (cf. [63]), we have

(72)  lim_{p→∞} A_{p}(x)/p = −\overline{λ}(x),  lim_{p→∞} A_{p}^{∗}(x)/p = −\underline{λ}(x)

almost everywhere and in L^{1}. Hence letting c → 1 and p → ∞ in (71) and using Theorem 34.2 in [7], we obtain that g_{∗}(x) ≥ −\overline{λ}(x) and g^{∗}(x) ≤ −\underline{λ}(x) almost everywhere. This finishes the proof of (i).

To see (ii), assume that {S_{i}}_{i=1}^{ℓ} is weakly conformal and m ∈ M_{σ}(Σ). Then |A_{p}(x) − A_{p}^{∗}(x)|/p converges to 0 uniformly as p tends to infinity. This together with (72) yields \overline{λ}(x) = \underline{λ}(x) for m-a.e. x ∈ Σ. This proves (ii). □

6. Estimates for local dimensions of invariant measures for C^{1} IFS

In this section, we prove a general version of Theorem 2.6, which is also needed in the proof of Theorem 2.11. Let {T_{i}}_{i=1}^{ℓ} be a C^{1} IFS on ℝ^{d}, and {S_{i}}_{i=1}^{ℓ} a C^{1} IFS on ℝ^{k}. Let ϕ : Σ → ℝ^{d} and π : Σ → ℝ^{k} denote the canonical projections associated with {T_{i}}_{i=1}^{ℓ} and {S_{i}}_{i=1}^{ℓ} respectively. Let η and ξ be two partitions of Σ defined respectively by

η = {ϕ^{−1}(z) : z ∈ ℝ^{d}}, ξ = σ^{−1}η.

Let P be the partition of Σ given as in (3) and let \overline{ρ}(x), \underline{ρ}(x) be defined as in (58). Applying Lemma 5.3 to the IFS {S_{i}}_{i=1}^{ℓ}, we have for any c > 1 there exist 0 < δ < c − 1 and r_{0} > 0 such that for any r ∈ (0, r_{0}) and x ∈ Σ, (73) B^{π}(x, (c−δ)^{−1}\underline{ρ}(x)r)∩P(x) ⊂ B^{πσ}(x, r)∩P(x) ⊂ B^{π}(x, (c−δ)\overline{ρ}(x)r)∩P(x).

The following technical proposition is substantial in our proof.

Proposition 6.1. Let m ∈ M_{σ}(Σ) and c > 1. Let δ, r_{0} be given as above. Then there exists Λ ⊂ Σ with m(Λ) = 1 such that for all x ∈ Λ and r ∈ (0, r_{0}),

(74)  \frac{m_{x}^{η}(B^{π}(x, c\overline{ρ}(x)r) ∩ P(x))}{m_{σx}^{η}(B^{π}(σx, r))} ≥ f(x) · \frac{m_{x}^{ξ}(B^{πσ}(x, r) ∩ P(x))}{m_{x}^{ξ}(B^{πσ}(x, r))}

and (75)

\frac{m_{x}^{η}(B^{π}(x, c^{−1}\underline{ρ}(x)r) ∩ P(x))}{m_{σx}^{η}(B^{π}(σx, r))} ≤ f(x) · \frac{m_{x}^{ξ}(B^{πσ}(x, (1 − cδ/2)r) ∩ P(x))}{m_{x}^{ξ}(B^{πσ}(x, (1 − cδ/2)r))},

where f := ∑_{A∈P} χ_{A} \frac{E_{m}(χ_{A}|ϕ^{−1}γ)}{E_{m}(χ_{A}|σ^{−1}ϕ^{−1}γ)}, γ = B(ℝ^{d}).

Proof. Write R_{t,x}(z) = T_{x_{1}}^{−1}B(T_{x_{1}}z, t) for t > 0, x = (x_{i})_{i=1}^{∞} ∈ Σ and z ∈ ℝ^{d}. It is direct to check that

(76)  σ^{−1}ϕ^{−1}R_{t,x}(ϕσx) ∩ P(x) = B^{ϕ}(x, t) ∩ P(x).

Hence for m-a.e. x,

\frac{m(ϕ^{−1}R_{t,x}(ϕσx))}{m(B^{ϕ}(x, t))} = \frac{m(B^{ϕ}(x, t) ∩ P(x))}{m(B^{ϕ}(x, t))} · \frac{m(ϕ^{−1}R_{t,x}(ϕσx))}{m(B^{ϕ}(x, t) ∩ P(x))}
= \frac{m(B^{ϕ}(x, t) ∩ P(x))}{m(B^{ϕ}(x, t))} · \frac{m(σ^{−1}ϕ^{−1}R_{t,x}(ϕσx))}{m(σ^{−1}ϕ^{−1}R_{t,x}(ϕσx) ∩ P(x))}.

Letting t → 0 and applying Proposition 3.5 and Remark 3.6, we have

(77) lim_{t→0} \frac{m(ϕ^{−1}R_{t,x}(ϕσx))}{m(B^{ϕ}(x, t))} = ∑_{A∈P} χ_{A}(x)\frac{E_{m}(χ_{A}|ϕ^{−1}γ)(x)}{E_{m}(χ_{A}|σ^{−1}ϕ^{−1}γ)(x)} =: f(x).

for m-a.e. x. Let Λ̃ denote the set of x ∈ Σ such that the following properties (1)-(4) hold:

(1) lim_{t→0} \frac{m(B^{ϕ}(x, t) ∩ P(x))}{m(B^{ϕ}(x, t))} = ∑_{A∈P} χ_{A}E_{m}(χ_{A}|ϕ^{−1}γ)(x) > 0.

(2) lim_{t→0} \frac{m(σ^{−1}ϕ^{−1}R_{t,x}(ϕσx) ∩ P(x))}{m(σ^{−1}ϕ^{−1}R_{t,x}(ϕσx))} = ∑_{A∈P} χ_{A}E_{m}(χ_{A}|σ^{−1}ϕ^{−1}γ)(x) > 0.

(3) For all q ∈ ℚ^{+},

m_{x}^{η}(B^{π}(x, q) ∩ P(x)) ≥ lim sup_{t→0} \frac{m (B^{π}(x, q) ∩ P(x) ∩ B^{ϕ}(x, t))}{m (B^{ϕ}(x, t))},

m_{x}^{η}(U^{π}(x, q) ∩ P(x)) ≤ lim inf_{t→0} \frac{m (B^{π}(x, q) ∩ P(x) ∩ B^{ϕ}(x, t))}{m (B^{ϕ}(x, t))},

m_{x}^{ξ}(B^{πσ}(x, q) ∩ P(x)) ≥ lim sup_{t→0} \frac{m (B^{πσ}(x, q) ∩ P(x) ∩ σ^{−1}ϕ^{−1}R_{t,x}(ϕσx))}{m (σ^{−1}ϕ^{−1}R_{t,x}(ϕσx))},

m_{x}^{ξ}(U^{πσ}(x, q) ∩ P(x)) ≤ lim inf_{t→0} \frac{m (B^{πσ}(x, q) ∩ P(x) ∩ σ^{−1}ϕ^{−1}R_{t,x}(ϕσx))}{m (σ^{−1}ϕ^{−1}R_{t,x}(ϕσx))},

where U^{π}(x, q) := π^{−1}U(πx, q), U^{πσ}(x, q) := σ^{−1}π^{−1}U(πσx, q) and U(z, q) denotes the open ball in ℝ^{k} of radius q centered at z.

(4) lim_{t→0} \frac{m(ϕ^{−1}R_{t,x}(ϕσx))}{m(B^{ϕ}(x, t))} = f(x).

Then we have m(Λ̃) = 1 by Proposition 3.5, Lemma 3.7, Remarks 3.6, 3.8 and

Now let Λ = Λ̃ ∩ σ^{−1}Λ̃. Then m(Λ) = 1. Fix x ∈ Λ and r ∈ (0, r_{0}). Let q_{1} ∈ ℚ^{+} ∩(r, cr/(c−δ)). Choose q_{2}, q_{3} ∈ ℚ^{+} such that q_{1} < q_{2} < cr/(c−δ) and q_{2}(c−δ)\overline{ρ}(x) < q_{3} < c\overline{ρ}(x)r. By (73), we have B^{π}(x, q_{3})∩P(x) ⊃ B^{πσ}(x, q_{2})∩ P(x). It together with (76) yields

(78) B^{π}(x, q_{3}) ∩ P(x) ∩ B^{ϕ}(x, t)) ⊃ B^{πσ}(x, q_{2}) ∩ P(x) ∩ σ^{−1}ϕ^{−1}R_{t,x}(ϕσx)).

Hence we have

\frac{m_{x}^{η}(B^{π}(x, c\overline{ρ}(x)r) ∩ P(x))}{m_{σx}^{η}(B^{π}(σx, r))}

≥ \frac{m_{x}^{η}(B^{π}(x, q_{3}) ∩ P(x))}{m_{σx}^{η}(U^{πσ}(x, q_{1}))}
≥ \frac{lim sup_{t→0} m(B^{π}(x, q_{3}) ∩ P(x) ∩ B^{ϕ}(x, t))/m(B^{ϕ}(x, t))}{lim inf_{t→0} m(B^{π}(σx, q_{1}) ∩ ϕ^{−1}R_{t,x}(ϕσx))/m(ϕ^{−1}R_{t,x}(ϕσx))}

(by Lemma 3.7 and Remark 3.8)

≥ lim_{t→0} \frac{m(ϕ^{−1}R_{t,x}(ϕσx))}{m(B^{ϕ}(x, t))} · lim sup_{t→0} \frac{m(B^{π}(x, q_{3}) ∩ P(x) ∩ B^{ϕ}(x, t))}{m(σ^{−1}B^{π}(σx, q_{1}) ∩ σ^{−1}ϕ^{−1}R_{t,x}(ϕσx))}
= lim_{t→0} \frac{m(ϕ^{−1}R_{t,x}(ϕσx))}{m(B^{ϕ}(x, t))} · lim sup_{t→0} \frac{m(B^{π}(x, q_{3}) ∩ P(x) ∩ B^{ϕ}(x, t))}{m(B^{πσ}(x, q_{1}) ∩ σ^{−1}ϕ^{−1}R_{t,x}(ϕσx))}.

Denote

X_{t} := m(B^{πσ}(x, q_{2}) ∩ P(x) ∩ σ^{−1}ϕ^{−1}R_{t,x}(ϕσx)),
Y_{t} := m(B^{πσ}(x, q_{1}) ∩ σ^{−1}ϕ^{−1}R_{t,x}(ϕσx)),
Z_{t} := m(σ^{−1}ϕ^{−1}R_{t,x}(ϕσx)).

Using the property (4), we have

\frac{m_{x}^{η}(B^{π}(x, c\overline{ρ}(x)r) ∩ P(x))}{m_{σx}^{η}(B^{π}(σx, r))}

≥ f(x) · lim sup_{t→0} \frac{m(B^{π}(x, q_{3}) ∩ P(x) ∩ B^{ϕ}(x, t))}{m(B^{πσ}(x, q_{1}) ∩ σ^{−1}ϕ^{−1}R_{t,x}(ϕσx))}
≥ f(x) · lim sup_{t→0} X_{t}/Y_{t}  (by (78))
≥ f(x) · lim sup_{t→0} \frac{X_{t}/Z_{t}}{Y_{t}/Z_{t}} ≥ f(x) · \frac{lim inf_{t→0} X_{t}/Z_{t}}{lim sup_{t→0} Y_{t}/Z_{t}}

≥ f(x) · \frac{m_{x}^{ξ}(U^{πσ}(x, q_{1}) ∩ P(x))}{m_{x}^{ξ}(B^{πσ}(x, q_{1}))} (by Lemma 3.7 and Remark 3.8)

≥ f(x) · \frac{m_{x}^{ξ}(B^{πσ}(x, r) ∩ P(x))}{m_{x}^{ξ}(B^{πσ}(x, q_{1}))}.

Letting q_{1} ↓ r, we obtain (74). (75) follows from an analogous argument. □

Let (ϕ, π) denote the map Σ → ℝ^{d} × ℝ^{k}, x ↦ (ϕx, πx). It is easy to see that (ϕ, π) is the canonical projection w.r.t. the direct product of {T_{i}}_{i=1}^{ℓ} and {S_{i}}_{i=1}^{ℓ}. In the following we give a general version of Theorem 2.6.

Theorem 6.2. Let m ∈ M_{σ}(Σ). Then for m-a.e. x ∈ Σ, we have

(79)  lim sup_{r→0} \frac{log m_{x}^{η}(B^{π}(x, r))}{log r} ≤ \frac{E_{m}(g|I)(x)}{−\underline{λ}(x)} and
(80)  lim inf_{r→0} \frac{log m_{x}^{η}(B^{π}(x, r))}{log r} ≥ \frac{E_{m}(g|I)(x)}{−\overline{λ}(x)},

where

g := I_{m}(P|σ^{−1}ϕ^{−1}B(ℝ^{d})) − I_{m}(P|ϕ^{−1}B(ℝ^{d}))
+I_{m}(P|(ϕ, σ)^{−1}B(ℝ^{d} × ℝ^{k})) − I_{m}(P|σ^{−1}(ϕ, π)^{−1}B(ℝ^{d} × ℝ^{k})),

and \overline{λ}(x), \underline{λ}(x) denote the upper and lower Lyapunov exponents of \{S_{i}\}_{i=1}^{ℓ} at x (see Definition 2.5). In particular, if {S_{i}}_{i=1}^{ℓ} is m-conformal, we have

lim_{r→0} \frac{log m_{x}^{η}(B^{π}(x, r))}{log r} = \frac{h_{(ϕ,π)}(σ, m, x) − h_{ϕ}(σ, m, x)}{λ(x)}.

Proof. It suffices to prove (79) and (80). For short we only prove (79). The proof of (80) is analogous.

We first prove the following inequality

(81)  lim sup_{r→0} \frac{log m_{x}^{η}(B^{π}(x, r))}{log r} ≤ \frac{E_{m}(g|I)(x)}{E_{m}(log \overline{ρ}|I)(x)}  m-a.e.,

where \overline{ρ}(x) = ∥S′_{x_{1}}(σx)∥ for x = (x_{i})_{i=1}^{∞}. To see it, let c > 1 so that

c sup_{x∈Σ} \overline{ρ}(x) < 1.

Let r_{0} and f be given as in Proposition 6.1. For n ∈ ℕ and x ∈ Σ, define

\overline{ρ}_{n}(x) = \overline{ρ}(x)\overline{ρ}(σx) ⋯ \overline{ρ}(σ^{n−1}x).

Write

H_{n}(x) := log \frac{m_{x}^{η} (B^{π}(x, c^{n}\overline{ρ}_{n}(x)r_{0}))}{m_{σx}^{η} (B^{π}(σx, c^{n−1}\overline{ρ}_{n−1}(σx)r_{0}))},
G_{n}(x) := log \frac{m_{x}^{η} (B^{π}(x, c^{n}\overline{ρ}_{n}(x)r_{0}) ∩ P(x))}{m_{x}^{η} (B^{π}(x, c^{n}\overline{ρ}_{n}(x)r_{0}))},

W_{n}(x) := log \frac{m_{x}^{ξ} (B^{πσ}(x, c^{n−1}\overline{ρ}_{n−1}(σx)r_{0}) ∩ P(x))}{m_{x}^{ξ} (B^{πσ}(x, c^{n−1}\overline{ρ}_{n−1}(σx)r_{0}))}.

Then by Proposition 6.1 we have for m-a.e. x, H_{n}(x) + G_{n}(x) ≥ log f(x) + W_{n}(x), that is,

H_{n}(x) ≥ log f(x) − G_{n}(x) + W_{n}(x).

However

log m_{x}^{η} (B^{π}(x, c^{n}\overline{ρ}_{n}(x)r_{0})) = ∑_{j=0}^{n−1} H_{n−j}(σ^{j}x) + log m_{σ^{n}x}^{η} (B^{π}(σ^{n}x, r_{0})) .

Hence for m-a.e. x,

\frac{log m_{x}^{η} (B^{π}(x, c^{n}\overline{ρ}_{n}(x)r_{0}))}{n} ≥ \frac{1}{n} ∑_{j=0}^{n−1} [log f(σ^{j}x) − G_{n−j}(σ^{j}x) + W_{n−j}(σ^{j}x)]

+ \frac{1}{n} log m_{σ^{n}x}^{η} (B^{π}(σ^{n}x, r_{0})) .

Note that by Proposition 3.5,

G_{n} → G := −I_{m}(P|η̂ ∨ π^{−1}B(ℝ^{k})),
W_{n} → W := −I_{m}(P|σ^{−1}η̂ ∨ σ^{−1}π^{−1}B(ℝ^{k}))

pointwise and in L^{1}. By Lemma 4.13 and Proposition 3.9, we have for m-a.e. x,

lim inf_{n→∞} \frac{log m_{x}^{η} (B^{π}(x, c^{n}\overline{ρ}_{n}(x)r_{0}))}{n} ≥ E_{m}((log f − G + W)|I)(x)
= E_{m}(g|I)(x).

In the meantime, by Birkhoff ergodic Theorem, we have

lim_{n→∞} \frac{1}{n} log(c^{n}\overline{ρ}_{n}(x)r_{0}) = log c + E_{m}(log \overline{ρ}|I)(x) m-a.e.

Hence we have

lim sup_{r→0} \frac{log m_{x}^{η} (B^{π}(x, r))}{log r} = lim sup_{n→∞} \frac{log m_{x}^{η} (B^{π}(x, c^{n}\overline{ρ}_{n}(x)r_{0}))}{log(c^{n}\overline{ρ}_{n}(x)r_{0})}
≤ \frac{E_{m}(g|I)(x)}{log c + E_{m}(log \overline{ρ}|I)(x)}.

Taking c → 1, we obtain (81).

Let q ∈ ℕ. Considering the IFS {T_{i_{1}...i_{q}} : 1 ≤ i_{j} ≤ ℓ, 1 ≤ j ≤ q} and {S_{i_{1}...i_{q}} : 1 ≤ i_{j} ≤ ℓ, 1 ≤ j ≤ q}, analogous to (81) we have

(82)  lim sup_{r→0} \frac{log m_{x}^{η}(B^{π}(x, r))}{log r} ≤ \frac{E_{m}(g_{q}|I)(x)}{E_{m}(log h_{q}|I)(x)},

where

g_{q} := I_{m}(P_{0}^{q−1}|σ^{−q}ϕ^{−1}B(ℝ^{d})) − I_{m}(P_{0}^{q−1}|ϕ^{−1}B(ℝ^{d}))
+I_{m}(P_{0}^{q−1}|(ϕ, π)^{−1}B(ℝ^{d} × ℝ^{k})) − I_{m}(P_{0}^{q−1}|σ^{−q}(ϕ, π)^{−1}B(ℝ^{d} × ℝ^{k}))

and h_{q}(x) := ∥S′_{x_{1}...x_{q}}(σ^{q}x)∥ for x = (x_{i})_{i=1}^{∞}.

Due to (23), we have E_{m}(g_{q}|I)(x) = qE_{m}(g|I)(x). It is easily seen that h_{q}(x) is sub-multiplicative in the sense that h_{p+q}(x) ≤ h_{p}(x)h_{q}(σ^{p}x). Thus by Kingman sub-additive ergodic theorem (cf. [63]), we have

lim_{q→∞} \frac{1}{q}E_{m}(log h_{q}|I)(x) = −\underline{λ}(x) for m-a.e. x.

Hence letting q → ∞ in (82) we obtain (79). This finishes the proof of Theorem 6.2. □

Proof of Theorem 2.6. In Theorem 6.2, we take T_{i}(x) = x/2 for all 1 ≤ i ≤ ℓ to obtain Theorem 2.6. To see it, we know that the attractor of {T_{i}}_{i=1}^{ℓ} is just the singleton {0}. Hence η is the trivial partition {Σ, ∅} of Σ, and thus we have m_{x}^{η} ≡ m. □

7. Proofs of Theorem 2.11 and Theorem 2.12

7.1. Proof of Theorem 2.11. Let Φ = {S_{i}}_{i=1}^{ℓ} be the direct product of k C^{1} IFS Φ_{1}, …, Φ_{k}, which are defined respectively on compact X_{i} ⊂ ℝ^{q_{i}} (i = 1, …, k). For each i, let Γ_{i} denote the canonical projection w.r.t. Φ_{i}, and let λ_{i}(x) denote the Lyapunov exponent of Φ_{i} at x provided it exists.

Let m ∈ M_{σ}(Σ). Assume that Φ_{1}, …, Φ_{k} are m-conformal. Let Ω denote the collection of all permutations of {1, …, k}. For τ ∈ Ω, we denote

Λ_{τ} := {x ∈ Σ : λ_{i}(x) exists for all i, λ_{τ(1)}(x) ≤ λ_{τ(2)}(x) ≤ ⋯ ≤ λ_{τ(k)}(x)} .

Then m (⋃_{τ∈Ω} Λ_{τ} ) = 1. Let π denote the canonical projection associated with the IFS Φ. In the following we show that the local dimension d(m ∘ π^{−1}, πx) exists for m-a.e. x ∈ Σ.

Without loss of generality we only show that d(m ∘ π^{−1}, πx) exists for m-a.e. x ∈ Λ_{e}, where e denotes the identity in Ω. Here we may assume m(Λ_{e}) > 0. For other Λ_{τ} ’s, the proof is essentially identical under a change of coordinates.

For i = 1, …, k, let π_{i} denote the canonical projection w.r.t. Φ_{1} × ⋯ × Φ_{i}. It is clear that π = π_{k}. Bear in mind that

λ_{1}(x) ≤ λ_{2}(x) ≤ ⋯ ≤ λ_{k}(x) (x ∈ Λ_{e}).

For i = 1, …, k, we use {m_{x}^{i}} to denote the family of conditional measures {m_{x}^{η_{i}}} of m associated with the partition

η_{i} = {π_{i}^{−1}(z) : z ∈ ∏_{t=1}^{i} ℝ^{q_{t}}} .

For convenience, we use {m_{x}^{0}} denote the family of conditional measures of m with the trivial partition {Σ, ∅}. It is clear that m_{x}^{0} = m for all x ∈ Σ.

For i = 1, …, k, we give a metric d_{i} on ∏_{t=1}^{i} ℝ^{q_{t}} by

d_{i}((z_{1}, …, z_{i}), (w_{1}, …, w_{i})) = sup_{1≤t≤i} |z_{t} − w_{t}|_{ℝ^{q_{t}}}.

and define d = d_{k}. We claim that for any x ∈ Λ_{e} and ϵ > 0,

(83)  η_{i}(x) ∩ P_{0}^{n}(x) ⊂ B^{π}(x, e^{−n(λ_{i+1}(x)−ϵ)})

when n is large enough. Here B^{π}(x, r) is defined as in (12). To see the claim, let x ∈ Λ_{e} and y ∈ η_{i}(x). Then π_{i}y = π_{i}x. Thus

d(πy, πx) = sup_{1≤t≤k} |Γ_{t}y, Γ_{t}x|_{ℝ^{q_{t}}} = sup_{i+1≤t≤k} |Γ_{t}y, Γ_{t}x|_{ℝ^{q_{t}}}.

Since y ∈ P_{0}^{n}(x) and λ_{i+1}(x) ≤ … ≤ λ_{k}(x), by Proposition 5.6, we have

d(πy, πx) ≤ e^{−n(λ_{i+1}(x)−ϵ)}

when n is large enough, and (83) follows.

For i = 0, 1, …, k and x ∈ Σ, denote

h_{i}(x) = lim_{n→∞} \frac{− log m_{x}^{i}(P_{0}^{n}(x))}{n + 1}

provided that the limit exists. By Proposition 4.14,

(84)  h_{i}(x) = h(σ, m, x) − h_{π_{i}}(σ, m, x) for m-a.e.x ∈ Σ.

For i = 0, 1, …, k − 1 and x ∈ Σ, denote

ϑ_{i}(x) = lim inf_{r→0} \frac{log m_{x}^{i}(B^{Γ_{i+1}}(x, r))}{log r}.

By Theorem 6.2 and (84), we have

(85)  ϑ_{i}(x) = \frac{h_{π_{i+1}}(σ, m, x) − h_{π_{i}}(σ, m, x)}{λ_{i+1}(x)} = \frac{h_{i}(x) − h_{i+1}(x)}{λ_{i+1}(x)}

for m-a.e. x ∈ Σ.

For i = 0, 1, …, k and x ∈ Σ, define

\overline{δ}_{i}(x) = lim sup_{r→0} \frac{log m_{x}^{i}(B^{π}(x, r))}{log r}, \underline{δ}_{i}(x) = lim inf_{r→0} \frac{log m_{x}^{i}(B^{π}(x, r))}{log r}.

We claim that (C1) \overline{δ}_{k}(x) = \underline{δ}_{k}(x) = 0 for all x ∈ Σ. (C2) h_{i}(x) − h_{i+1}(x) ≥ λ_{i+1}(\overline{δ}_{i}(x) − \overline{δ}_{i+1}(x)) for m-a.e. x ∈ Λ_{e} and i =

0, 1 …, k − 1; (C3) \underline{δ}_{i+1}(x) + ϑ_{i}(x) ≤ \underline{δ}_{i}(x) for m-a.e. x ∈ Λ_{e} and i = 0, 1 …, k − 1; It is easy to see that (C1)-(C3) together with (84)-(85) force that for m-a.e.

x ∈ Λ_{e}, \underline{δ}_{i}(x) = \overline{δ}_{i}(x) (we denoted the common value as δ_{i}(x)) for i = 0, …, k and, furthermore

(86)  d(m ∘ π^{−1}, πx) = δ_{0}(x) = ∑_{i=0}^{k−1} ϑ_{i}(x) = ∑_{i=0}^{k−1} \frac{h_{i}(x) − h_{i+1}(x)}{λ_{i+1}(x)}.

which is the desired result in Theorem 2.11. In the following we prove (C1)-(C3) respectively.

Proof of (C1). Since η_{k} = {π^{−1}(z) : z ∈ ∏_{t=1}^{k} ℝ^{q_{t}}}, we have

m_{x}^{k}(B^{π}(x, r)) = m_{x}^{k}(η_{k}(x)) = 1

for all x ∈ Σ. Thus \overline{δ}_{k}(x) = \underline{δ}_{k}(x) = 0 for all x ∈ Σ. □

Proof of (C2). We give a proof by contradiction, which is modified from [40, §10.2]. Assume that (C2) is not true. Then there exists 0 ≤ i ≤ k such that

h_{i}(x) − h_{i+1}(x) < λ_{i+1}(x)(\overline{δ}_{i}(x) − \overline{δ}_{i+1}(x))

on a subset of Λ_{e} with positive measure. Hence there exist α > 0 and real numbers h_{i}, h_{i+1}, λ_{i+1}, \overline{δ}_{i}, \overline{δ}_{i+1} with λ_{i+1} > 0 such that

(87)  h_{i} − h_{i+1} < λ_{i+1}(\overline{δ}_{i} − \overline{δ}_{i+1}) − α

and for any ϵ > 0, there exists B_{ϵ} ⊂ Λ_{e} with m(B_{ϵ}) > 0 so that for x ∈ B_{ϵ},

|h_{i}(x) − h_{i}| < ϵ/2, |h_{i+1}(x) − h_{i+1}| < ϵ/2, |λ_{i+1}(x) − λ_{i+1}| < ϵ/2

and

|\overline{δ}_{i}(x) − \overline{δ}_{i}| < ϵ/2, |\overline{δ}_{i+1}(x) − \overline{δ}_{i+1}| < ϵ/2.

Fix ϵ > 0. There exists n_{0} : B_{ϵ} → ℕ such that for m-a.e. x ∈ B_{ϵ} and n > n_{0}(x), we have

(1) \frac{log m_{x}^{i+1} (B^{π}(x, e^{−n(λ_{i+1}−2ϵ)}))}{−n(λ_{i+1} − 2ϵ)} ≤ \overline{δ}_{i+1} + ϵ;

(2) −\frac{1}{n} log m_{x}^{i+1}(P_{0}^{n}(x)) ≥ h_{i+1} − ϵ (by (84));

(3) η_{i}(x) ∩ P_{0}^{n}(x) ⊂ B^{π}(x, e^{−n(λ_{i+1}−2ϵ)}) (by (83));

(4) −\frac{1}{n} log m_{x}^{i}(P_{0}^{n}(x)) ≤ h_{i} + ϵ (by (84)).

Take N_{0} such that

Δ := {x ∈ B_{ϵ}: n_{0}(x) ≤ N_{0}}

has the positive measure. By Lemma 3.3 and Lemma 3.10, there exist c > 0 and Δ′ ⊂ Δ with m(Δ′) > 0 such that for x ∈ Δ′, there exists n = n(x) ≥ N_{0} such that

(5) \frac{m_{x}^{i+1}(L ∩ Δ)}{m_{x}^{i+1}(L)} ≥ c, where

L := B^{π}(x, e^{−n(λ_{i+1}−2ϵ)});

(6) \frac{log m_{x}^{i} (B^{π}(x, 2e^{−n(λ_{i+1}−2ϵ)}))}{−n(λ_{i+1} − 2ϵ)} > \overline{δ}_{i} − ϵ;

(7) \frac{log(1/c)}{n} < ϵ.

Take x ∈ Δ′ such that (1)–(7) are satisfied with n = n(x). Denote C = η_{i+1}(x) and C′ = η_{i}(x). Then by (5) and (1),

m_{x}^{i+1}(L ∩ Δ) ≥ cm_{x}^{i+1}(L) ≥ ce^{−n(λ_{i+1}−2ϵ)(\overline{δ}_{i+1}+ϵ)}.

But for each y ∈ L ∩ Δ, we have by (2), m_{y}^{i+1}(P_{0}^{n}(y)) ≤ e^{−n(h_{i+1}−ϵ)}. It follows that the number of distinct P_{0}^{n}-atoms intersecting C ∩ L ∩ Δ is larger than

m_{x}^{i+1}(L ∩ Δ)e^{n(h_{i+1}−ϵ)}.

However each such a P_{0}^{n}-atom, say P_{0}^{n}(y), intersects C′ ∩ L ∩ Δ, and this together with (3) guarantees that C′ ∩ P_{0}^{n}(y) is contained in C′ ∩ B^{π}(x, 2e^{−n(λ_{i+1}−2ϵ)}).

To see this, let z ∈ P_{0}^{n}(y) ∩ C′ ∩ L ∩ Δ. Since z ∈ Δ, we have d(πz, πx) ≤ e^{−n(λ_{i+1}−2ϵ)}. Thus

C′ ∩ P_{0}^{n}(y) = η_{i}(z) ∩ P_{0}^{n}(z) ⊂ B^{π}(x, e^{−n(λ_{i+1}−2ϵ)}) ⊂ B^{π}(x, 2e^{−n(λ_{i+1}−2ϵ)}).

Meanwhile by (4), m_{x}^{i}(P_{0}^{n}(y)) ≥ e^{−n(h_{i}+ϵ)} (for w ∈ P_{0}^{n}(y) ∩ C′ ∩ L, we have η_{i}(x) = η_{i}(w) and thus m_{x}^{i}(P_{0}^{n}(y)) = m_{w}^{i}(P_{0}^{n}(w))). Hence we have

m_{x}^{i}(B^{π}(x, 2e^{−n(λ_{i+1}−2ϵ)})) ≥ #{P_{0}^{n}-atoms intersecting C′ ∩ L ∩ Δ} · e^{−n(h_{i}+ϵ)}
≥ m_{x}^{i+1}(L ∩ Δ)e^{n(h_{i+1}−ϵ)}e^{−n(h_{i}+ϵ)}
≥ ce^{−n(λ_{i+1}−2ϵ)(\overline{δ}_{i+1}+ϵ)}e^{n(h_{i+1}−ϵ)}e^{−n(h_{i}+ϵ)}.

Comparing this with (6), we have

(λ_{i+1} − 2ϵ)(\overline{δ}_{i} − ϵ)

≤ (λ_{i+1} − 2ϵ)(\overline{δ}_{i+1} + ϵ)(λ_{i} − 2ϵ) + \frac{log(1/c)}{n} + h_{i} − h_{i+1} + 2ϵ

≤ (λ_{i+1} − 2ϵ)(\overline{δ}_{i+1} + ϵ)(λ_{i} − 2ϵ) + h_{i} − h_{i+1} + 3ϵ.

Taking ϵ → 0 yields h_{i} − h_{i+1} ≥ λ_{i+1}(\overline{δ}_{i} − \overline{δ}_{i+1}), which leads to a contradiction with (87). □

Proof of (C3). Here we give a proof by contradiction, adopting an idea from the proof of [40, Lemma 11.3.1]. Assume that (C3) is not true. Then there exists 0 ≤ i ≤ k − 1 such that \underline{δ}_{i+1}(x) + ϑ_{i}(x) > \underline{δ}_{i}(x) on a subset of Λ_{e} with positive measure. Hence there exists β > 0 and real numbers \underline{δ}_{i}, \underline{δ}_{i+1}, λ_{i} such that

(88)  \underline{δ}_{i+1} + ϑ_{i} > \underline{δ}_{i} + β,

and for any ϵ > 0, there exists A_{ϵ} ⊂ Λ_{e} with m(A_{ϵ}) > 0 so that for x ∈ A_{ϵ},

(89) |\underline{δ}_{i}(x) − \underline{δ}_{i}| < ϵ/2, |\underline{δ}_{i+1}(x) − \underline{δ}_{i+1}| < ϵ/2, |ϑ_{i}(x) − ϑ_{i}| < ϵ/2.

Let 0 < ϵ < β/4. Find N_{1} and a set A′_{ϵ} ⊂ A_{ϵ} with m(A′_{ϵ}) > 0 such that for x ∈ A′_{ϵ} and n > N_{1},

(90)  m_{x}^{i+1} (B^{π}(x, 2e^{−n})) ≤ e^{−n(\underline{δ}_{i+1}−ϵ)}.

By Lemma 3.3 and Lemma 3.10, we can find c > 0 and A′′_{ϵ} ⊂ A′_{ϵ} with m(A′′_{ϵ}) > 0 and N_{2} such that for all x ∈ A′′_{ϵ} and n ≥ N_{2},

\frac{m_{x}^{i}(A′_{ϵ} ∩ B^{π}(x, e^{−n}))}{m_{x}^{i}(B^{π}(x, e^{−n}))} > c.

For x ∈ A′′_{ϵ} and n ≥ N_{2}, we have

\begin{aligned}m_{x}^{i}(B^{π}(x, e^{−n})) & ≤ c^{−1}m_{x}^{i}(A′_{ϵ} ∩ B^{π}(x, e^{−n})) \\ & = c^{−1} ∫ m_{y}^{i+1}(A′_{ϵ} ∩ B^{π}(x, e^{−n})) dm_{x}^{i}(y) \\ & = c^{−1} ∫_{B^{Γ_{i+1}}(x,e^{−n})} m_{y}^{i+1}(A′_{ϵ} ∩ B^{π}(x, e^{−n})) dm_{x}^{i}(y). \end{aligned} ; (91)

Let y ∈ η_{i}(x) such that η_{i+1}(y) ∩ A′_{ϵ} ∩ B^{π}(x, e^{−n}) ≠ ∅. Then there exists w ∈ A′_{ϵ} ∩ B^{π}(x, e^{−n}) such that π_{i+1}y = π_{i+1}w. Hence A′_{ϵ} ∩ B^{π}(x, e^{−n}) ⊂ B^{π}(w, 2e^{−n}) and by (90)

m_{y}^{i+1}(A′_{ϵ} ∩ B^{π}(w, e^{−n})) = m_{w}^{i+1}(A′_{ϵ} ∩ B^{π}(w, e^{−n}))
≤ m_{w}^{i+1}(B^{π}(w, 2e^{−n}))
≤ e^{−n(\underline{δ}_{i+1}−ϵ)}.

Combining it with (91), we have

m_{x}^{i}(B^{π}(x, e^{−n})) ≤ c^{−1}e^{−n(\underline{δ}_{i+1}−σ)}m_{x}^{i}(B^{Γ_{i+1}}(x, e^{−n})) (x ∈ A′′_{ϵ}, n ≥ N_{2}).

Letting n → ∞, we obtain \underline{δ}_{i}(x) ≥ \underline{δ}_{i+1} − ϵ + ϑ_{i}(x) for x ∈ A′′_{ϵ}. Combining it with (89) yields

\underline{δ}_{i} ≥ \underline{δ}_{i+1} + ϑ_{i} − 4ϵ ≥ \underline{δ}_{i+1} + ϑ_{i} − β,

which contradicts (88). □

7.2. Proof of Theorem 2.12.

Definition 7.1. A real square matrix A is called asymptotically similar if all the (complex) eigenvalues of A are equal in modulus. Correspondingly, a linear transformation T on a finite-dimensional vector space V is called asymptotically similar if its representation matrix (associated with some basis of V) is asymptotically similar.

Lemma 7.2. Let (A_{1}, …, A_{ℓ}) be an ℓ-tuple of commuting linear transformations on ℝ^{d}. Then there are subspaces V_{1}, …, V_{k} of ℝ^{d} such that

(i) ℝ^{d} = V_{1} ⊕ ⋯ ⊕ V_{k}; (ii) V_{i} is A_{j}-invariant for 1 ≤ i ≤ k and 1 ≤ j ≤ ℓ; (iii) The restriction of A_{j} on V_{i} is asymptotically similar for 1 ≤ i ≤ k and

1 ≤ j ≤ ℓ.

Proof. For brevity, we only prove the lemma in the case ℓ = 2. The reader will see that the idea works for all cases.

Let S, T be two commuting linear transformations on ℝ^{d}. Let f denote the real minimal polynomial of S. Suppose f = f_{1}^{t_{1}} ⋯ f_{p}^{t_{p}} is the decomposition of f into powers of distinct, real irreducible monic factors f_{i}. Let W_{i} denote the null space of [f_{i}(S)]^{t_{i}}, i = 1, …, p. Then W_{i}’s are S-invariant and ℝ^{d} = W_{1} ⊕ ⋯ ⊕ W_{p} (cf. [62, Theorem 7.3]). Moreover S_{W_{i}}, the restriction of S on W_{i}, is asymptotically similar.

Since ST = T S, W_{i} is also T -invariant for each i. But T_{W_{i}} may be not asymptotically similar. However, as above, for each i, we can decomposed W_{i} into W_{i} = W_{i,1} ⊕ ⋯ ⊕ W_{i,u_{i}} such that W_{i,j} are the null spaces corresponding to some factors of the minimal polynomial of T_{W_{i}}. Again, W_{i,j} is T_{W_{i}}-invariant and S_{W_{i}}-invariant. Furthermore T_{W_{i,j}} and S_{W_{i,j}} are asymptotically similar. Hence ℝ^{d} = ⨁_{i,j} W_{i,j} is the desired decomposition for S and T. □

Proof of Theorem 2.12. Let {S_{i}}_{i=1}^{ℓ} be the IFS given in the theorem. By Lemma 7.2, there is a non-singular linear transformation Q on ℝ^{d} such that {QS_{i}Q^{−1}}_{i=1}^{ℓ} is the direct product of k asymptotically conformal IFS. Hence the desired result follows from Theorem 2.11. □

8. A variational principle about dimensions of self-conformal sets

In this section, we assume that K is the attractor of a C^{1} weakly conformal IFS Φ = {S_{i}}_{i=1}^{ℓ} on a compact set X ⊂ ℝ^{d}. The main result of this section is the following variational principle.

Theorem 8.1. Under the above setting, we have

(92) dim_{H} K = dim_{B} K
(93)  = sup {dim_{H} μ : μ = m ∘ π^{−1}, m ∈ M_{σ}(Σ), m is ergodic}
(94)  = max {dim_{H} μ : μ = m ∘ π^{−1}, m ∈ M_{σ}(Σ)}
(95)  = sup \{\frac{h_{π}(σ, m)}{∫ λ dm} : m ∈ M_{σ}(Σ)\} .

Proof. Without loss of generality we assume that \overline{dim}_{B}(K) > 0, where \overline{dim}_{B} denotes the upper box-counting dimension (cf. [13]). Let

0 < t_{3} < t_{2} < t_{1} < \overline{dim}_{B}(K).

We first prove that there is an ergodic measure m ∈ M_{σ}(Σ) such that dim_{H} m ∘ π^{−1} ≥ t_{3}. To achieve this, let α = \frac{t_{2}}{t_{3}} − 1 and let r_{0} be given as in Corollary 5.5. Since \overline{dim}_{B}(K) > t_{1}, for any 0 < ϵ < r_{0}, there exist r ∈ (0, ϵ) and integer N ≥ r^{−t_{1}} such that there are disjoint closed balls B(z_{i}, r) (i = 1, …, N) with centers z_{i} ∈ K. By Corollary 5.5, we can find words w_{i} ∈ Σ^{∗} (i = 1, …, N) such that S_{w_{i}}(K) ⊂ B(z_{i}, r) and

(96)  |S_{w_{i}}(x) − S_{w_{i}}(y)| ≥ r^{1+α}|x − y|  (x, y ∈ K).

This implies r^{1+α}diam(K) ≤ diam(S_{w_{i}}(K)) ≤ 2r. According to this fact and (59), there exist two positive constants A, B (independent of r) such that

B log(1/r) ≤ |w_{i}| ≤ A log(1/r) for all 1 ≤ i ≤ N.

Hence by the pigeon hole principle, there is a subset J of {1, …, N} with cardinality

#J ≥ \frac{N}{(A − B) log(1/r) + 1} ≥ \frac{r^{−t_{1}}}{(A − B) log(1/r) + 1} ≥ r^{−t_{2}}

such that the words w_{i} (i ∈ J) have the same length, say n.

Now we adopt an argument from the proof of [12, Theorem 4]. Let

δ = min{d(B(z_{i}, r), B(z_{j}, r)) : i, j ∈ J , i ≠ j}.

For any positive integer q and distinct sequences i_{1}, …, i_{q} and j_{1}, …, j_{q} taking values in J, let k be the least integer such that i_{k} ≠ j_{k}. Applying (96) (k − 1) times, we have

d(S_{w_{i_{1}}} ∘ ⋯ ∘ S_{w_{iq}} (K), S_{w_{j_{1}}} ∘ ⋯ ∘ S_{w_{jq}} (K))
≥ r^{(1+α)(k−1)}d(B(z_{i_{k}}, r), B(z_{j_{k}}, r)) ≥ r^{q(1+α)}δ.

Define a measure η on the class of finite unions of sets S_{w_{i_{1}}} ∘ ⋯ ∘ S_{w_{iq}} (K) by letting η(S_{w_{i_{1}}} ∘ ⋯ ∘ S_{w_{iq}} (K)) = (#J)^{−q}. This extends to a measure η on the σ-algebra generating by these sets. Let U be any subset of K with diam(U) < δ and let q be the least integer such that

r^{(q+1)(1+α)}δ ≤ diam(U) < r^{q(1+α)}δ.

Then U intersects at most one set S_{w_{i_{1}}} ∘ ⋯ ∘ S_{w_{iq}} (K), hence

η(U) ≤ (#J)^{−q} ≤ r^{t_{2}q} ≤ r^{−t_{2}}δ^{−t_{2}/(1+α)}diam(U)^{t_{2}/(1+α)}
= r^{−t_{2}}δ^{−t_{3}}diam(U)^{t_{3}}.

This implies dim_{H} η ≥ t_{3}.

We point out that the measure η constructed as above is, indeed, the projection of a σ^{n}-invariant and ergodic measure ν under π. Actually ν is the unique measure on Σ satisfying

ν([w_{i_{1}} … w_{i_{q}}]) = (#J)^{−q} (q ∈ ℕ, i_{1}, …, i_{q} ∈ J).

Applying Theorem 2.8 to the IFS {S_{w_{i}} : i ∈ J }, We have

dim_{H} η = dim_{H} ν ∘ π^{−1} = \frac{h_{π}(σ^{n}, ν)}{− ∫ log ∥S′_{x_{1}...x_{n}}(πσ^{n}x)∥dν} .

Take m = \frac{1}{n} ∑_{i=0}^{n−1} ν ∘σ^{−i}. Then m is σ-invariant and ergodic. Applying Theorem 2.8 and Proposition 4.3, we have

dim_{H} m ∘ π^{−1} = \frac{h_{π}(σ, m)}{− ∫ log ∥S′_{x_{1}}(πσx)∥dm} = \frac{h_{π}(σ^{n}, ν)}{− ∫ log ∥S′_{x_{1}...x_{n}}(πσ^{n}x)∥dν}
= dim_{H} η ≥ t_{3}.

Since t_{3} < \overline{dim}_{B}K is arbitrarily given, we obtain (92) and (93). To show (94), let (m_{i}) be a sequence of measures in M_{σ}(Σ) with

lim_{i→∞} dim_{H} m_{i} ∘ π^{−1} = dim_{H} K.

Take a sequence of positive numbers (a_{i}) such that ∑_{i=1}^{∞} a_{i} = 1. Then m = ∑_{i=1}^{∞} a_{i}m_{i} is an element in M_{σ}(Σ) with

dim_{H} m ∘ π^{−1} = sup_{i} dim_{H} m_{i} ∘ π^{−1} = dim_{H} K.

To show (95), according to (93), it suffices to show that

(97) dim_{H} m ∘ π^{−1} ≥ \frac{h_{π}(σ, m)}{− ∫ log ∥S′_{x_{1}}(πσx)∥ dm(x)}  (m ∈ M_{σ}(Σ)).

Fix m and let μ = m ∘ π^{−1}. Denote by Λ the righthand side of (97). By Theorem 2.8, d(μ, z) exists for μ-a.e.z ∈ ℝ^{d}. Hence to show (97), we only need to show that for any ϵ > 0, there is a Borel set E ⊂ ℝ^{d} such that μ(E) > 0 and d(μ, z) ≥ Λ − ϵ for z ∈ E. Assume this is false. Then d(μ, z) < Λ − ϵ for μ-a.e. z ∈ ℝ^{d}. Thus by Theorem 2.8 again, we have

h_{π}(σ, m, x) < λ(x)(Λ − ϵ) for m-a.e. x ∈ Σ.

Taking integration w.r.t. m on both sides yields

h_{π}(σ, m) < (Λ − ϵ) ∫ λ dm,

which leads to a contradiction. □

Remark 8.2. Assume that {S_{i}}_{i=1}^{ℓ} is a weakly conformal IFS which satisfies the AWSC (see Definition 2.14). Then the supremum in (93) and (95) can be attained by ergodic measures. To see this, by Proposition 4.20, the map m ↦ h_{π}(σ, m) is upper semi-continuous on M_{σ}(Σ), hence the supremum in (95) is attained at some member, say m_{0}, in M_{σ}(Σ). Let m_{0} = ∫ ν dℙ(ν) be the ergodic decomposition of m_{0}. By Theorem 2.2(ii), we have

dim_{H} K = \frac{h_{π}(σ, m_{0})}{∫ λ dm_{0}} = \frac{∫ h_{π}(σ, ν) dℙ(ν)}{∫ ∫ λ dν dℙ(ν)}.

Since \frac{h_{π}(σ,ν)}{∫ λ dν} ≤ dim_{H} K for each ν, the above equality implies that \frac{h_{π}(σ,ν)}{∫ λ dν} = dim_{H} K for ℙ-a.e. ν. Hence the supremum in (95) can be attained at some ergodic measure, so do the supremum in (93).

9. Proof of Theorem 2.15

We first present some lemmas.

Lemma 9.1. Let {S_{i}}_{i=1}^{ℓ} be an IFS with attractor K. For n ∈ ℕ, write Σ_{n} = {1, …, ℓ}^{n} and denote

N_{n} = #{S_{u} : u ∈ Σ_{n}}.

Then (i) sup\{h_{π}(σ, m) : m ∈ M_{σ}(Σ)\} ≤ \frac{log N_{n}}{n}. (ii) Let t_{n} = sup_{x∈ℝ^{d}} #{S_{u} : u ∈ Σ_{n}, x ∈ S_{u}(K)}. Then

sup\{h_{π}(σ, m) : m ∈ M_{σ}(Σ), m is ergodic\} ≥ \frac{log N_{n} − log t_{n}}{n} .

Proof. We first show (i). Let n ∈ ℕ and m ∈ M_{σ}(Σ). By the definition of N_{n}, we can construct a subset Ω of Σ_{n} with #Ω = N_{n} such that for any u ∈ Σ_{n}, there exists w ∈ Ω so that S_{u} = S_{w}. Hence there is a map g : Σ_{n} → Ω such that S_{u} = S_{g(u)} for each u ∈ Σ_{n}. Let (Ω^{ℕ}, T) denote the one-sided full shift over Ω. Define G : Σ → Ω^{ℕ} by

G((x_{i})_{i=0}^{∞}) = (w_{j})_{j=1}^{∞} ((x_{i})_{i=1}^{∞} ∈ Σ),

where w_{j} = g(x_{(j−1)n+1}x_{(j−1)n+2} ⋯ x_{jn}). Let π̃ : Ω^{ℕ} → ℝ^{d} denote the canonical projection w.r.t. the IFS {S_{u} : u ∈ Ω}. Then by Lemma 4.23(ii), we have

h_{π}(σ^{n}, m) = hπ̃(T, m ∘ G^{−1}) ≤ log(#Ω) = log N_{n}.

It follows that h_{π}(σ, m) ≤ log N_{n}/n. This proves (i).

To show (ii), let ν be the Bernoulli measure on Ω^{ℕ} with probability weight (1/N_{n}, …, 1/N_{n}). Then ν can be viewed as a σ^{n}-invariant measure on Σ. By Lemma 4.23(ii), we have h_{π}(σ^{n}, ν) = hπ̃(T, ν). Note that for x ∈ ℝ^{d}, there are at most t_{n} words u in Ω such that x ∈ S_{u}(K). By Corollary 4.22, we have

hπ̃(T, ν) ≥ h(T, ν) − log t_{n} = log N_{n} − log t_{n}.

Let μ = \frac{1}{n} ∑_{i=0}^{n−1} ν ∘ σ^{−i}. Then μ is σ-invariant and ergodic, furthermore

h_{π}(σ, μ) = \frac{1}{n}h_{π}(σ^{n}, ν) = \frac{1}{n}hπ̃(T, ν) ≥ (log N_{n} − log t_{n})/n,

as desired. □

Lemma 9.2. Let Φ = {S_{i}}_{i=1}^{ℓ} be an affine IFS on ℝ^{d} given by

S_{i}(x_{1}, …, x_{d}) = (ρ_{1}x_{1}, ⋯, ρ_{d}x_{d}) + (a_{i,1}, …, a_{i,d}),

where 1 > ρ_{1} > ρ_{2} > ⋯ > ρ_{d} > 0 and a_{i,j} ∈ ℝ. Let K denote the attractor of Φ, and write λ_{j} = log(1/ρ_{j}) for j = 1, …, d and λ_{d+1} = ∞. View Φ as the direct product of Φ_{1}, …, Φ_{d}, where Φ_{j} = {S_{i,j}(x_{j}) = ρ_{j}x_{j} + a_{i,j}}_{i=1}^{ℓ}. Let π_{j} denote the canonical projection w.r.t. the IFS Φ_{1} × ⋯ × Φ_{j}. Then we have (98)

∑_{j=1}^{d} (\frac{1}{λ_{j}} − \frac{1}{λ_{j+1}} ) H_{j} ≤ \underline{dim}_{B}(K) ≤ \overline{dim}_{B}(K) ≤ ∑_{j=1}^{d} (\frac{1}{λ_{j}} − \frac{1}{λ_{j+1}} ) H̃_{j},

with H_{j} = sup {h_{π_{j}}(σ, m) : m ∈ M_{σ}(Σ)} and

H̃_{j} = lim_{n→∞} \frac{log # \{S_{u}^{(j)} : u ∈ Σ_{n}\}}{n},

where {S_{i}^{(j)}}_{i=1}^{ℓ} is the IFS Φ_{1} × ⋯ × Φ_{j} on ℝ^{j}.

Proof. Without loss of generality we assume that

S_{i}([0, 1]^{d}) ⊂ [0, 1]^{d} (i = 1, …, ℓ).

For n ∈ ℕ, we write

N_{n}^{(j)} = #{S_{u}^{(j)} : u ∈ Σ_{n}} (j = 1, …, d),

and

q_{d}(n) = n, q_{j}(n) = [(\frac{log ρ_{d}}{log ρ_{j}} − \frac{log ρ_{d}}{log ρ_{j+1}} ) n] for 1 ≤ j ≤ d − 1,

where [x] denotes the integral part of x.

Construct Ω_{n,j} ⊂ Σ_{q_{j}(n)} (j = 1, …, d) such that #Ω_{n,j} = N_{q_{j}(n)}^{(j)} and for each

u ∈ Σ_{q_{j}(n)}, there is w ∈ Ω_{n,j} so that S_{u}^{(j)} = S_{w}^{(j)}. Then the family of following rectangles

(99)  ∏_{j=1}^{d} S_{w_{d}w_{d−1}···w_{j},j}([0, 1])  (w_{1} ∈ Ω_{n,1}, …, w_{d} ∈ Ω_{n,d})

is a cover of K. To see it, let u_{j} ∈ Σ_{q_{j}(n)} (j = 1, …, d). Then we can find

w_{j} ∈ Ω_{n,j} (j = 1, …, d) such that S_{u_{j}}^{(j)} = S_{w_{j}}^{(j)}. Hence

S_{u_{d}u_{d−1}...u_{1}}(K) ⊂ S_{u_{d}u_{d−1}...u_{1}}([0, 1]^{d}) ⊂ ∏_{j=1}^{d} S_{u_{d}u_{d−1}···u_{1},j}([0, 1])

⊂ ∏_{j=1}^{d} S_{u_{d}u_{d−1}···u_{j},j}([0, 1]) = ∏_{j=1}^{d} S_{w_{d}w_{d−1}···w_{j},j}([0, 1]).

It follows that the family of rectangles in (99) covers K. One can check that each rectangle in (99) is an almost (ρ_{d})^{n}-cube. Hence by the definition of box-counting dimension, we have

\overline{dim}_{B}K ≤ lim sup_{n→∞} \frac{∏_{j=1}^{d} #Ω_{n,j}}{− log(ρ_{d})^{n}} = lim sup_{n→∞} \frac{∏_{j=1}^{d} N_{q_{j}(n)}^{(j)}}{− log(ρ_{d})^{n}}

= ∑_{j=1}^{d} (\frac{1}{λ_{j}} − \frac{1}{λ_{j+1}} ) H̃_{j}.

This proves one part of (98).

To see the other part of (98), for j = 1, …, d, let Q_{j} denote the collection {[0, 1)^{j} + α : α ∈ ℤ^{j}}, and define

M_{n}^{(j)} = #{Q ∈ Q_{j} : diag(ρ_{1}^{n}, …, ρ_{j}^{n})Q ∩ K_{j} ≠ ∅},

where K_{j} denotes the attractor of Φ_{1} × ⋯ × Φ_{j}. Then by Proposition 4.18(ii),

we have H_{j} = lim_{n→∞} \frac{log M_{n}^{(j)}}{n}. We claim that for n ∈ ℕ, there exists a subset \overline{Ω}_{n,j} ⊂ Σ_{n} with cardinality ≥ 7^{−j}M_{n}^{(j)} such that

(100)  S_{w}^{(j)}([0, 1]^{j}) ∩ S_{w′}^{(j)}([0, 1]^{j}) = ∅ for all w, w′ ∈ \overline{Ω}_{n,j} with w ≠ w′.

To show the claim, we construct a finite subset of Q_{j}, denoted by W_{n}^{(j)}, such that (i) #W_{n}^{(j)} ≥ 7^{−j}M_{n}^{(j)}; (ii) diag(ρ_{1}^{n}, …, ρ_{j}^{n})Q ∩ K_{j} ≠ ∅ for each Q ∈ W_{n}^{(j)}; (iii)

2Q ∩ 2Q̃ = ∅ for Q, Q̃ ∈ W_{n}^{(j)} with Q ≠ Q̃, where 2Q := ⋃_{Q′∈Q_{j}: Q′∩Q≠∅} Q′.

For each Q ∈ W_{n}^{(j)}, since diag(ρ_{1}^{n}, …, ρ_{j}^{n})Q ∩ K_{j} ≠ ∅, we can pick a word

w(Q) ∈ Σ_{n} such that diag(ρ_{1}^{n}, …, ρ_{j}^{n})Q ∩ S_{w(Q)}^{(j)}K_{j} ≠ ∅ and hence

diag(ρ_{1}^{n}, …, ρ_{j}^{n})Q ∩ S_{w(Q)}^{(j)}([0, 1]^{j}) ≠ ∅.

Denote \overline{Ω}_{n,j} = \{w(Q) : Q ∈ W_{n}^{(j)}\}. The separation condition (iii) for the elements in W_{n}^{(j)} guarantees (100). This finishes the proof of the claim.

As above, we can construct \overline{Ω}_{n,j} well for each j = 1, …, d and n ∈ ℕ. Now fix n and consider the following collection of rectangles:

∏_{j=1}^{d} S_{w_{d}w_{d−1}···w_{j},j}([0, 1]) (w_{j} ∈ \overline{Ω}_{q_{j}(n),j}, 1 ≤ j ≤ d).

It is clear that the above rectangles are almost (ρ_{d})^{n}-cubes and each of them intersects with K. Furthermore they are disjoint due to (100). Hence by the definition of box-counting dimension, we have

\underline{dim}_{B}(K) ≥ lim inf_{n→∞} \frac{∏_{j=1}^{d} #\overline{Ω}_{q_{j}(n),j}}{− log(ρ_{d})^{n}} ≥ lim inf_{n→∞} \frac{∏_{j=1}^{d} 7^{−j}M_{q_{j}(n)}^{(j)}}{− log(ρ_{d})^{n}}

= ∑_{j=1}^{d} (\frac{1}{λ_{j}} − \frac{1}{λ_{j+1}} ) H_{j}.

This finishes the proof of (98). □

Proof of Theorem 2.15. We divide the proof into two steps: Step 1. Show the variational principle for dim_{H} K.

We first give an upper bound for dim_{H} K. Fix n ∈ ℕ. Define

N_{j} = #{S_{u}^{(j)} : u ∈ Σ_{n}} (j = 1, …, d),

where {S_{i}^{(j)}}_{i=1}^{ℓ} denotes the IFS Φ_{1} × ⋯ × Φ_{j}. Then we can construct

Ω_{j} ⊂ Σ_{n} (j = d, d − 1, …, 1)

such that #Ω_{j} = N_{j}, Σ_{n} ⊃ Ω_{d} ⊃ Ω_{d−1} ⊃ ⋯ ⊃ Ω_{1} and furthermore, for each u ∈ Σ_{n} and 1 ≤ j ≤ d, there is w_{j} ∈ Ω_{j} such that S_{u}^{(j)} = S_{w_{j}}^{(j)}. Hence there are natural maps θ_{d}, θ_{d−1}, …, θ_{1} with

Σ_{n} ⟶^{θ_{d}} Ω_{d} ⟶^{θ_{d−1}} Ω_{d−1} ⟶^{θ_{d−2}} ⋯ ⟶^{θ_{2}} Ω_{2} ⟶^{θ_{1}} Ω_{1}

such that S_{u}^{(j)} = S_{θ_{j}(u)}^{(j)} for any 1 ≤ j ≤ d and u ∈ Ω_{j+1}, with convention Ω_{d+1} = Σ_{n}.

Let Z_{d} : Ω_{d} → ℝ be the indicator of Ω_{d}, i.e., Z_{d}(u) = 1 for all u ∈ Ω_{d}. Define

Z_{d−1}(w) = ∑_{u∈θ_{d−1}^{−1}(w)} Z_{d}(u) (w ∈ Ω_{d−1}).

Define inductively

Z_{j}(w) = ∑_{u∈θ_{j}^{−1}(w)} Z_{j+1}(u)^{\frac{log ρ_{j+1}}{log ρ_{j+2}}} (w ∈ Ω_{j}, j = d − 2, …, 1).

In particular, define

Z_{0} = ∑_{u∈Ω_{1}} Z_{1}(u)^{\frac{log ρ_{1}}{log ρ_{2}}} .

Using the technique by Kenyon & Peres [33] (which is an extension of McMullen [44]), we have

(101)  dim_{H} K ≤ \frac{log Z_{0}}{−n log ρ_{1}} .

More precisely, define a probability vector (p(u))_{u∈Ω_{d}} by

p(u) = \frac{Z_{d}(u)}{Z_{d−1}(θ_{d−1}(u))} · ∏_{j=1}^{d−1} \frac{Z_{j}(θ_{j}θ_{j+1} ⋯ θ_{d−1}(u))^{\frac{log ρ_{j}}{log ρ_{j+1}}}}{Z_{j−1}(θ_{j−1}θ_{j} ⋯ θ_{d−1}(u))}

with convention Z_{0}(θ_{0} … θ_{d−1}(u)) = Z_{0} for any u ∈ Ω_{d}. Let ν be the product measure on (Ω_{d})^{ℕ} by assigning probability p(u) to each digit u ∈ Ω_{d}. The measure ν can be viewed as a measure on Σ, which is σ^{n}-invariant and ergodic. Let μ = ν ∘ π^{−1}. Then

(102)  lim inf_{r→0} \frac{log μ(B(πx, r))}{log r} ≤ \frac{log Z_{0}}{−n log ρ_{1}}  (x ∈ Σ).

A detailed proof of (102) was given by Shmerkin (see the proof of (4.3) in [60]) for the case d = 2, whilst a slight modification of the proof of [33, Theorem 1.2] provides a proof of (102) for d ≥ 2. Then (101) follows from (102) and Billingsley’s lemma.

Now we want to indicate certain connection between the upper bound \frac{log Z_{0}}{−n log ρ_{1}} and the projection entropies. First we define the projections θ_{j}^{∗} : Ω_{j+1}^{ℕ} → Ω_{j}^{ℕ} (j = d − 1, …, 1) by

θ_{j}^{∗} ((u_{k})_{k=1}^{∞}) = (θ_{j}(u_{k}))_{k=1}^{∞} ((u_{k})_{k=1}^{∞} ∈ Ω_{j+1}^{ℕ}).

Then it is easy to see that for each 1 ≤ j ≤ d − 1, the measure

ν_{j} := ν ∘ (θ_{j}^{∗} ∘ θ_{j+1}^{∗} ∘ ⋯ ∘ θ_{d−1}^{∗})^{−1}

is a product measure on Ω_{j}^{ℕ}. Let T_{j} denote the left shift operator on Ω_{j}^{ℕ}. By a direct calculation, we have

\frac{log Z_{0}}{−n log ρ_{1}} = ∑_{j=1}^{d} (\frac{1}{λ_{j}} − \frac{1}{λ_{j+1}} ) \frac{h(T_{j}, ν_{j})}{n}.

Thus we have

(103)  dim_{H} K ≤ ∑_{j=1}^{d} (\frac{1}{λ_{j}} − \frac{1}{λ_{j+1}} ) \frac{h(T_{j}, ν_{j})}{n}.

Let π̃_{j} (j = 1, …, d) denote the canonical projection from Ω_{j}^{ℕ} to ℝ^{j} w.r.t. the

IFS {S_{u}^{(j)}}_{u∈Ω_{j}} ( remember that π_{j} denotes the canonical projection from Σ to ℝ^{d} w.r.t. {S_{u}^{(j)} : u ∈ Σ_{n}}). According to Lemma 4.23(ii), we have

(104)  hπ̃_{j}(T_{j}, ν_{j}) = h_{π_{j}}(σ^{n}, ν)  (j = 1, …, d).

Since Φ_{1} × ⋯ × Φ_{j} (j = 1, …, d) satisfy the AWSC, there is a sequence (t_{n}) of positive integers with lim_{n} log t_{n}/n = 0, such that

(105)  sup_{x∈ℝ^{j}} #{S_{u}^{(j)} : u ∈ Ω_{j}, x ∈ S_{u}^{(j)}(K_{j})} ≤ t_{n}  (j = 1, …, d),

where K_{j} denotes the attractor of Φ_{1} × ⋯ × Φ_{j}. By Corollary 4.22, we have

hπ̃_{j}(T_{j}, ν_{j}) ≥ h(T_{j}, ν_{j}) − log t_{n} ≥ h(T_{j}, ν_{j}) − log t_{n}.

It together with (104) yields h_{π_{j}}(σ^{n}, ν) ≥ h(T_{j}, ν_{j}) − log t_{n}. Now applying Theorem 2.11 to the IFS {S_{u} : u ∈ Σ_{n}}, we have

dim_{H} ν ∘ π^{−1} = \frac{1}{n} ∑_{j=1}^{d} (\frac{1}{λ_{j}} − \frac{1}{λ_{j+1}} ) h_{π_{j}}(σ^{n}, ν)

≥ \frac{1}{n} ∑_{j=1}^{d} (\frac{1}{λ_{j}} − \frac{1}{λ_{j+1}} ) (h(T_{j}, ν_{j}) − log t_{n})

≥ dim_{H} K − \frac{log t_{n}}{n} · ∑_{j=1}^{d} (\frac{1}{λ_{j}} − \frac{1}{λ_{j+1}} )  (by (103)).

Let m = \frac{1}{n} ∑_{i=1}^{n} ν ∘ σ^{−i}. Then m is ergodic and dim_{H} m ∘ π^{−1} = dim_{H} ν ∘ π^{−1}. Letting n tend to ∞, we obtain

(106)  sup{dim_{H} m ∘ π^{−1} : m ∈ M_{σ}(Σ), m is ergodic } ≥ dim_{H} K.

It is clear the “≥” in above inequality can be replaced by “=” since m ∘ π^{−1} is supported on K. Note that h_{π_{j}}(σ, ·) (j = 1, …, d) are upper semi-continuous on M_{σ}(Σ) (see Proposition 4.20 and (105)). By Theorem 2.2(ii) and Theorem 2.11, we see that the supremum in (106) is attained at some ergodic element in M_{σ}(Σ). This finishes the proof of the variational principle for dim_{H} K.

Step 2. Show the variational principle for dim_{B} K.

By Lemma 9.2, we only need to show that under the assumption of Theorem 2.15,

(107)  H_{j} ≥ H̃_{j}  (j = 1, …, d),

where

H_{j} = sup\{h_{π_{j}}(σ, m) : m ∈ M_{σ}(Σ)\}, H̃_{j} = lim_{n→∞} \frac{log #\{S_{u}^{(j)} : u ∈ Σ_{n}\}}{n}.

To see (107), by (105) and Lemma 9.1, we have

H_{j} ≥ \frac{log # \{S_{u}^{(j)} : u ∈ Σ_{n}\} − log t_{n}}{n} (n ∈ ℕ).

Letting n → ∞, we obtain (107) by the assumption log t_{n}/n → 0. This finishes the proof of the theorem. □

Remark 9.3. With an essentially identical proof, Theorem 2.15 can be extended to the following class of IFS Φ = Φ_{1} × ⋯ × Φ_{k} on ℝ^{q_{1}} × ⋯ × ℝ^{q_{k}}, where Φ_{j} has the form {A_{j}z_{j} + c_{i,j}}_{i=1}^{ℓ} such that A_{j} is the inverse of an integral matrix and all the eigenvalues of A_{j} equals ρ_{j} in modulus, ρ_{1} > ⋯ > ρ_{k}, c_{i,j} ∈ ℚ^{q_{j}}.

This together with Lemma 7.2 and the proof of Theorem 2.12 yields

Theorem 9.4. Let Φ = {S_{i}}_{i=1}^{ℓ} be an IFS on ℝ^{d} of the form

S_{i}(x) = Ax + c_{i} (i = 1, …, ℓ),

where A is the inverse of an integral expanding d × d matrix, c_{i} ∈ ℤ^{d}. Let K be the attractor of the IFS. Then there is an ergodic measure on K of full Hausdorff dimension.

10. A final remark about infinite non-contractive IFS

In the previous sections, we have made the restriction that an IFS consists of finitely many contractive maps. We remark that part of our results can be extended to certain infinite non-contractive IFS.

Let Φ = {S_{i}}_{i=1}^{∞} be a family of maps on ℝ^{d} of the form

S_{i}(x) = ρ_{i}R_{i}(x) + a_{i} (i = 1, 2, …),

where ρ_{i} > 0, R_{i} are orthogonal d × d matrices, a_{i} ∈ ℝ^{d}.

Let (X, σ) be the left shift over the alphabet {i : i ∈ ℕ}, and let m be an ergodic measure on X satisfying H_{m}(P_{∞}) < ∞, where P_{∞} denotes the partition of X given by

P_{∞} = {[i] : i ∈ ℕ},

where [i] = {(x_{i})_{i=1}^{∞} ∈ X : x_{1} = i}. Assume that Φ is m-contractive in the sense that

(108)  ∑_{i=1}^{∞}(log ρ_{i})m([i]) < 0,  ∑_{i=1}^{∞}(log |a_{i}|)m([i]) < ∞.

Denote

λ = − ∑_{i=1}^{∞}(log ρ_{i})m([i]).

Let X′ denote the set of points x = (x_{i})_{i=1}^{∞} ∈ X such that

lim_{n→∞}(1/n) log(ρ_{x_{1}}ρ_{x_{2}} … ρ_{x_{n}}) = −λ, lim_{n→∞}(1/n) log |a_{x_{n}}| = 0.

Then X′ satisfies σ^{−1}(X′) = X′. Furthermore by Birkhoff’s ergodic theorem,

m(X′) = 1.

Define the projection map π : X′ → ℝ^{d} by

π(x) = lim_{n→∞} S_{x_{1}} ∘ S_{x_{2}} ∘ ⋯ ∘ S_{x_{n}}(0) (x ∈ X′).

It is easily checked that π is well defined. Let μ = m ∘ π^{−1} be the projection of m under π. We have the following theorem

Theorem 10.1. Under the above setting, μ = m ∘ π^{−1} is exactly dimensional and

dim_{H} μ = \frac{h_{π}(σ, m)}{λ},

where H_{π}(σ, m) = H_{m}(P_{∞}|σ^{−1}π^{−1}γ) − H_{m}(P_{∞}|π^{−1}γ), γ = B(ℝ^{d}).

We remark that when m is a Bernoulli product measure, μ = m ∘ π^{−1} is the stationary measure of certain affine random walk determined by Φ and m, and the decay property of μ at infinity has been extensively studied in the literature (cf.

[24] and references therein).

The proof of Theorem 10.1 is essentially identical to that given in Section 6. Indeed we only need to replace Σ in Section 6 by X′, and replace ‘let c > 1 so that c sup_{x∈Σ} \overline{ρ}(x) < 1’ in the proof of Theorem 6.2 by ‘let 1 < c < e^{λ}’.

Acknowledgement.The authors are grateful to François Ledrappier for his encouragement and helpful comments. They are indebted to Eric Olivier for stimulating discussions of the variational principle about the Hausdorff dimension of self-affine sets, and to Wen Huang for the discussions of the entropy theory. They also thank Quansheng Liu, Emile Le Page and Yong-Luo Cao for valuable comments, and Guo-Hua Zhang for critical reading of the manuscript. The first author was partially supported by the RGC grant in CUHK, Fok Ying Tong Education Foundation and NSFC (Grant 10571100). The second author was partially supported by NSF under grants DMS-0240097 and DMS-0503870.

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