Some Ergodic Properties of Commuting Diffeomorphisms

Huyi Hu

Department of Mathematics, University of Arizona, Tucson, AZ 85721

Abstract. For a smooth ZZ^{2}−action on a C^{∞} compact Riemannian manifold M, we discuss its ergodic properties which include the decomposition of the tangent space of M into subspaces related to Lyapunov exponents, the existence of Lyapunov charts, and the subaddtivity of entropies.

§0 Introduction

In this paper we discuss some ergodic properties of commuting diffeomorphisms on a C^{∞} compact Riemannian manifold concerning Lyapunov exponents and entropies. Let M be a compact C^{∞} Riemannian manifold without boundary, f, g ∈ Diff^{2}(M) with fg = gf, where fg denote the composition of f and g. In fact f and g generate a smooth ZZ^{2}−action on M. We will give a decomposition of the tangent space of M into subspaces related to the Lyapunov exponents of both actions f and g, and construct a family of Lyapunov charts. We will show that for almost every x in M, if f and g have same unstable subspace, then they have same unstable manifold at x. We will investigate the subaddtivity of entropies of commuting diffeomorphisms, i.e. the entropy of the composition fg is less than or equal to the sum of the entropies of f and g. In the circumstances for measure-theoretic entropies the subaddtivity always holds whenever the measure is invariant under the actions f and g, and becomes additive if the unstable subspace of one map does not intersect with the stable subspace of another map at almost every point, but for topological entropies additional condition is needed to obtain the subaddtivity.

We denote by M(M, f) the set of f−invariant Borel probability measures on M. It is known by many authors (for example, see Proposition 1.2 for the proof,) that f and g have common invariant measures, i.e. M(M, f) ∩ M(M, g) ≠ ∅. We let M(M, f, g) = M(M, f) ∩ M(M, g).

Throughout the paper, we always assume that M is a compact C^{∞} Riemannian manifold without boundary, f and g are C^{2} diffeomorphism on M with fg = gf, μ is an f− and g−invariant Borel probability measure on M, i.e. μ ∈ M(M, f, g).

Let T_{x}M be the tangent space of M at x ∈ M. The diffeomorphism f induces a map Df_{x} : T_{x}M → T_{fx}M. It is well known (see [O]) that there exists a measurable set Γ_{f} with νΓ_{f} = 1, ∀ ν ∈ M(M, f), such that for all x ∈ Γ_{f}, u ∈ T_{x}M, the limit

χ(x, u, f) = lim_{n→∞} \frac{1}{n} log ∥Df_{x}^{n}u∥

exists and is called Lyapunov exponent of u at x. Let λ_{1}(x, f) > ⋯ > λ_{r(x,f)}(x, f) denote all Lyapunov exponents of f at x with multiplicities m_{1}(x, f), ⋯, m_{r(x,f)}(x, f) respectively,

and T_{x}M = ⨁_{i=1}^{r(x,f)} E_{i}(x, f) be the corresponding decomposition of tangent space at x ∈ M.

Similarly, for diffeomorphism g we have Lyapunov exponents λ_{1}(x, g) > ⋯ > λ_{r(x,g)}(x, g) with multiplicities m_{1}(x, g), ⋯, m_{r(x,g)}(x, g) respectively, and the corresponding decomposi-

tion T_{x}M = ⨁_{j=1}^{r(x,g)} E_{j}(x, g).

Suppose f and g are commuting diffeomorphisms. The spectrum {λ_{i}(x, f), m_{i}(x, f)} of f is f−invariant. We will show that it is also g−invariant, i.e. ∀u ∈ E_{i}(x, f),

χ(gx, Dg_{x}u, f) = χ(x, u, f) = λ_{i}(x, f).

and therefore Dg_{x}E_{i}(x, f) = E_{i}(gx, f). Thus we can redecompose each E_{i}(x, f) according to diffeomorphism g and get the following.

Theorem A. let M be a C^{∞} compact Riemannian manifold without boundary, f, g ∈ Diff^{2}(M) with fg = gf. Then there exists a measurable set Γ with f^{s}g^{t}Γ = Γ, ∀ s, t ∈ ZZ and μΓ = 1, ∀ μ ∈ M(M, f, g), satisfying that for all x ∈ Γ, there is a decomposition of the tangent space into

T_{x}M = ⨁_{i=1}^{r(x,f)} ⨁_{j=1}^{r(x,g)} E_{ij}(x),

such that ∀ s, t ∈ ZZ, if E_{ij}(x) ≠ {0}, then ∀ 0 ≠ u ∈ E_{ij}(x),

lim_{n→∞} \frac{1}{n} log ∥D(f^{s}g^{t})_{x}^{n}u∥ = sλ_{i}(x, f) + tλ_{j}(x, g),

and if (i_{1}, j_{1}) ≠ (i_{2}, j_{2}), E_{i_{1}j_{1}}(x) ≠ {0}, E_{i_{2}j_{2}}(x) ≠ {0}, then

lim_{n→∞} \frac{1}{n} log |sin(E_{i_{1}j_{1}}((f^{s}g^{t})^{n}x), E_{i_{2}j_{2}}((f^{s}g^{t})^{n}x))| = 0.

Moreover, ∀ s, t ∈ ZZ, D(f^{s}g^{t})_{x}(E_{ij}(x)) = E_{ij}(f^{s}g^{t}x), λ_{i}(f^{s}g^{t}x, f) = λ_{i}(x, f), λ_{j}(f^{s}g^{t}x, g) = λ_{j}(x, g).

Probably, this result is known. However, because of its importance for our discussion, we state it here and give the proof in §2 for completeness. In particular, if we take s = 1 and t = 0, then we have

lim_{n→∞} \frac{1}{n} log ∥Df_{x}^{n}u∥ = λ_{i}(x, f), ∀ 0 ≠ u ∈ E_{ij}(x),

lim_{n→∞} \frac{1}{n} log |sin(E_{i_{1}j_{1}}(f^{n}x), E_{i_{2}j_{2}}(f^{n}x))| = 0.

and Df_{x}E_{ij}(x) = E_{ij}(fx), λ_{i}(fx, f) = λ_{i}(x, f), and λ_{j}(fx, g) = λ_{j}(x, g). Symmetrically, we have similar results concerning diffeomorphism g, if we take s = 0 and t = 1. The explicit statement is given in Proposition 2.8.

By the definition of Lyapunov exponents, given γ > 0, ∀ n, k ∈ ZZ, ∥Df_{x}^{n}u∥e^{−nλ_{i}(x,f)} and ∥Dg_{x}^{k}u∥e^{−kλ_{j}(x,g)}, u ∈ E_{ij}(x), are dominated by C(x)^{±1}e^{±nγ}∥u∥ and C(x)^{±1}e^{±kγ}∥u∥, respectively. We will show in §3 that C(x) can be chosen such that C(f^{±1}x), C(g^{±1}x) ≤ C(x)e^{γ}. This is a generalization of Pesin’s theory ([P]) to commuting diffeomorphisms.

Based on the facts we can construct a family of Lyapunov charts, on which the maps f̃ and g̃, induced by f and g respectively, act approximately to the linear maps with eigenvalues e^{λ_{i}(x,f)} and e^{λ_{j}(x,g)} respectively. The result is stated in Proposition 4.1.

Let E^{s}(x, f) = ⨁_{λ_{i}(x,f)<0} E_{i}(x, f), E^{u}(x, f) = ⨁_{λ_{i}(x,f)>0} E_{i}(x, f), and if λ_{i}(x, f) = 0 for

some i then let E^{c}(x, f) = E_{i}(x, f). Also, let E^{sc}(x, f) = E^{s}(x, f) ⊕ E^{c}(x, f), E^{uc}(x, f) = E^{u}(x, f) ⊕ E^{c}(x, f). The unstable manifold for diffeomorphism f, say, is defined by

w^{u}(x, f) = \{y ∈ M : lim sup_{n→∞} \frac{1}{n} log d(f^{−n}x, f^{−n}y) < 0\},

which is f−invariant. It is easy to see by the definition that it is also g− invariant. From Theorem A we know that E^{u}(x, f) = E^{u}(x, g) or E^{sc}(x, f) = E^{sc}(x, g) is equivalent to E^{u}(x, f) ∩ E^{sc}(x, g) = {0} = E^{u}(x, g) ∩ E^{sc}(x, f). If it holds at some x ∈ Γ, we will prove in §5 that the unstable manifolds w^{u}(x, f) and w^{u}(x, g) coincide at x.

The next topic in this paper is concerning the relationship among the entropies of f, g and fg. We will prove that if f and g are C^{2} diffeomorphisms on a smooth compact manifold preserving a Borel probability measure μ, then h_{μ}(fg) ≤ h_{μ}(f) + h_{μ}(g). It is not true for general measure preserving transformations on a probability space. There are some unpublished counterexamples, for instance, due to D. S. Ornstein and B. Weiss, and due to J.-P. Thouvenot, of two commuting measure preserving automorphisms S, T of probability space (X, B, ν) with h_{ν}(S) = h_{ν}(T) = 0, but such that h_{ν}(ST) > 0. However, in the smooth dynamical systems the subadditivity of measure-theoretic entropies holds.

From Ledrappier and Young’s formula relating entropies, exponents and dimensions, we know that the entropy of a diffeomorphism on a smooth manifold is determined by the behavior of the map on its unstable manifold. If two commuting diffeomorphisms have same family of unstable manifolds, i.e. w^{u}(x, f) = w^{u}(x, g), μ−a.e., then we can construct an increasing partition subordinate to the unstable manifolds (see §6 ), and use it to get equality in above subadditivity formula. We combine the two results in the following theorem which we will prove in §7 and §9.

Theorem B. Let M be a C^{∞} compact Riemannian manifold without boundary, f, g ∈ Diff^{2}(M), and fg = gf. Then

h_{μ}(fg) ≤ h_{μ}(f) + h_{μ}(g), ∀μ ∈ M(M, f, g),

where h_{μ}(·) denotes the measure-theoretic entropy. Moreover, if E^{u}(x, f) ∩ E^{s}(x, g) = {0} and E^{s}(x, f) ∩ E^{u}(x, g) = {0}, μ − a.e, then the equality holds.

For topological entropies the the answer to the question whether h(fg) ≤ h(f) + h(g) is also negative in general. L. Wayne Goodwyn has a counterexample for the case, i.e. there exists a compact metric space X and two homeomorphisms S and T with ST = T S such that h(S) = h(T) = 0 and h(ST) > 0(see [G]). Since topological entropy is the supremum of measure-theoretic entropy, from Theorem B we can prove the formula if some additional hypotheses are given on diffeomorphism fg.

Theorem C. Let M,f,g is same as in Theorem B. If for fg, one of the following conditions holds:

i) fg ∈ Diff^{∞}(M), or ii) fg is expensive, or iii) fg has finite number of ergodic measures with maximal entropy,

then

h(fg) ≤ h(f) + h(g).

This theorem will be proved in §10. In the section we will also present a counterexample to show that if f and g are homeomorphisms on a compact smooth manifold, then the result of Theorem C fails. We don’t know whether the result still holds if the additional hypotheses on fg are removed.

§1. Ergodicity

In this section we will give the definition of ergodicity and discuss the properties for two commuting continuous maps on a compact metric space.

For a map T from a set X to itself, we denote by Fix(T) the set of fixed points of T.

Proposition 1.1. If T and S are commuting maps on a set X, then S(Fix(T)) ⊂ Fix(T). Proof. Take x ∈ Fix(T). Since T (Sx) = S(T x) = Sx, we have Sx ∈ Fix(T).

Let M(X) be the set of Borel probability measures on a compact metric space X, and T be a map on X. T induces a map T^{∗} on M(X) by putting T^{∗}μ = μ ∘ T^{−1}, ∀μ ∈ M(X). Thus Fix(T^{∗}) = M(X, T).

Proposition 1.2. If T and S are commuting continuous maps on a compact metric space X, then M(X, T) ∩ M(X, S) ≠ ∅.

Proof. By Proposition 1.1, S^{∗}(M(X, T)) ⊂ M(X, T). Since M(X, T) is a nonempty compact convex set in weak ^{∗} topology and S^{∗} is continuous, we know that S^{∗} has a fixed point in M(X, T).

We write M(X, T, S) = M(X, T) ∩ M(X, S). Since both M(X, T) and M(X, S) are convex sets, M(X, T, S) is convex.

Definition. Suppose T and S are continuous maps on a compact metric space X with T S = ST. A measure μ ∈ M(X, T, S) is said to be (T, S)−ergodic if for any measurable set B with μ(T^{−1}B△B) = 0 = μ(S^{−1}B△B), μ(B) = 0 or μ(B) = 1.

(T, S)−ergodicity shares some properties with those of single transformation. For example, we give the following propositions whose proof is parallel to the case of one transformation (See [W], Chapter 1 and 6).

Proposition 1.3. μ is (T, S)−ergodic iff any measurable function ϕ on X with ϕ(T x) = ϕ(x) = ϕ(Sx), μ − a.e. is constant μ − a.e.

Proof. ”⇒” is based on the fact that for such function ϕ, the set {x : ϕ(x) > C}, C ∈ IR, is invariant under the actions T and S. ”⇐” holds because the characteristic function χ_{B}, where B is a set with μ(T^{−1}B△B) = 0 = μ(S^{−1}B△B), satisfies χ_{B}(T x) = χ_{B}(x) = χ_{B}(Sx), μ − a.e. and equals to 1 or 0 almost everywhere.

Proposition 1.4. μ is (T, S)−ergodic iff μ is an extreme point of M(X, T, S).

Proof. Suppose μ is not (T, S)−ergodic. We can find a measurable set E with T E = E = SE and 0 < μ(E) < 1. Let

μ_{1}(·) = \frac{μ(· ∩ E)}{μ(E)}, μ_{2}(·) = \frac{μ(· ∩ X\E)}{μ(X\E)}.

Then μ = pμ_{1} + (1 − p)μ_{2}, where p = μ_{1}(E). So μ can be expressed as a convex combination of μ_{1}, μ_{2} ∈ M(X, T, S) and is not an extreme point of M(X, T, S).

Suppose μ is (T, S)−ergodic, and μ = pμ_{1} + (1 − p)μ_{2} for some μ_{1}, μ_{2} ∈ M(X, T, S) and p ∈ (0, 1). Then μ_{1} is absolutely continuous with respect to μ, and the Radon-Nikodym derivative ϕ = dμ_{1}/dμ satisfies that ϕ(T x) = ϕ(x) = ϕ(Sx), μ − a.e. So it must follow that ϕ(x) = 1, μ − a.e, because ∫_{X} ϕdμ = 1. Thus, μ = μ_{1}. Similarly, μ = μ_{2}. Hence μ is an extreme point of M(X, T, S).

Proposition 1.4 shows the existence of (T, S)−ergodic measure and gives rise to the possibility of (T, S)−ergodic decomposition.

Notice if T and S commute, then so do T and T S.

Proposition 1.5. Suppose T and S are homeomorphisms on X.

i) M(X, T, T S) = M(X, T, S).

ii) μ is (T, S)−ergodic iff μ is (T, T S)− ergodic.

Proof. i) is clear. ii) follows from i) and Proposition 1.4.

We denote by E(X, T) the set of ergodic measures under action T. Then by our notation, M(E(X, T)) is the set of Borel probability measures on E(X, T). It is known that for any ν ∈ M(X, T), there exists a unique element π ∈ M(E(X, T)), such that ν has the ergodic decomposition ([W],Chapter 6)

ν = ∫_{E(X,T)} ν_{e}dπ(ν_{e}),  (Δ)

It means that ∀ ϕ ∈ C(X),

∫_{X} ϕ(x)dν(X) = ∫_{E(X,T)}(∫_{X} ϕ(x)dν_{e}(x))dπ(ν_{e}).  (ΔΔ)

In fact, given any π ∈ M(E(X, T)), this formula can determine a unique T −invariant measure ν on X as well. Thus we obtain a 1−1 map τ : M(X, T) → M(E(X, T)) defined by τ(ν) = π.

Now we give following remarks which may be helpful for understanding (T, S)−ergodic measures. Here we need assume that both T and S are homeomorphisms on a compact metric space X.

Remark 1.6. S^{∗} induces a map S^{∗∗} = (S^{∗})^{∗} on the set M(E(X, T)) by S^{∗∗}(ν) = ν ∘ (S^{∗})^{−1}.

Proof. S^{∗} is invertible. By Proposition 1.1, S^{∗}(M(X, T)) = M(X, T). Since S^{∗} is affine and E(X, T) is the set of extreme points of M(X, T), S^{∗}(E(X, T)) = E(X, T). Then the result follows.

Remark 1.7.

i) μ ∈ M(X, T) is S−invariant iff τ(μ) is S^{∗}−invariant. In other words, μ ∈ Fix(S^{∗}) iff τ(μ) ∈ Fix(S^{∗∗}).

ii) μ is (T, S)−ergodic iff τ(μ) is ergodic with respect to S^{∗}. Proof. i) Denote π = τ(μ). Since S^{∗} is affine and maps E(X, T) to itself, we have that

S^{∗}μ = ∫_{E(X,T)} S^{∗}μ_{e}dπ(μ_{e}) = ∫_{E(X,T)} μ_{e}dπ((S^{∗})^{−1}μ_{e}) = ∫_{E(X,T)} μ_{e}d(S^{∗∗}π)(μ_{e}).

Comparing it with (Δ) we know that S^{∗}μ = μ iff S^{∗∗}π = π.

ii) Notice that i) means τ(M(X, T, S)) = M(E(X, T), S^{∗}). Since τ is 1 − 1 and affine, the extreme points of the two sets are corresponding under the action τ.

§2. Lyapunov Exponents

From now on we back our discussion on the smooth dynamical systems. The suppositions on M, f, g and μ are as before. Recall that Γ_{f} is a subset of M such that fΓ_{f} = Γ_{f}, νΓ_{f} = 1, ∀ ν ∈ M(M, f), and for any x ∈ Γ_{f},

χ(x, u, f) = lim_{n→∞} \frac{1}{n} log ∥ Df_{x}^{n}u ∥, ∀u ∈ T_{x}M.

Lemma 2.1. χ(gx, Dg_{x}u, f) = χ(x, u, f). Proof. There exists C > 0 such that ∀x ∈ M, u ∈ T_{x}M,

C^{−1} ∥ u ∥≤∥ Dg_{x}u ∥≤ C ∥ u ∥ .

Thus

C^{−1} ∥ Df_{x}^{n}u ∥≤∥ Dg_{f^{n}x}Df_{x}^{n}u ∥≤ C ∥ Df_{x}^{n}u ∥ .

So

lim_{n→∞} \frac{1}{n} log ∥ Df_{gx}^{n}Dg_{x}u ∥= lim_{n→∞} \frac{1}{n} log ∥ Dg_{f^{n}x}Df_{x}^{n}u ∥= lim_{n→∞} \frac{1}{n} log ∥ Df_{x}^{n}u ∥ .

We know that the set Γ_{f} and the spectrum {λ_{i}(x, f), m_{i}(x, f)} are f−invariant, and Df_{x}(E_{i}(x, f)) = E_{i}(fx, f), i = 1, ⋯, r(x, f). From above lemma we have the following.

Corollary 2.2.

i) The set Γ_{f} is g−invariant, i.e. gΓ_{f} = Γ_{f}. ii) The spectrum {λ_{i}(x, f), m_{i}(x, f), i = 1, ⋯, r(x, f)} is g−invariant. iii) Dg_{x}(E_{i}(x, f)) = E_{i}(gx, f), i = 1, ⋯, r(x, f).

Next proposition is a special case of Theorem A.

Proposition 2.3. There exists a measurable set Γ_{1} with fΓ_{1} = Γ_{1} = gΓ_{1} and μΓ_{1} = 1, ∀ μ ∈ M(M, f, g), satisfying that for all x ∈ Γ_{1}, there is a decomposition of tangent space into

T_{x}M = ⨁_{i=1}^{r(x,f)} ⨁_{j=1}^{r(x,g)} E_{ij}(x)

such that if E_{ij}(x) ≠ {0}, then ∀ 0 ≠ u ∈ E_{ij}(x),

lim_{n→∞} \frac{1}{n} log ∥Df_{x}^{n}u∥ = λ_{i}(x, f), lim_{n→∞} \frac{1}{n} log ∥Dg_{x}^{n}u∥ = λ_{j}(x, f),

and

lim_{n→∞} \frac{1}{n} log |sin(E_{i_{1}j_{1}}(f^{n}x), E_{i_{2}j_{2}}(f^{n}x))| = 0, lim_{n→∞} \frac{1}{n} log |sin(E_{i_{1}j_{1}}(g^{n}x), E_{i_{2}j_{2}}(g^{n}x))| = 0.

Moreover,the following invariant properties hold.

i) Df_{x}E_{ij}(x) = E_{ij}(fx), Dg_{x}E_{ij}(x) = E_{ij}(gx).

ii) λ_{i}(fx, f) = λ_{i}(x, f) = λ_{i}(gx, f), λ_{j}(fx, g) = λ_{j}(x, g) = λ_{j}(gx, g).

Proof. For any point x ∈ Γ_{f}, let T_{x}M = E_{i}(x, f) ⊕ ⋯ ⊕ E_{r(x,f)}(x, f) be the decomposition of tangent space for diffeomorphism f. By Corollary 2.2, Dg_{x}(E_{i}(x, f)) = E_{i}(gx, f). Restricted on {E_{i}(x, f)}, {Dg_{x}^{n}} is a cocycle on M with respect to g (see [Ru]), where we take E_{i}(x, f) = {0} if i > r(x, f) or x is not in Γ_{f}. Now we use the Multiplicative Ergodic Theorem for each i to get a subset Γ^{(i)} ⊂ Γ, such that ∀ x ∈ Γ^{(i)}, after relabelling the subscript, if necessary, E_{ij}(x) has desired properties. Since for each i, μΓ^{(i)} = 1, ∀ μ ∈ M(M, f, g), and

by Corollary 2.2.i), fΓ^{(i)} = Γ^{(i)}, we can take Γ_{1} = ⋂_{i=1}^{r(x,f)} Γ^{(i)}.

The Proof of Theorem A.

First we claim that ∀ s, t ∈ ZZ, i = 1, ⋯, r(x, f), j = 1, ⋯, r(x, g), the set

A_{γ} = {x : ∃u_{x} ∈ E_{ij}(x), s.t. χ(x, u_{x}, f^{s}g^{t}) − sλ_{i}(x, f) − tλ_{j}(x, g) > 4γ}

satisfies μA_{γ} = 0 for all μ ∈ M(M, f, g).

Suppose it is not true. Then there exists a μ ∈ M(M, f, g) with μA_{γ} > 0. Choose l > 0 such that the sets

A′ = {x ∈ A_{γ} : ∥D(f^{s}g^{t})_{x}^{n}u_{x}∥ ≥ l^{−1}∥u_{x}∥ exp n(χ(x, u_{x}, f^{s}g^{t}) − γ), ∀n ≥ 0},
A′′ = {x ∈ A_{γ} : ∥Dg_{x}^{tn}u∥ ≤ l∥u∥ exp n(tλ_{j}(x, g) + γ), ∀u ∈ E_{ij}(x), n ≥ 0}

have measure larger than \frac{1}{2}μA_{γ}. Then A′ ∩A′′ ≠ ∅. By Poincaré Recurrence Theorem we can take x ∈ A′∩A′′ such that there exists a sufficient large integer n > \frac{2 log l}{γ} with f^{sn}x ∈ A′∩A′′ and

∥Df_{x}^{sn}u∥ ≤ ∥u∥ exp n(sλ_{i}(x, f) + γ), ∀u ∈ E_{ij}(x).

Since Df_{x}^{sn}u ∈ E_{ij}(f^{sn}x) and f^{sn}x ∈ A′′,

∥D(f^{s}g^{t})_{x}^{n}u∥ = ∥Dg_{f^{sn}x}^{tn}Df_{x}^{sn}u∥ ≤ l∥Df_{x}^{sn}u∥ exp n(tλ_{j}(f^{sn}x, g) + γ)
≤ l∥u∥ exp n(sλ_{i}(x, f) + tλ_{j}(x, g) + 2γ)
< l^{−1}∥u∥ exp n(χ(x, u, f^{s}g^{t}) − γ), ∀u ∈ E_{ij}(x).

In particular, take u = u_{x}, then

∥D(f^{s}g^{t})_{x}^{n}u_{x}∥ < l^{−1}∥u_{x}∥ exp n(χ(x, u_{x}, f^{s}g^{t}) − γ).

This contradicts the fact x ∈ A′.

Similar claim for the set

B_{γ} = {x : ∃u_{x} ∈ E_{ij}(x), s.t. χ(x, u_{x}, f^{s}g^{t}) − sλ_{i}(x, f) − tλ_{j}(x, g) < 4γ}

is also true. It is easy to see by the claims that for any μ ∈ M(M, f, g), μ−a.e.x ∈ M,∀ s, t ∈ ZZ, i = 1, ⋯, r(x, f), j = 1, ⋯, r(x, g), if E_{ij}(x) ≠ {0}, then ∀ 0 ≠ u ∈ E_{ij}(x),

χ(x, u, f^{s}g^{t}) = lim_{n→∞} \frac{1}{n} log ∥D(f^{s}g^{t})_{x}^{n}u∥ = sλ_{i}(x, f) + tλ_{j}(x, g).

Using the same idea, with some modification, we can prove the result concerning the equality

lim_{n→∞} \frac{1}{n} log |sin(E_{i_{1}j_{1}}((f^{s}g^{t})^{n}x), E_{i_{2}j_{2}}((f^{s}g^{t})^{n}x))| = 0.

The rest of the results of the theorem follow directly from Proposition 2.3.

§3. A Version of Pesin’s Theory

In this section we give a version of Pesin’s theory in the case of commuting diffeomorphisms. The main result is stated in Proposition 3.6.

Lemma 3.1. Let A(x) be a positive measurable function on Γ such that there exist positive measurable functions P_{1}(x) and P_{2}(x) on Γ satisfying that for all x ∈ Γ,

P_{1}(x)e^{−6(|n|+|k|)γ} ≤ A((f^{n}g^{k})x) ≤ P_{2}(x)e^{6(|n|+|k|)γ}, ∀ u, k ∈ ZZ.

Then a measurable function C : Γ → [1, ∞) can be found such that, ∀ x ∈ Γ,

C(x)^{−1} ≤ A(x) ≤ C(x),

and

C(f^{±1}x) ≤ C(x)e^{8γ}, C(g^{±1}x) ≤ C(x)e^{8γ}.

Proof. For any x ∈ Γ, except for finite number of pairs (n, k),

A((f^{n}g^{k})x)e^{−8(|n|+|k|)γ} ≤ 1, A((f^{n}g^{k})x)^{−1}e^{−8(|n|+|k1|)γ} ≤ 1.

Thus,

C(x) = max{1, A(f^{n}g^{k}x)e^{−8(|n|+|k|)γ}, A(f^{n}g^{k}x)^{−1}e^{−8(|n|+|k|)γ}, ∀ n, k ∈ ZZ}

is a required function.

Lemma 3.2. For any γ > 0, there exists a measurable function Q : Γ → [1, ∞) such that ∀ n, k ∈ ZZ, 0 ≠ u ∈ E_{ij}(x), x ∈ Γ, i = 1, ⋯, r(x, f), j = 1, ⋯, r(x, g),

\begin{aligned}Q(x)^{−1}∥u∥e^{nλ_{i}(x,f)+kλ_{j}(x,g)−3(|n|+|k|)γ} & ≤ ∥D(f^{n}g^{k})_{x}u∥ \\ & ≤ Q(x)∥u∥e^{nλ_{i}(x,f)+kλ_{j}(x,g)+3(|n|+|k|)γ}. \end{aligned}

Proof. In the proof we always assume that ∥u∥ = 1.

We assert that for each x ∈ Γ, if |n| or |k| is sufficiently large, then

e^{−3(|n|+|k|)γ} ≤ ∥D(f^{n}g^{k})_{x}u∥e^{−nλ_{i}(x,f)−kλ_{j}(x,g)} ≤ e^{3(|n|+|k|)γ}.

First we suppose n ≥ k ≥ 0.

Take l = l(x) > 0, such that ∀ y ∈ ⋃_{n,k∈Z Z}(f^{n}g^{k})x, v ∈ E_{ij}(y), i = 1, ⋯, r(x, f),

i = 1, ⋯, r(x, g),

∥Df_{y}v∥ ≤ ∥v∥e^{λ_{i}(x,f)+lγ}, ∥Dg_{y}v∥ ≤ ∥v∥e^{λ_{j}(x,g)+lγ}.

By Theorem A, ∀ 0 ≤ t ≤ l, u ∈ E_{ij}(x),

lim_{s→∞} \frac{1}{s} log ∥D(f^{l}g^{t})_{x}^{s}u∥ = lλ_{i}(x, f) + tλ_{j}(x, g).

We can choose a positive integer s_{0} > l, such that ∀s > s_{0}, l ≥ t ≥ 0,

e^{slλ_{i}(x,f)+stλ_{j}(x,g)−(sl+st)γ} ≤ ∥D(f^{l}g^{t})_{x}^{s}u∥ ≤ e^{slλ_{i}(x,f)+stλ_{j}(x,g)+(sl+st)γ}.
Denote N_{I} = ls_{0}.

Take n ≥ N_{I}. For 0 ≤ k ≤ n, we can write n = sl + p, k = ts + q, where 0 ≤ p < l, 0 ≤ q < s. Notice that λ_{i}(x, f) and λ_{j}(x, g) are both f and g−invariant, we have

∥D(f^{n}g^{k})_{x}u∥ = ∥Df^{p} ∘ Dg^{q} ∘ D(f^{l}g^{t})_{x}^{s}u∥
≤ e^{p(λ_{i}(x,f)+lγ)} · e^{q(λ_{j}(x,g)+lγ)} · e^{slλ_{i}(x,f)+stλ_{j}(x,g)+(sl+st)γ}
= e^{nλ_{i}(x,f)+kλ_{j}(x,g)+(pl+ql+sl+st)γ}
≤ e^{nλ_{i}(x,f)+kλ_{j}(x,g)+3(n+k)γ}.
Also, we can write n = sl − p, k = ts − q, where 0 ≤ p < l, 0 ≤ q < s. Then

\begin{aligned}∥D(f^{l}g^{t})_{x}^{s}u∥ & = ∥Df^{p} ∘ Dg^{q} ∘ D(f^{n}g^{k})_{x}u∥ \\ & ≤ e^{p(λ_{i}(x,f)+lγ)} · e^{q(λ_{j}(x,g)+lγ)}∥D(f^{n}g^{k})_{x}u∥. \end{aligned}

So

∥D(f^{n}g^{k})_{x}u∥ ≥ e^{−p(λ_{i}(x,f)+lγ)} · e^{−q(λ_{j}(x,g)+lγ)} · e^{slλ_{i}(x,f)+stλ_{j}(x,g)−(sl+st)γ}
= e^{nλ_{i}(x,f)+kλ_{j}(x,g)−(pl+ql+sl+st)γ}
≥ e^{nλ_{i}(x,f)+kλ_{j}(x,g)−3(n+k)γ}.

These inequalities show that in the case of n ≥ k ≥ 0, our assertion is true if n ≥ N_{I}. It is also true for the case k ≥ n ≥ 0 because we can find K_{I} > 0 similarly such that the inequality holds if k ≥ K_{I}. Since the iterations of f and g in positive and negative directions are symmetrical, the assertion is true if one or two of n and k are negative.

Now we know that

\begin{matrix}Q(x) = max\{1, & ∥D(f^{n}g^{k})_{x}u∥^{−1}e^{nλ_{i}(x,f)+kλ_{j}(x,g)−3(|n|+|k|)γ}, \\ ∥D(f^{n}g^{k})_{x}u∥e^{−nλ_{i}(x,f)−kλ_{j}(x,g)−3(|n|+|k|)γ}, ∀n, k ∈ ZZ, u ∈ E_{ij}(x), ∥u∥ = 1\} \end{matrix}

is a required function.

Remark 3.3. In the proof of Lemma 3.2, Q(x) is chosen to be the minimal function satisfying our requirement, i.e. ∀ x ∈ Γ,

\begin{matrix}Q(x) = inf\{q ≥ 1 : q^{−1}e^{−3(|n|+|k|)γ} ≤ ∥D(f^{n}g^{k})_{x}u∥ & e^{−nλ_{i}(x,f)−kλ_{j}(x,g)} ≤ qe^{3(|n|+|k|)γ} \\ ∀n, k ∈ ZZ, u ∈ E_{ij}(x)\}. \end{matrix}

Lemma 3.4. For any γ > 0, there exists a measurable function R : Γ → [1, ∞), such that ∀ n, k ∈ ZZ, i_{1}, i_{2} = 1, ⋯, r(x, f), j_{1}, j_{2} = 1, ⋯, r(x, g), (i_{1}, j_{1}) ≠ (i_{2}, j_{2}),

|sin(E_{i_{1}j_{1}}((f^{n}g^{k})x), E_{i_{2}j_{2}}((f^{n}g^{k})x))| ≥ R(x)^{−1}e^{−3(|n|+|k|)γ}.

Proof. The method is similar as in the proof for the left inequality of Lemma 3.2 if we use

max\{\frac{| sin(E_{i_{1}j_{1}}(fx), E_{i_{2}j_{2}}(fx))|}{| sin(E_{i_{1}j_{1}}(x), E_{i_{2}j_{2}}(x))|}\}, max\{\frac{| sin(E_{i_{1}j_{1}}(gx), E_{i_{2}j_{2}}(gx))|}{| sin(E_{i_{1}j_{1}}(x), E_{i_{2}j_{2}}(x))|}\}

instead of Df_{x} and Dg_{x} respectively, where the maximums run over i_{1}, i_{2} = 1, ⋯, r(x, f), j_{1}, j_{2} = 1, ⋯, r(x, g), (i_{1}, j_{1}) ≠ (i_{2}, j_{2}).

Lemma 3.5. The function Q(x) determined by the proof of Lemma 3.2 satisfies that

Q(f^{s}g^{t}x) ≤ Q(x)^{2}e^{6(|s|+|t|)γ}, ∀ s, t ∈ ZZ.

Proof. Take s, t ∈ ZZ, ∀ u ∈ E_{ij}(f^{t}g^{t}x), where i = 1, ⋯, r(x, f), j = 1, ⋯, r(x, g). We can choose u′ ∈ E_{ij}(x) with u = D(f^{s}g^{t})_{x}u′. By Lemma 3.2,

Q(x)^{−1}∥u′∥e^{sλ_{i}(x,f)+tλ_{j}(x,g)−3(|s|+|t|)γ} ≤ ∥u∥ ≤ Q(x)∥u′∥e^{sλ_{i}(x,f)+tλ_{j}(x,g)+3(|s|+|t|)γ}.

Since ∀ n, k ∈ ZZ, D(f^{n}g^{k})_{f^{s}g^{t}x}u = D(f^{n+s}g^{k+t})_{x}u′,

\begin{aligned}& Q(x)^{−1}∥u′∥e^{(n+s)λ_{i}(x,f)+(k+t)λ_{j}(x,g)−3(|n+s|+|k+t|)γ} \\ ≤ ∥D(f^{n}g^{k} & )_{f^{s}g^{t}x}u∥ ≤ Q(x)∥u′∥e^{(n+s)λ_{i}(x,f)+(k+t)λ_{j}(x,g)+3(|n+s|+|k+t|)γ}. \end{aligned}

From these inequalities we have that

\begin{aligned}& Q(x)^{−2}∥u∥e^{nλ_{i}(x,f)+kλ_{j}(x,g)−3(|n|+|k|)γ−6(|s|+|t|)γ} \\ ≤ ∥D(f^{n}g^{k} & )_{f^{s}g^{t}x}u∥ ≤ Q(x)^{2}∥u∥e^{nλ_{i}(x,f)+kλ_{j}(x,g)+3(|n|+|k|)γ+6(|s|+|t|)γ}. \end{aligned}

By Remark 3.3 the value of Q at f^{s}g^{t}x must satisfy

Q(f^{s}g^{t}x) ≤ Q(x)^{2}e^{6(|s|+|t|)γ}.

Proposition 3.6. Given M, f, g, Γ as in Theorem A, then for any γ > 0, there exists a measurable function C : Γ → [1, ∞) such that

i) ∀n, k ∈ ZZ, u ∈ E_{ij}(x), i = 1, ⋯, r(x, f), j = 1, ⋯, r(x, g),

C(x)^{−1}∥u∥e^{nλ_{i}(x,f)+kλ_{j}(x,g)−(|n|+|k|)γ} ≤ ∥D(f^{n}g^{k})_{x}u∥ ≤ C(x)∥u∥e^{nλ_{i}(x,f)+kλ_{j}(x,g)+(|n|+|k|)γ},

ii) ∀i_{1}, i_{2} = 1, ⋯, r(x, f), j_{1}, j_{2} = 1, ⋯, r(x, g), (i_{1}, j_{1}) ≠ (i_{2}, j_{2}),

|sin(E_{i_{1}j_{1}}(x), E_{i_{2}j_{2}}(x))| ≥ C(x)^{−1},

iii) C(f^{±1}x) ≤ C(x)e^{γ}, C(g^{±1}x) ≤ C(x)e^{γ}.

Proof. In Lemma 3.5 we replace s,t by −n,−k, respectively, then replace x by f^{n}g^{k}x to get

Q(x) ≤ Q(f^{n}g^{k}x)^{2}e^{6(|n|+|k|)γ},

i.e.

Q(f^{n}g^{k}x) ≥ \sqrt{Q(x)}e^{−3(|n|+|k|)γ}.

We let P_{1}(x) = \sqrt{Q(x)} and P_{2}(x) = Q(x)^{2}. By Lemma 3.1 there exists a measurable function C_{1}(x) > 0 with Q(x) ≤ C_{1}(x) and C_{1}(f^{±1}x) ≤ C_{1}(x)e^{8γ}, C_{1}(g^{±1}x) ≤ C_{1}(x)e^{8γ}.

Take

\begin{matrix}A(x) = max\{ & | sin(E_{i_{1}j_{1}}(x), E_{i_{2}j_{2}}(x))| : \\ i_{1}, i_{2} = 1, ⋯, r(x, f), j_{1}, j_{2} = 1, ⋯, r(x, g), (i_{1}, j_{1}) ≠ (i_{2}, j_{2})\}. \end{matrix}

By Lemma 3.4 and Lemma 3.1, there exists a measurable function C_{2}(x) > 0 such that A(x) ≥ C_{2}(x)^{−1} and C_{2}(f^{±1}x) ≤ C_{2}(x)e^{8γ}, C_{2}(g^{±1}x) ≤ C_{2}(x)e^{8γ}.

Now we use γ instead of 8γ and put C(x) = max{C_{1}(x), C_{2}(x)}. Then C(x) is a required function.

§4. Lyapunov Charts

We have already had the decomposition of tangent space into subspaces corresponding Lyapunov exponents for both f and g. In this section we construct Lyapunov charts for the diffeomorphisms by the same method used in [LY]. For simplicity our discussion just concerns the difference and skips the rest.

Let ⟨⟨·, ·⟩⟩ be the inner product on T_{x}M given by the Riemannian structure and ∥ · ∥ be the induced norm. Let ⟨·, ·⟩ and |·, ·| denote the usual inner product and norm in IR^{m} respectively. Also, for ρ > 0, let B̃(ρ) be the ball in IR^{m} centered at origin of radius ρ.

Let

λ_{+}(x, f) = min{λ_{i}(x, f) : λ_{i}(x, f) > 0}, λ_{−}(x, f) = max{λ_{i}(x, f) : λ_{i}(x, f) < 0},

Δλ(x, f) = min{λ_{i}(x, f) − λ_{i+1}(x, f), i = 1, ⋯, r(x, f) − 1},

and define λ_{±}(x, g), Δλ(x, g) in similar way. Take

0 < γ = γ(x) ≤ \frac{1}{200m} min\{Δλ(x, f), Δλ(x, g), ±λ_{±}(x, f), ±λ_{±}(x, g)\},

where m = dim M. Next proposition shows the existence and properties of Lyapunov charts for f and g.

Proposition 4.1. For the γ defined as above, there exists a measurable function l : Γ → [0, ∞) with l(f^{±1}x) ≤ l(x)e^{γ}, l(g^{±1}x) ≤ l(x)e^{γ}, and a set of embedings Φ_{x} : B̃(l(x)^{−1}) → M at each point x ∈ Γ such that the following holds.

i) Φ_{x}(0) = x, and the preimages R_{ij}(x) = DΦ_{x}(0)^{−1}(E_{ij}(x)) of E_{ij}(x) are mutually orthogonal in IR^{m}, where i = 1, ⋯, r(x, f), j = 1, ⋯, r(x, g).

ii) Let f̃_{x} = Φ_{fx}^{−1} ∘ f ∘ Φ_{x} be the connecting map between the chart at x and the chart at fx. g̃_{x}, (f̃g)_{x} and their inverses are defined similarly. Then (f̃g)_{x} = f̃_{gx}g̃_{x} = g̃_{fx}f̃_{x} and (f̃g)_{x}^{−1} = f̃_{g^{−1}x}^{−1}g̃_{x}^{−1} = g̃_{f^{−1}x}^{−1}f̃_{x}^{−1}.

iii) For any 1 ≤ i ≤ r(x, f), l ≤ j ≤ r(x, g),and u ∈ R_{ij}(x),

|u|e^{λ_{i}(x,f)−γ} ≤ |Df̃_{x}(0) u| ≤ |u|e^{λ_{i}(x,f)+γ},
|u|e^{λ_{j}(x,g)−γ} ≤ |Dg̃_{x}(0) u| ≤ |u|e^{λ_{j}(x,g)+γ},
|u|e^{λ_{i}(x,f)+λ_{j}(x,g)−γ} ≤ |D(f̃g)_{x}(0) u| ≤ |u|e^{λ_{i}(x,f)+λ_{j}(x,g)+γ}.

iv) Let L(Ψ) be the Lipschitz constant of the function Ψ. Then for F_{x} = f̃_{x}, g̃_{x}, (f̃g)_{x} and their inverses,

L(F_{x} − DF_{x}(0)) ≤ γ, L(DF_{x}) ≤ l(x).

v) There exists a number λ > 0 depending on γ and the exponents such that ∀x ∈ Γ,

|f̃_{x}^{±1}u| ≤ e^{λ}|u|, |g̃_{x}^{±1}u| ≤ e^{λ}|u|, |(f̃g)_{x}^{±1}u| ≤ e^{λ}|u|, ∀u ∈ B̃(e^{−λ−γ}l(x)^{−1}).

vi) For all u, v ∈ B̃(l(x)^{−1}), we have

K^{−1}d(Φ_{x}u, Φ_{x}v) ≤ |u − v| ≤ l(x)d(Φ_{x}u, Φ_{x}v),

for some universal constant K.

We shall refer to any system of local charts {Φ_{x} : x ∈ Γ} satisfying i) − vi) as (γ, l)−charts for f and g. Obviously, if {Φ_{x} : x ∈ Γ} is a system of (γ, l)−charts for both f and g, then it is a system for either f or g as well.

The Proof of Proposition 4.1.

By Proposition 3.6 there exists a measurable function C : Γ → [1, ∞) such that for all x ∈ Γ, we have the following.

i) ∀n, k ∈ ZZ, u ∈ E_{ij}(x), i = 1, ⋯, r(x, f), j = 1, ⋯, r(x, g),

C(x)^{−1}∥u∥e^{nλ_{i}(x,f)+kλ_{j}(x,g)−(|n|+|k|)γ} ≤ ∥D(f^{n}g^{k})_{x}u∥ ≤ C(x)∥u∥e^{nλ_{i}(x,f)+kλ_{j}(x,g)+(|n|+|k|)γ}.

ii) ∀i_{1}, i_{2} = 1, ⋯, r(x, f), j_{1}, j_{2} = 1, ⋯, r(x, g), (i_{1}, j_{1}) ≠ (i_{2}, j_{2}),
|sin(E_{i_{1}j_{1}}(x), E_{i_{2}j_{2}}(x))| ≥ C(x)^{−1}.
iii) C(f^{±1}x) ≤ C(x)e^{γ}, C(g^{±1}x) ≤ C(x)e^{γ}.
We define a new inner product ⟨⟨⟨·, ·⟩⟩⟩ on T_{x}M. First, for u, v ∈ E_{ij}(x), let

⟨⟨⟨u, v⟩⟩⟩ = ∑_{n=−∞}^{+∞} ∑_{k=−∞}^{+∞} \frac{⟨⟨D(f^{n}g^{k})_{x}u, D(f^{n}g^{k})_{x}v⟩⟩}{exp 2[nλ_{i}(x, f) + kλ_{j}(x, g) + 2(|n| + |k|)γ]}.

Then we extend ⟨⟨⟨·, ·⟩⟩⟩ to T_{x}M by demanding that all subspaces {E_{ij}(x)} be mutually orthogonal with respect to ⟨⟨⟨·, ·⟩⟩⟩. Let ||| · ||| be the corresponding norm. A calculation shows that ∀ 0 ≠ u ∈ E_{ij}(x),

∥u∥ ≤ |||u||| ≤ C_{0}C(x)∥u∥,

where C_{0} = ( ∑_{n=−∞}^{+∞} ∑_{k=−∞}^{+∞} e^{−2(|n|+|k|)γ})^{\frac{1}{2}} , and

|||u|||e^{λ_{i}(x,f)−2γ} ≤ |||Df_{x}u||| ≤ |||u|||e^{λ_{i}(x,f)+2γ},
|||u|||e^{λ_{j}(x,g)−2γ} ≤ |||Dg_{x}u||| ≤ |||u|||e^{λ_{j}(x,g)+2γ},

and then

|||u|||e^{λ_{i}(x,f)+λ_{j}(x,g)−4γ} ≤ |||Df_{gx} · Dg_{x}u||| ≤ |||u|||e^{λ_{i}(x,f)+λ_{j}(x,g)+4γ}.

For arbitrary 0 ≠ u ∈ T_{x}M, we can write u = ∑_{i=1}^{r(x,f)} ∑_{j=1}^{r(x,g)} u_{ij}, where u_{ij} ∈ E_{ij}(x). It is

clear that

∥u∥ ≤ ∑_{i} ∑_{j} ∥u_{ij}∥ ≤ ∑_{i} ∑_{j} |||u_{ij}||| ≤ m|||u|||,

because there are at most m different subspaces E_{ij}(x) in T_{x}M. With similar argument in [LY, Appendix] we obtain that

∥u∥ ≥ ∥u_{ij}∥C(x)^{−m+1}, ∀i = 1, ⋯, r(x, f), j = 1, ⋯, r(x, g).

Therefore

|||u||| ≤ ∑_{i=1}^{r(x,f)} ∑_{j=1}^{r(x,g)} |||u_{ij}||| ≤ C_{0}C(x) ∑_{i=1}^{r(x,f)} ∑_{j=1}^{r(x,g)} ∥u_{ij}∥ ≤ mC_{0}C(x)^{m}∥u∥.

Define a linear map L_{x} : T_{x}M → IR^{m} such that
⟨L_{x}u, L_{x}v⟩ = ⟨⟨⟨u, v⟩⟩⟩, ∀u, v ∈ T_{x}M,

and let Φ_{x} = exp_{x} ∘L_{x}^{−1}, then i)−iii) hold, if we use γ instead of max{mγ, 4γ}. To get iv)−vi) we take l(x) = CC(x)^{m}, where C is large enough and chosen in a way similar as in [LY, Appendix], and take K = 2m for vi). The proof is finished.

Corollary 4.2. With above notation, ∀ 0 < ϵ < e^{−γ},

f̃_{x}^{±1}B̃(ϵe^{−λ}l(x)^{−1}) ⊂ B̃(ϵl(fx)^{−1}), g̃_{x}^{±1}B̃(ϵe^{−λ}l(x)^{−1}) ⊂ B̃(ϵl(gx)^{−1}).

Proof. It is Proposition 4.1.v).

Now we introduce a new norm | · |′ on IR^{m}. For any v ∈ IR^{m}, we may write v =

∑_{i=1}^{r(x,f)} ∑_{j=1}^{r(x,g)} v_{ij}, where v_{ij} ∈ R_{ij} as R_{ij} ≠ {0}. Then let

|v|′ = max{|v_{ij}|, ∀i = 1, ⋯, r(x, f), j = 1, ⋯, r(x, g)}.

It is clear that \frac{1}{\sqrt{m}}|v| ≤ |v|′ ≤ |v|, where m = dim M, and Proposition 4.1 still holds for | · |′, but the universal constant K = 2m may change to K = 2m\sqrt{m}. From here on, we use | · |′ as the norm in IR^{m} and still write it as | · |.

Suppose {Φ_{x} : x ∈ Γ} is a system of (γ, l)−charts for f. For x ∈ Γ, let R^{u} = L_{x}^{−1}E^{u}(x, f),R^{sc} = L_{x}^{−1}E^{sc}(x, f) and so on. Then IR^{m} = R^{s} × R^{c} × R^{u}, or IR^{m} = R^{sc} × R^{u}. By Proposition 4.1.ii), DΦ_{x}(0) takes R^{u},R^{c},R^{s} to E^{u}(x, f),E^{c}(x, f),E^{s}(x, f) respectively. The u−coordinate of a point v ∈ IR^{m} is denoted by v_{u}. Other notations such as v_{c}, v_{s}, v_{sc} are understood in obvious way. Clearly

|v| = max{|v_{s}|, |v_{c}|, |v_{u}|}.

We should remember that the notations v_{u},v_{sc} depend on the diffeomorphism, but we will not indicate it in our notation, because there is no ambiguity from context.

Corollary 4.3. Let x ∈ Γ and 0 < ϵ < e^{−λ−γ}. We have the following.

i) If v, v′ ∈ B(ϵl(x)^{−1}) and |v − v′| = |v_{u} − v′_{u}|, then

|f̃_{x}v − f̃_{x}v′| = |(f̃_{x}v)_{u} − (f̃_{x}v′)_{u}| ≥ e^{λ_{+}(x,f)−2γ}|v − v′|.

ii) If v, v′ ∈ B(ϵl(x)^{−1}) and |v − v′| = |v_{sc} − v′_{sc}|, then

|f̃_{x}^{−1}v − f̃_{x}^{−1}v′| = |(f̃_{x}^{−1}v)_{sc} − (f̃_{x}^{−1}v′)_{sc}| ≥ e^{−2γ}|v − v′|.

Proof. It follows from Proposition 4.1.iii)-v).

§5. Unstable Manifold

Recall that the global unstable manifold of a diffeomorphism f at x is the set w^{u}(x, f) =

\{y ∈ M : lim sup_{n→∞} \frac{1}{n} log d(f^{−n}x, f^{−n}y) < 0\}.

Let {Φ_{x} : x ∈ Γ} be a system of (γ, l)− charts for f. The local unstable manifold, denoted by w_{α}^{u}(x, f), of f at x associated with {Φ_{x}} and α is defined to be the component of w^{u}(x, f) ∩ Φ_{x}R(αl(x)^{−1}) that contains x. The Φ_{x}^{−1}− image of this set in the x−chart is denoted by W_{α}^{u}(x, f).

The following result is well-known.

Proposition 5.1. For 0 < α ≤ 1 and x ∈ Γ, W^{u}(x, f) is the graph of a function

ψ_{x}^{f} : B^{u}(αl(x)^{−1}) ⟶ B^{sc}(αl(x)^{−1})

with ψ_{x}^{f}(0) = 0 and ∥ Dψ_{x}^{f} ∥≤ \frac{1}{3}, where B^{u}(β) and B^{sc}(β) denote the balls centered at the origin of radius β in R^{u} and R^{sc} respectively.

The family of global unstable manifolds {w^{u}(x, f) : x ∈ Γ} is f−invariant, that is, fw^{u}(x, f) = w^{u}(fx, f). If f and g commute, we have the following.

Lemma 5.2. fw^{u}(x, g) = w^{u}(fx, g). Proof. There exists C > 0,such that ∀x, y ∈ M,

C^{−1}d(x, y) ≤ d(fx, fy) ≤ Cd(x, y).

Therefore

fw^{u}(x, g) = \{y ∈ M : lim sup_{n→∞} \frac{1}{n} log d(g^{−n}x, g^{−n}(f^{−1}y) ) < 0\}

= \{y ∈ M : lim sup_{n→∞} \frac{1}{n} log d(g^{−n}(fx), g^{−n}y) < 0\}

= w^{u}(fx, g).

Now we consider the case that E^{u}(x, f) = E^{u}(x, g), μ − a.e. Let {Φ_{x} : x ∈ Γ} be a system of (γ, l)−charts for both f and g. Then by Proposition 5.1 there are two functions ψ_{x}^{f} and ψ_{x}^{g} defined on the balls of radius αl(x)^{−1} centered at origin in R^{u}, where R^{u} = (DΦ_{x}(0))^{−1}E^{u}(x, f) = (DΦ_{x}(0))^{−1}E^{u}(x, g). The graphs of ψ_{x}^{f} and ψ_{x}^{g} are W_{α}^{u}(x, f) and W_{α}^{u}(x, g) respectively.

Lemma 5.3. Suppose E^{u}(x, f) = E^{u}(x, g), μ − a.e. and 0 < α < e^{−λ−γ}. Then ∀ x ∈ Γ,

i) f̃_{x}^{−1}W_{α}^{u}(x, f) ⊂ W_{α}^{u}(f^{−1}x, f), f̃_{x}^{−1}W_{α}^{u}(x, g) ⊂ W_{α}^{u}(f^{−1}x, g); ii) ∀v ∈ W_{α}^{u}(x, f) or W_{α}^{u}(x, g),

|f̃_{x}^{−n}v| = |(f̃_{x}^{−n}v)_{u}| ≤ |v| exp[−n(λ_{+}(x, f) − 2γ)].

Proof. We only need prove the result related to W_{α}^{u}(x, g). Take v ∈ W_{α}^{u}(x, g) arbitrary. Then |v| ≤ αl(x)^{−1}. By Proposition 4.1.v), |f̃_{x}^{−1}v| ≤ e^{−γ}l(x)^{−1} ≤ l(f^{−1}x)^{−1}, i.e. f̃_{x}^{−1}v ∈ B(l(f^{−1}x)^{−1}). It is easy to know by Lemma 5.2 and Proposition 5.1 that f̃_{x}^{−1}v ∈ W_{α′}^{u}(f^{−1}x, g) for some 0 < α′ < 1. Thus, Proposition 5.1 gives that |f̃_{x}^{−1}v| = |(f̃_{x}^{−1}v)_{u}|. By Corollary 4.3.i),

|v| = |f̃_{f^{−1}x}f̃_{x}^{−1}v| ≥ e^{λ_{+}(x,f)−2γ}|f̃_{x}^{−1}v|,

i.e.

|f̃_{x}^{−1}v| ≤ e^{−(λ_{+}(x,f)−2γ)}|v| ≤ |v|e^{−γ} ≤ αl(f^{−1}x).

Thus, |f̃_{x}^{−1}v| ∈ l(f^{−1}x)^{−1} and i) holds. Continuing this process we obtain ii).

Proposition 5.4. Suppose M is a C^{∞} compact Riemannian manifold without boundary, f, g ∈ Diff^{2}(M) with fg = gf, and Γ is as in Theorem A. For any x ∈ Γ, if E^{u}(x, f) = E^{u}(x, g), then w^{u}(x, f) = w^{u}(x, g).

Proof. We only need show that for some 0 < α < e^{−λ−γ}, W_{α}^{u}(x, f) = W_{α}^{u}(x, g).

Suppose it is not true. Then we can find u ∈ W^{u}(x, f), v ∈ W^{u}(x, g), such that
|u − v| = |u_{sc} − v_{sc}| > 0.

where u_{sc},v_{sc} are the sc−coordinates of u and v respectively. By Lemma 5.3, we have that

|(f̃_{x}^{−n}u)_{u}| = |f̃_{x}^{−n}u| ≤ e^{−n(λ_{+}(x,f)−2γ)}|u| ≤ αe^{−n(λ_{+}(x,f)−2γ)},
|(f̃_{x}^{−n}v)_{u}| = |f̃_{x}^{−n}v| ≤ e^{−n(λ_{+}(x,f)−2γ)}|v| ≤ αe^{−n(λ_{+}(x,f)−2γ)}

and f̃_{x}^{−n}u,f̃_{x}^{−n}v ∈ B(αl(f^{−n}x)^{−1}) for any n > 0. Applying Corollary 4.3.ii) repeatedly, we get

|(f̃_{x}^{−n}u)_{sc} − (f̃_{x}^{−n}v)_{sc}| ≥ e^{−2nγ}|u − v|.

Without loss generality we may assume that λ_{+}(x, f) ≥ λ_{+}(x, g). Since Proposition 5.1 implies |(f̃_{x}^{−n}u)_{sc}| ≤ \frac{1}{3}|(f̃_{x}^{−n}u)_{u}|, we get ∀ n ≥ 0,

\frac{|(f̃_{x}^{−n}v)_{sc}|}{|(f̃_{x}^{−n}v)_{u}|} ≥ \frac{|(f̃_{x}^{−n}u)_{sc} − (f̃_{x}^{−n}v)_{sc}| − |(f̃_{x}^{−n}u)_{sc}|}{|(f̃_{x}^{−n}v)_{u}|}

≥ \frac{e^{−2nγ}|u − v| − \frac{1}{3}αe^{−n(λ_{+}(x,f)−2γ)}}{αe^{−n(λ_{+}(x,g)−2γ)}}

= α^{−1}|u − v|e^{n(λ_{+}(x,g)−4γ)} − \frac{1}{3}e^{n(λ_{+}(x,g)−λ_{+}(x,f))}.

By our assumption, λ_{+}(x, g) − λ_{+}(x, f) ≤ 0. Hence the right hand side in above inequality tends to infinite as n → ∞. But by Proposition 5.1, |(f̃_{x}^{−n}v)_{sc}| ≤ \frac{1}{3}|(f̃_{x}^{−n}v)_{u}|. This is a contradiction.

§6. Local Entropies

Suppose ν ∈ M(M, f), but not necessary in M(M, g). Let B(x, ϵ) be a closed ball in M centered at x of radius ϵ. We call the set

B_{n}(x, ϵ, f) = ⋂_{i=0}^{n} f^{−i}B(f^{i}x, ϵ)

to be an (n, ϵ, f)−ball of f at x ∈ M. The local entropy h_{ν}(x, f) of f at x is defined as (see [BK])

h_{ν}(x, f) = lim_{ϵ→0} lim sup_{n→∞} −\frac{1}{n} log νB_{n}(x, ϵ, f)

= lim_{ϵ→0} lim inf_{n→∞} −\frac{1}{n} log νB_{n}(x, ϵ, f)

which holds for ν −a.e. x ∈ M, and satisfies that h_{ν}(fx, f) = h_{ν}(x, f) and ∫ h_{ν}(x, f)dν(x) = h_{ν}(f).

Recall g^{∗}ν = ν ∘ g^{−1}. Since ν is a fixed point of f^{∗} and f^{∗}g^{∗} = g^{∗}f^{∗}, by Proposition 1.1, g^{∗}ν is also a fixed point of f^{∗}, i.e. g^{∗}ν ∈ M(M, f).

Lemma 6.1. ∀ ν ∈ M(M, f), h_{g^{∗}ν}(x, f) = h_{ν}(g^{−1}x, f), ν − a.e.x ∈ M. Therefore, h_{g^{∗}ν}(f) = h_{ν}(f).

Proof. Since g ∈ Diff^{2}(M), there exists C > 1, such that ∀ x ∈ M, ϵ > 0,

B(g^{−1}x, C^{−1}ϵ) ⊂ g^{−1}B(x, ϵ) ⊂ B(g^{−1}x, Cϵ).

Hence

B_{n}(g^{−1}x, C^{−1}ϵ, f) ⊂ g^{−1}B_{n}(x, ϵ, f) ⊂ B_{n}(g^{−1}x, Cϵ, f).

So, by the definition of local entropy,

h_{g^{∗}ν}(x, f) = lim_{ϵ→0} lim sup_{n→∞} \frac{1}{n} log ν(g^{−1}B_{n}(x, ϵ, f))

= lim_{ϵ→0} lim sup_{n→∞} \frac{1}{n} log νB_{n}(g^{−1}x, ϵ, f) = h_{ν}(g^{−1}x, f).

This is the first result. Also we have

h_{g^{∗}ν}(f) = ∫ h_{g^{∗}ν}(x, f)d(g^{∗}ν)(x) = ∫ h_{ν}(g^{−1}x, f)dν(g^{−1}x) = ∫ h_{ν}(y, f)dν(y) = h_{ν}(f).

Since ∀ μ ∈ M(M, f, g), g^{∗}μ = μ, the following fact can be induced directly from Lemma 6.1.

Corollary 6.2. h_{μ}(x, f) is both f and g−invariant. Consequently, if μ is (f, g)−ergodic, then h_{μ}(x, f) = h_{μ}(f), μ − a.e. x ∈ Γ.

Suppose a measure μ is given. For δ ∈ (0, 1), we denote by N_{n}(ϵ, δ, f) the minimal number of (n, ϵ, f)−balls covering a set of the measure more than or equal to 1 − δ. A. Katok has proved (see [K]) that if μ is an ergodic measure for f, then for every δ ∈ (0, 1), h_{μ}(f) = lim_{ϵ→0} lim sup_{n→∞} \frac{1}{n} log N_{n}(ϵ, δ, f). Now we have same result for the measure μ which is

(f, g)−ergodic.

Proposition 6.3. If μ is an (f, g)−ergodic measure on M, then ∀ δ ∈ (0, 1),

h_{μ}(f) = lim_{ϵ→0} lim sup_{n→∞} \frac{1}{n} log N_{n}(ϵ, δ, f).

Proof. Take γ > 0 arbitrary.

Since

h_{μ}(x, f) = lim_{ϵ→0} lim inf_{n→∞} −\frac{1}{n} log μB_{n}(x, ϵ, f), μ − a.e. x ∈ M,

we can choose an ϵ > 0 and an n_{0} > 0 such that ∀ n > n_{0}, the set

A_{n} = {x ∈ M : μB_{n}(x, 2ϵ, f) ≤ exp −n(h_{μ}(f) − γ) }

has measure larger than or equal to \frac{1}{2}(1 + δ).

Let A′_{n} be a set that can be covered by N_{n}(ϵ, δ, f) (n, ϵ, f)−balls. A′_{n} ∩ A_{n} ≠ ∅ because μ(A′_{n} ∩ A_{n}) ≥ \frac{1}{2}(1 − δ). Thus N_{n}(ϵ, δ, f) (n, ϵ, f)−balls can cover A′_{n} ∩ A_{n}. On the other hand, to cover A′_{n} ∩ A_{n} by (n, 2ϵ, f)−balls centered at points in A′_{n} ∩ A_{n}, the number of such balls can not be less than \frac{1−δ}{2} exp n(h_{μ}(f) − γ). Since each (n, ϵ, f)−ball whose intersection with set A′_{n} ∩ A_{n} is nonempty must be contained in an (n, 2ϵ, f)−ball centered at a point in A′_{n} ∩ A_{n}, we have

N_{n}(ϵ, δ, f) ≥ \frac{1 − δ}{2} exp n(h_{μ}(f) − γ).

It is true for any n > n_{0}. Hence

lim sup_{n→∞} \frac{1}{n} log N_{n}(ϵ, δ, f) ≥ h_{μ}(f) − γ,

and therefore

lim_{ϵ→0} lim sup_{n→∞} \frac{1}{n} log N_{n}(ϵ, δ, f) ≥ h_{μ}(f).

The inequality in another direction can be obtained in similar way.

In our discussion we only need the inequality with the proved direction.

Suppose ρ : M → IR_{+} is a measurable function. Define an (n, ρ, f)−ball at x ∈ M by

B_{n}(x, ρ, f) = ⋂_{i=0}^{n} f^{−i}B(f^{i}x, ρ(f^{i}x) ).

Proposition 6.4. Let {ρ_{ϵ} : ϵ > 0} be a family of functions on M satisfying that

i) 0 < ρ_{ϵ} ≤ ϵ, ∀ x ∈ M, ii) ∫ log ρ_{ϵ}dμ < ∞, ∀ ϵ > 0, iii) ρ_{ϵ} monotonously decreases as ϵ → 0.

Then

h_{μ}(x, f) = lim_{ϵ→0} lim sup_{n→∞} −\frac{1}{n} log μB(x, ρ_{ϵ}, f), μ − a.e. x ∈ Γ.

Proof. Clearly B_{n}(x, ρ_{ϵ}, f) ⊂ B_{n}(x, ϵ, f). Hence

h_{μ}(x, f) ≤ lim_{ϵ→0} lim sup_{n→∞} −\frac{1}{n} log μB_{n}(x, ρ_{ϵ}, f).

By the results of Mañé [M], Brin and Katok [BK],

h_{μ}(f) ≥ ∫ lim_{ϵ→0} lim sup_{n→∞} −\frac{1}{n} log μB_{n}(x, ρ_{ϵ}, f)dμ ≥ ∫ h_{μ}(x, f)dμ = h_{μ}(f).

So the equalities hold everywhere and the result follows.

§7. The Subadditivity of Entropies of Commuting Diffeomorphisms

We devote this section to the proof of the first part of Theorem B. The method we use here is estimating the number of (n, ϵ)−balls which cover the set of measure more than or equal to 1 − δ for some constant δ ∈ (0, 1).

Let {Φ_{x} : x ∈ Γ} be a system of (γ, l)−charts for both f and g. Recall that B̃(ρ) denote the ball in IR^{m} centered at the origin of radius ρ, and the maps Φ_{x} and f̃_{x} are defined on B̃(l(x)^{−1}). Sometimes we will omit the subscript x.

For l > 0, let Γ_{l} = {x ∈ Γ : l(x) < l}.

Lemma 7.1. Suppose y ∈ Γ_{l} ∩ f^{−n}Γ_{l} and 0 < ϵe^{−2nγ} < l^{−1}. Let E ⊂ IR^{m}.
i) If E ⊂ B̃(ϵe^{−2nγ}), f̃_{y}^{n}E ⊂ B̃(ϵ),and f̃_{y}^{i}E ⊂ B̃(l(f^{i}y)^{−1}), ∀i = 1, ⋯, n, then
f̃_{y}^{i}E ⊂ B̃(ϵe^{−2(n−i)γ}), ∀i = 0, 1, ⋯, n.
ii) If E,f̃_{y}^{n}E ⊂ B̃(ϵe^{−2nγ}), and f̃_{y}^{i}E ⊂ B̃(l(f^{i}y)^{−1}), ∀i = 1, ⋯, n − 1, then
f̃_{y}^{i}E ⊂ B̃(ϵe^{−2 max{n−i,i}γ}), ∀i = 0, 1, ⋯, n.

Proof. i) Suppose there is v ∈ E with |f̃_{y}^{i}v| > ϵe^{−2(n−i)γ} for some i ∈ (0, n). If |f̃_{y}^{i}v| = |(f̃_{y}^{i}v)_{u}|, then by Corollary 4.3.i),

|f̃_{y}^{n}v| = |f̃^{n−i}(f̃_{y}^{i}v)| ≥ e^{(n−i)(λ_{+}(x,f)−2γ)}|f̃_{y}^{i}v| > e^{2(n−i)γ} · ϵe^{−2(n−i)γ} = ϵ.

If |f̃_{y}^{i}v| = |(f̃_{y}^{i}v)_{cs}|, then by Corollary 4.3.ii),

|v| = |f̃^{−i}(f̃_{y}^{i}v)| ≥ e^{−2iγ}|f̃_{y}^{i}v| > e^{−2iγ} · ϵe^{−2(n−i)γ} = ϵe^{−2nγ}.

Both cases are impossible.

ii) By part i), f̃_{y}^{i}E ⊂ B̃(ϵe^{−2(n−i)γ}). Again, using part i) on the set f̃_{y}^{n}E for f^{−1}, we get f̃_{y}^{i}E ⊂ B̃(ϵe^{−2iγ}). The proof is finished by combining the inclusions.

Take l > 0 such that μΓ_{l} > 0. For any x ∈ Γ_{l}, let τ_{f}(x) be the smallest positive integer k such that f^{k}x ∈ Γ_{l}. By Poincaré Recurrence Theorem for μ − a.e x ∈ Γ_{l}, τ_{f}(x) < ∞. We extend τ_{f}(x) to M by putting τ_{f}(x) = 0 if x ∈ M\Γ_{l}.

For any ϵ > 0, define a function ρ_{ϵ,f} : M → IR_{+} by

ρ_{ϵ,f}(x) = min{ϵ, l^{−2}e^{−(λ+γ)τ_{f} (x)}}.

Now log ρ_{ϵ,f} is integrable for any ϵ > 0 because ∫_{Γ_{l}} τ_{f}(x) ≤ 1. So the family of functions {ρ_{ϵ,f} : ϵ > 0 } satisfies the conditions in Proposition 6.4.

Lemma 7.2. Let 0 < ϵ < l^{−2} and ρ_{ϵ,f} be defined as above. If y ∈ Γ_{l} ∩ f^{−n}Γ_{l}, then
f̃_{y}^{i}Φ_{y}^{−1}[B_{n}(y, ρ_{ϵ,f}, f) ∩ B(y, ϵe^{−2nγ})] ⊂ B̃(ϵle^{−2(n−i)γ}), ∀i = 0, ⋯, n.

Proof. Denote

E = Φ_{y}^{−1}[B_{n}(y, ρ_{ϵ,f}, f) ∩ B(y, ϵe^{−2nγ})].

Take v ∈ E arbitrary and let z = Φ_{y}v.

Clearly d(y, z) ≤ ϵe^{−2nγ}, d(f^{n}y, f^{n}z) ≤ ρ_{ϵ,f}(f^{n}y) ≤ ϵ. Therefore by Proposition 4.1.vi),

|v| = |Φ_{y}^{−1}(z)| ≤ ϵle^{−2nγ}, |f̃_{y}^{n}v| = |Φ_{f^{n}y}^{−1}(f^{n}z)| ≤ ϵl.
For i = 1, ⋯, n, we have that
d(f^{i}y, f^{i}z) ≤ ρ_{ϵ,f}(f^{i}y) = min{ϵ, l^{−2}e^{−(λ+γ)τ_{f} (f^{i}y)}}.

If f^{i}y ∈ Γ_{l}, then d(f^{i}y, f^{i}z) ≤ l^{−2} ≤ l(f^{i}y)^{−2} and therefore |f̃_{y}^{i}v| = |Φ^{−1}(f^{i}z)| ≤ l(f^{i}y)^{−1}. Suppose f^{i}y ∉ Γ_{l}. Let j < i be the largest integer such that f^{j}y ∈ Γ_{l}, then τ_{f}(f^{j}y) ≥ i − j and d(f^{j}y, f^{j}z) ≤ l^{−2}e^{−(λ+γ)(i−j)} ≤ l(f^{j}y)^{−2}e^{−(λ+γ)(i−j)}. So we have

|f̃_{y}^{j}v| = |Φ_{f^{j}y}^{−1}(f^{j}z)| ≤ l(f^{j}y)^{−1}e^{−(λ+γ)(i−j)}.

Thus, by Proposition 4.1.v),

|f̃_{y}^{i}v| = |f̃^{i−j}(f̃_{y}^{j}v)| ≤ e^{(i−j)λ}|f̃_{y}^{j}v| ≤ l(f^{j}y)^{−1}e^{−(i−j)γ} ≤ l(f^{i}y)^{−1}.

Now we know that E satisfies the conditions of Lemma 7.1.i) and our result follows.

Lemma 7.3. Suppose 0 < ϵ < l^{−2}e^{−λ−2γ}. For y ∈ Γ_{l} ∩ f^{−n}Γ_{l} ∩ (fg)^{−n}Γ_{l}, if we set

Δ = B_{n}(y, ρ_{ϵ,f}, f) ∩ B(y, ϵe^{−2nγ}) ⋂(fg)^{−n}[B_{n}((fg)^{n}y, ρ_{ϵ,g^{−1}}, g^{−1}) ∩ B((fg)^{n}y, ϵe^{−2nγ})],

then

Δ ⊂ B_{n}(y, ϵlK, fg),

where K is as in Proposition 4.1.vi).

Proof. Let

E = Φ_{y}^{−1}Δ.

By Lemma 7.2, ∀ k = 0, 1, ⋯, n,

f̃_{y}^{k}E ⊂ f̃_{y}^{k}Φ_{y}^{−1}[B_{n}(y, ρ_{ϵ,f}, f) ∩ B(y, ϵe^{−2nγ})] ⊂ B̃(ϵle^{−2(n−k)γ}).

Similarly, since (f̃g)^{n}g̃^{−k}Φ_{y}^{−1} = g̃^{−k}Φ_{(fg)^{n}y}^{−1}(fg)^{n}, ∀ k = 0, 1, ⋯, n,

(f̃g)^{n}g̃_{y}^{−k}E ⊂ g̃^{−k}Φ_{(fg)^{n}y}^{−1}[B_{n}((fg)^{n}y, ρ_{ϵ,g^{−1}}, g^{−1}) ∩ B((fg)^{n}y, ϵe^{−2nγ})] ⊂ B̃(ϵle^{−2(n−k)γ}).

Using n − k instead of k, and noticing (f̃g)^{n}g̃_{y}^{−(n−k)} = (f̃g)^{k}f̃_{y}^{n−k}, we have

f̃_{y}^{n−k}E ⊂ B̃(ϵle^{−2kγ}), (f̃g)^{k}f̃_{y}^{n−k}E ⊂ B̃(ϵle^{−2kγ}), ∀ k = 0, 1, ⋯, n.  (∗)

Now we claim that ∀ k = 0, 1, ⋯, n,

(f̃g)^{i}f̃_{y}^{n−k}E ⊂ B̃(ϵle^{−2 max{k−i,i}γ}), ∀ i = 0, 1, ⋯, k.

For k = 0, the claim is true because by (∗) we have already had f̃_{y}^{n}E ⊂ B(ϵl). We suppose the claim is true for k − 1, i.e.

(f̃g)^{i}f̃_{y}^{n−k+1}E ⊂ B̃(ϵle^{−2 max{k−i−1,i}γ}), ∀ i = 0, 1, ⋯, k − 1.

Thus

(f̃g)^{i}f̃_{y}^{n−k}E ⊂ f̃^{−1}B̃(ϵle^{−2 max{k−i−1,i}γ}),  ∀ i = 0, 1, ⋯, k − 1.  (∗∗)

Since 0 < ϵ < l^{−2}e^{−λ−2γ}, ϵle^{−2 max{k−i−1,i}γ}e^{λ} < l^{−1}e^{−2 max{k−i,i}γ}. Also,

l((fg)^{i}f^{n−k}y) = l(f^{−(k−i)}g^{i}(f^{n}y)) ≤ l(f^{n}y)e^{(k−i)γ}e^{iγ} ≤ le^{kγ} ≤ le^{2 max{k−i,i}γ}.

Therefore ϵle^{−2 max{k−i−1,i}γ}e^{λ} < l((fg)^{i}f^{n−k}y)^{−1}. Now we can use Corollary 4.2 on the right hand side in (∗∗) and obtain that ∀ i = 0, 1, ⋯, k − 1,

(f̃g)^{i}f̃_{y}^{n−k}E ⊂ B̃(ϵle^{−2 max{k−i−1,i}γ}e^{λ}) ⊂ B̃(l((fg)^{i}f^{n−k}y)^{−1}).  (∗ ∗ ∗)

By Proposition 4.1, l(f^{±1}x) ≤ l(x)e^{γ}. Hence y ∈ Γ_{l} ∩ f^{−n}Γ_{l} implies that f^{n−k}y ∈ Γ_{le^{kγ}} ∩ f^{−k}Γ_{l} ⊂ Γ_{l′} ∩ f^{−k}Γ_{l′}, where l′ = le^{kγ}. Also, ϵle^{−2kγ} < (l′)^{−1}. Thus by (∗) and (∗ ∗ ∗) we can use Lemma 7.1.ii) on the set f̃_{y}^{n−k}E for diffeomorphism fg to obtain that

(f̃g)^{i}f̃_{y}^{n−k}E ⊂ B̃(ϵle^{−2 max{k−i,i}γ}), ∀ i = 0, 1, ⋯, k,

i.e. the claim is true for k.

By induction the claim is true for k = n. So

(f̃g)^{i}E ⊂ B̃(ϵle^{−2 max{n−i,i}γ}) ⊂ B̃(ϵl), ∀ i = 0, 1, ⋯, n.

Thus,

(fg)^{i}Φ_{y}E = Φ_{(fg)^{i}y}(f̃g)_{y}^{i}E ⊂ Φ_{(fg)^{i}y}B̃(ϵl) ⊂ B((fg)^{i}y, ϵlK), ∀ i = 0, 1, ⋯, n,

i.e.

Δ = Φ_{y}E ⊂ (fg)^{−i}B((fg)^{i}y, ϵlK), ∀ i = 0, 1, ⋯, n.

The result follows from the definition of B_{n}(y, ϵlK, fg).

The Proof of Theorem B (First Part).

Because the entropy map, which is defined on the set of invariant measures and has values in [0, +∞], is affine and any μ ∈ M(M, f, g) has (f, g)−ergodic decomposition, we only need prove the theorem if μ is an (f, g)−ergodic measure.

Take γ > 0 small sufficiently.

Take δ ∈ (0, 1).

Let {Φ_{x} : x ∈ Γ} be a system of (γ, l)−charts for both f and g. Let Γ_{l} = {x ∈ Γ : l(x) ≤ l\}. Fix an l > 1 such that μΓ_{l} > 1 − \frac{δ}{5}. We define two families of functions \{ρ_{ϵ,f}\} and {ρ_{ϵ,g^{−1}}} as above corresponding to f and g^{−1} respectively.

Let

A_{n,ϵ,γ}^{f} = {x ∈ Γ : μB_{k}(x, ρ_{ϵ,f}, f) ≥ exp −k(h_{μ}(f) + γ), ∀k ≥ n}.

Since ρ_{ϵ,f} is decreasing as ϵ → 0, by Proposition 6.4,

h_{μ}(f) = h_{μ}(x, f) ≥ lim sup_{n→∞} −\frac{1}{n} log μB_{n}(x, ρ_{ϵ,f}, f), μ − a.e. x ∈ Γ.

So ∀ ϵ > 0, μA_{n,ϵ,γ}^{f} → 1, as n → ∞. Then ∃ n_{f}(ϵ) > 0, such that ∀ n > n_{f}(ϵ),

μA_{n,ϵ,γ}^{f} ≥ 1 − \frac{δ}{5}.

By the definition of A_{n,ϵ,γ}^{f}, there are at most exp n(h_{μ}(f) + γ) disjoint (n, ρ_{ϵ,f}, f)−balls centered at points in A_{n,ϵ,γ}^{f}. So the same number of (n, 2ρ_{ϵ,f}, f)−balls centered at points in A_{n,ϵ,γ}^{f} can cover A_{n,ϵ,γ}^{f}. Suppose {B_{n}(x, 2ρ_{ϵ,f}, f) : x ∈ S_{f}} is a set of such balls. Then we have that

⋃_{x∈S_{f}} B_{n}(x, 2ρ_{ϵ,f}, f) ⊃ A_{n,ϵ,γ}^{f},

|S_{f}| ≤ exp n(h_{μ}(f) + γ).

Similarly, for diffeomorphism g^{−1}, ∀ ϵ > 0, ∃n_{g}(ϵ) > 0, such that ∀ n > n_{g}(ϵ), we have sets A_{n,ϵ,γ}^{g} and S_{g} satisfying the following.

μA_{n,ϵ,γ}^{g} ≥ 1 − \frac{δ}{5}.

⋃_{x∈S_{g}} B_{n}(x, 2ρ_{ϵ,g^{−1}}, g^{−1}) ⊃ A_{n,ϵ,γ}^{g},

|S_{g}| ≤ exp n(h_{μ}(g) + γ).

We denote by N(α) the minimal number of balls of radius α covering M. Since dim M = m, there exists a constant C > 0 such that N(α) < Cα^{−m}, ∀ α > 0. Let S_{0} be a set such that

⋃_{x∈S_{0}} B(x, 2ϵe^{−2nγ}) ⊃ M,

|S_{0}| = N(2ϵe^{−2nγ}) < C · (2ϵe^{−2nγ})^{−m}.
Now we take 0 < ϵ < \frac{1}{4}l^{−2}e^{−λ−2γ}. For each n > max\{n_{f}(ϵ), n_{g}(ϵ)\}, let
A_{n} = A_{n,ϵ,γ}^{f} ∩ A_{n,ϵ,γ}^{g} ∩ Γ_{l} ∩ f^{−n}Γ_{l} ∩ (fg)^{−n}Γ_{l}.

Clearly, μA_{n} ≥ 1 − δ. For any x_{f} ∈ S_{f}, x_{g} ∈ S_{g}, x′, x′′ ∈ S, if the intersection

A_{n} ∩ B_{n}(x_{f}, 2ρ_{ϵ,f}, f) ∩ B(x′, 2ϵe^{−2nγ}) ⋂(fg)^{−n}[B_{n}(x_{g}, 2ρ_{ϵ,g^{−1}}, g^{−1}) ∩ B(x′′, 2ϵe^{−2nγ})] is not empty, then for any y in it, the intersection is contained in the set

B_{n}(y, 4ρ_{ϵ,f}, f) ∩ B(y, 4ϵe^{−2nγ}) ⋂(fg)^{−n}[B_{n}((fg)^{n}y, 4ρ_{ϵ,g^{−1}}, g^{−1}) ∩ B((fg)^{n}y, 4ϵe^{−2nγ})].

Notice y ∈ A_{n} ⊂ Γ_{l} ∩ f^{−n}Γ_{l} ∩ (fg)^{−n}Γ_{l}, and 0 < 4ϵ < l^{−2}e^{−λ−2γ}, the result of Lemma 7.3 still holds if we use 4ϵ instead of ϵ. So the set is contained in B_{n}(y, 4ϵlK, fg).

There are at most

|S_{f}| · |S_{g}| · |S_{0}|^{2} ≤ exp n(h_{μ}(f) + γ) · exp n(h_{μ}(g) + γ) · C^{2} · (2ϵe^{−2nγ})^{−2m}
= C^{2} · (2ϵ)^{−2m} · exp n[h_{μ}(f) + h_{μ}(g) + (4m + 2)γ]

different such intersections. Each one is contained in an (n, 4ϵlK, fg)−ball. Since these intersections cover A_{n}, and μA_{n} > 1 − δ, we have

N_{n}(4ϵlK, δ, fg) ≤ C^{2} · (2ϵ)^{−2m} · exp n[h_{μ}(f) + h_{μ}(g) + (4m + 2)γ].

Thus,

h_{μ}(fg) = lim_{ϵ→0} lim sup_{n→∞} \frac{1}{n} log N_{n}(4ϵlK, δ, fg) ≤ h_{μ}(f) + h_{μ}(g) + (4m + 2)γ.

Since γ is arbitrary, we get

h_{μ}(fg) ≤ h_{μ}(f) + h_{μ}(g).

§8. A Partition Subordinating to W^{u}−foliations

In this section we assume that E^{u}(x, f) = E^{u}(x, g), μ − a.e. By Proposition 5.4, we have w^{u}(x, f) = w^{u}(x, g), μ − a.e. Therefore it can be written as w^{u}(x). We will construct a measurable partition η subordinating to w^{u} and increasing under the action of diffeomorphisms f and g, so that we can compute entropies of the diffeomorphisms and get the proof of the equality part in Theorem B.

In our discussion we also assume that μ is an (f, g)−ergodic measure on M.

A measurable partition ξ of M is a partition of M such that, up to a set of measure zero, the quotient space M/ξ is separated by a countable number of measurable sets(see[Ro]).

A measurable partition ξ of M is said to be subordinate to the w^{u}−foliation if for μ − a.e.x, ξ(x) ⊂ w^{u}(x) and ξ(x) contains a neighborhood of x open in the submanifold topology of w^{u}(x)(see [LY]).

For two partitions ξ_{1} and ξ_{2}, we say ξ_{1} refines ξ_{2}, denoted by ξ_{1} ≥ ξ_{2}, if ξ_{1}(x) ⊂ ξ_{2}(x), μ − a.e. We say that a partition ξ is f−increasing if fξ ≤ ξ, g− increasing is defined analogously. ξ is said to be (f, g)−increasing, if ξ is both f− and g−increasing.

Let B^{u} be the biggest sub−σ−algebra whose elements are unions of entire w^{u}−manifold.

Proposition 8.1. There is a measurable partition η on M with the following properties.

i) η is subordinate to w^{u}−foliation.

ii) η is (f, g)−increasing, i.e. fη ≤ η and gη ≤ η. iii) Both ⋁_{n=0}^{∞} f^{−n}η and ⋁_{k=0}^{∞} g^{−k}η are the partition into points (mod 0).

iv) The biggest σ−algebra contained in ⋂_{n=0}^{∞} ⋂_{k=0}^{∞} f^{−n}g^{−k}η is B^{u}.

To prove the proposition we introduce some lemmas.

Lemma 8.2. Let ρ_{0} > 0, 0 < a < 1 and ν be a finite non-negative Borel measure on [0, ρ_{0}]. Then the Lebesgue measure of the set

L_{a} = {ρ : 0 ≤ ρ ≤ ρ_{0}, ∑_{n=0}^{∞} ∑_{k=0}^{∞} ν([ρ − a^{n+k}, ρ + a^{n+k}]) < ∞}

is equal to ρ_{0}.

Proof. The idea is similar to the proof of Proposition 3.2 in [LS]. The modification is replacing set N_{a,n} in the proof by set

N_{a,n,k} = \{ρ : 0 ≤ ρ ≤ ρ_{0}, ν([ρ − a^{n+k}, ρ + a^{n+k}]) > \frac{ν([0, ρ])}{n^{2}k^{2}}\}.

Lemma 8.3. There is a constant b > 0, such that ∀x ∈ Γ, for Lebesgue almost every choice of ρ, 0 < ρ < l(x)^{−1}, μ − a.e.y ∈ M, the inequality

d(f^{−n}g^{−k}y, ∂B(x, ρ))e^{(n+k)(λ_{+}(x)−2γ)} < b^{−1}

holds at most for finite number of pairs (n, k), where λ_{+}(x) = min{λ_{+}(x, f), λ_{+}(x, g)}.

Proof. Take b > 0 such that d(z, ∂B(x, ρ)) ≤ τ implies |d(x, z) − ρ| ≤ bτ whenever 0 < τ < ρ ≤ l(x)^{−1}.

Define a non-negative Borel measure ν on IR by ν(A) = μ{y ∈ M : d(x, y) ∈ A} for any Borel set A ⊂ IR. Thus, by Lemma 8.2, we get, applied a = e^{−(λ_{+}(x)−2γ)}, that

P = {ρ : 0 ≤ ρ ≤ l(x)^{−1}, ∑_{n=0}^{∞} ∑_{k=0}^{∞} μ{y ∈ M : |d(x, y) − ρ| < e^{−(n+k)(λ_{+}(x)−2γ)}} < ∞}

has Lebesgue measure l(x)^{−1}. Since μ is f− and g−invariant,

P = {ρ : 0 ≤ ρ ≤ l(x)^{−1}, ∑_{n=0}^{∞} ∑_{k=0}^{∞} μ{y ∈ M : |d(x, f^{−n}g^{−k}y)−ρ| < e^{−(n+k)(λ_{+}(x)−2γ)}} < ∞}.

From the choice of b, ∀ρ ∈ P,

∑_{n=0}^{∞} ∑_{k=0}^{∞} μ\{y ∈ M : d(f^{−n}g^{−k}y, ∂B(x, ρ)) < \frac{1}{b}e^{−(n+k)(λ_{+}(x)−2γ)}\} < ∞.

By Borel-Cantelli Lemma, except for finite number of pairs (n, k),

μ\{y ∈ M : d(f^{−n}g^{−k}y, ∂B(x, ρ))e^{(n+k)(λ_{+}(x)−2γ)} < \frac{1}{b}\} = 0.

This finishes the proof of the lemma.

Define a metric d_{w}(·, ·) on M by

d_{w}(y, z) = \{ \begin{matrix}d_{w(x)}(y, z), & if y, z ∈ w^{u}(x) for some x ∈ M; \\ ∞, & otherwise, \end{matrix}

where d_{w(x)}(·, ·) is a metric on w^{u}(x) induced by the Riemannian structure on w^{u}(x). Clearly, d_{w}(·, ·) is independent of the choice of x ∈ M.

Lemma 8.4. For 0 < α < 1, z ∈ Γ, if y ∈ w_{α}^{u}(z), then ∀n, k ≥ 0,
d_{w}(f^{−n}g^{−k}y, f^{−n}g^{−k}z) ≤ 2Kl(z)d_{w}(y, z)e^{−(n+k)(λ_{+}(x)−2γ)}.

Proof. Take a system of (γ, l)−charts {Φ_{z} : z ∈ Γ} for both f and g. Let v = Φ_{z}^{−1}y, then v ∈ W_{α}^{u}(z). By Lemma 5.3,

|(f̃^{−n}g̃^{−k})_{z}v| ≤ |v|e^{−nλ_{+}(z,f)−kλ_{+}(z,g)−2(n+k)γ} ≤ |v|e^{−(n+k)(λ(z)−2γ)}.

By Proposition 4.1.vi) and Lemma 5.1, d_{w}(f^{−n}g^{−k}y, f^{−n}g^{−k}z) ≤ 2K|(f̃^{−n}g̃^{−k})_{z}v| and |v| ≤ l(z)d(y, z) ≤ l(z)d_{w}(y, z). Hence the result follows.

The Proof of Proposition 8.1. Take l > 0 with μΓ_{l} > 0. Fix 0 < α < 1. Take x ∈ Γ_{l} such that ∀ρ > 0, μ(B(x, ρ) ∩ Γ_{l}) > 0. Let

S(x, ρ) = ⋃_{y∈Γ_{l}∩B(x,ρ)} w_{α}^{u}(y) ∩ B(x, ρ).

Thus ∀ 0 < ρ < \frac{α}{4}l^{−1}, if two points z_{1}, z_{2} ∈ S(x, ρ) are not in the same local leaf w_{α}^{u}(y) ∩ B(x, ρ) for some y ∈ Γ_{l} ∩ B(x, ρ), then d_{w}(z_{1}, z_{2}) > 2ρ.

For any 0 < ρ < \frac{α}{4}l^{−1} we construct a partition ξ_{ρ} of M defined by all the sets

ξ_{ρ}(y) = \{ \begin{matrix}w_{α}^{u}(y) ∩ S(x, ρ), & if y ∈ S(x, ρ); \\ M\S(x, ρ), & otherwise, \end{matrix}

and then put

η_{ρ} = ⋁_{n=0}^{∞} ⋁_{k=0}^{∞} f^{n}g^{k}ξ_{ρ}.

Since μ( ⋃_{n=1}^{∞} ⋃_{k=1}^{∞} f^{n}g^{k}S(x, ρ)) = 1, it follows that μ − a.e.z ∈ Γ, η_{ρ}(z) ⊂ w^{u}(z). It is also

clear that η_{ρ} satisfies the properties ii)−iv) in the proposition. To complete the proof we have to choose a ρ > 0 such that μ − a.e.z, η_{ρ}(z) contains an open neighborhood of z in the submanifold topology of w^{u}(z).

Let

β_{ρ} = l(z)^{−1} · inf_{n≥0}\{α, \frac{1}{4}K^{−1}d(f^{−n}g^{−k}z, ∂B(x, ρ))e^{(n+k)(λ_{+}(z)−2γ)}, K^{−1}ρ\}.

By Lemma 8.3, there is a ρ > 0 such that β_{ρ}(z) > 0, μ − a.e. z ∈ M.

Now we only need prove that ∀z ∈ Γ, if y ∈ w^{u}(z), d_{w}(y, z) < β_{ρ}(z), then y ∈ η_{ρ}(z). In this circumstances, y ∈ w_{α}^{u}(z) and by Lemma 8.4, ∀ n, k ≥ 0,

d_{w}(f^{−n}g^{−k}y, f^{−n}g^{−k}z) ≤ 2Kl(z)e^{−(n+k)(λ_{+}(z)−2γ)} · β_{ρ}(z).

We have following cases to consider.

i) Both f^{−n}g^{−k}y and f^{−n}g^{−k}z belong to S(x, r). By the choice of β_{ρ}(z),

d_{w}(f^{−n}g^{−k}y, f^{−n}g^{−k}z) ≤ 2ρ.

So the two points in the same local leaf of S(x, ρ). Therefore ξ_{ρ}(f^{−n}g^{−k}y) = ξ_{ρ}(f^{−n}g^{−k}z).

ii) Neither f^{−n}g^{−k}y nor f^{−n}g^{−k}z belongs to S(x, r). By the construction of ξ_{ρ}, we have ξ_{ρ}(f^{−n}g^{−k}y) = ξ_{ρ}(f^{−n}g^{−k}y).

iii) One of f^{−n}g^{−k}y and f^{−n}g^{−k}z belongs to S(x, r) but the other does not. By the choice of β_{ρ}(z),

d_{w}(f^{−n}g^{−k}y, f^{−n}g^{−k}z) ≤ \frac{1}{2}d(f^{−n}g^{−k}z, ∂B(x, ρ)).

It is impossible.

Hence ∀ n, k ≥ 0, we always have ξ_{ρ}(f^{−n}g^{−k}y) = ξ_{ρ}(f^{−n}g^{−k}z). So y ∈ η_{ρ}(z).

We fix ρ > 0 such that η = ⋁_{n=0}^{∞} ⋁_{k=0}^{∞} f^{n}g^{k}ξ_{ρ} is the measurable partition of M satisfying

Proposition 8.1.

Lemma 8.5. Let η be a partition constructed as above. Then

h_{μ}(f, η) = h_{μ}(f), h_{μ}(g, η) = h_{μ}(g).

Proof. We only prove the first equality.

For any f−invariant measure ν, h_{ν}(f, η) = H_{ν}(η|fη) = ∫_{M} − log ν(η(x)|fη(x))dν(x), where ν(·|η(x)) is the system of conditional measures with respect to the σ−algebra generated by partition η. By (ΔΔ) in §1 we have h_{μ}(f, η) = ∫_{E(M,f)} h_{μ_{e}}(f, η)dπ(μ_{e}). Similarly, h_{ν}(f) = ∫_{M} h_{ν}(x, f)dν(x) implies h_{μ}(f) = ∫_{E(M,f)} h_{μ_{e}}(f)dπ(μ_{e}). Therefore we only need show that π − a.e. μ_{e}, h_{μ_{e}}(f, η) = h_{μ_{e}}(f).

For π − a.e. μ_{e}, μ_{e}( ⋃_{n=0}^{∞} ⋃_{k=0}^{∞} f^{n}g^{k}S(x, ρ)) = 1. Since μ_{e} ∈ E(M, f), we can take k =

k(μ_{e}) such that μ_{e}( ⋃_{n=0}^{∞} f^{n}g^{k}S(x, ρ)) = 1. Denote η_{e} = ⋁_{n=0}^{∞} f^{n}g^{k}ξ_{ρ}, then η_{e} is a partition

constructed as same as in the proof of Lemma 3.1.1 in [LY]. So we have h_{μ_{e}}(f, η_{e}) = h_{μ_{e}}(f) (see [LY], Corollary 5.3).

Now we need prove that h_{μ_{e}}(f, η_{e}) = h_{μ_{e}}(f, η). The argument is similar to the prove of Lemma 3.1.2 in [LY]. Notice η_{e} ≤ η and fη_{e} ≤ η_{e}, we have

\begin{aligned}h_{μ_{e}}(f, η) & = h_{μ_{e}}(f, η_{e} ∨ η) = h_{μ_{e}}(f, η_{e} ∨ f^{n}η) = H_{μ_{e}}(η_{e} ∨ f^{n}η|fη_{e} ∨ f^{n+1}η) \\ & = H_{μ_{e}}(η_{e}|fη_{e} ∨ f^{n+1}η) + H_{μ_{e}}(η_{e}|f^{−n}η_{e} ∨ fη). \end{aligned}

As n → ∞, the first term increases monotonously and tends to H_{μ_{e}}(η_{e}|fη_{e}) = h_{μ_{e}}(f, η_{e}), while the second term decreases and goes to 0. Since the formula is true for any n > 0, this finishes the proof.

§9. The Condition for Equality

We will prove the rest part of Theorem B in the section. First we consider a special case, for μ − a.e, x ∈ M, E^{u}(x, f) = E^{u}(x, g).

Proposition 9.1. If E^{u}(x, f) = E^{u}(x, g), μ − a.e. then h_{μ}(fg) = h_{μ}(f) + h_{μ}(g).

Proof. Take partition η as in the proof of Proposition 8.1. Since fη ≤ η and gη ≤ η, by Lemma 8.5, we have

\begin{aligned}h_{μ}(fg) ≥ h_{μ}(fg, η) & = H_{μ}(η|fgη) = H_{μ}(η ∨ gη|fgη) \\ = H_{μ}(gη|fgη) + H_{μ}(η|gη ∨ fgη) & = H_{μ}(η|fη) + H_{μ}(η|gη) = h_{μ}(f) + h_{μ}(g). \end{aligned}

Then the result follows from the first part of Theorem B.

Now we consider the general case, E^{u}(x, f)∩E^{s}(x, g) = {0} = E^{s}(x, f)∩E^{u}(x, g), μ−a.e.

Lemma 9.2. Suppose E^{u}(x, f) ∩ E^{s}(x, g) = {0} and E^{s}(x, f) ∩ E^{u}(x, g) = {0}, μ − a.e. Then there exists an N > 0, such that ∀ n > N,

h_{μ}(f^{n}g) ≥ h_{μ}(f^{n}), h_{μ}(fg^{n}) ≥ h_{μ}(g^{n}).

Proof. Take N_{f} > 0, such that

N_{f} · Δλ(x, f) > λ_{1}(x, g),

where Δλ(x, f) = min{λ_{i}(x, f) − λ_{i+1}(x, f) : i = 1, ⋯, s(x, f) − 1} is as in §4.

Let u(x, f) = min{i : λ_{i}(x, f) ≥ 0}, in other words, u(x, f) is defined such that λ_{u(x,f)}(x, f) is the smallest nonnegative exponent of f. u(x, g), u(x, f^{n}g) are understood in similar way.

For any n ≥ N_{f}, if we denote the Lyapunov exponents of f^{n}g by λ_{1}(x, f^{n}g) > ⋯ > λ_{r(x,f^{n}g)}(x, f^{n}g), then by Theorem A and the supposition of the lemma, ∀ 1 ≤ p ≤ u(x, f^{n}g), ∃ 1 ≤ i ≤ u(x, f), 1 ≤ j ≤ u(x, g) with

λ_{p}(x, f^{n}g) = nλ_{i}(x, f) + λ_{j}(x, g).

By the choice of N_{f} we know that nλ_{i_{1}}(x, f) + λ_{j_{1}}(x, g) > nλ_{i_{2}}(x, f) + λ_{j_{2}}(x, g) if and only

if i_{1} > i_{2}, or i_{1} = i_{2} and j_{1} > j_{2}. Thus the decomposition E^{u}(x, f^{n}g) = ⨁_{u=1}^{u(x,f^{n}g)} E_{p}(x, f^{n}g)

of unstable part in tangent space can be written as

E^{u}(x, f^{n}g) = ⨁_{i=1}^{u(x,f)} ⨁_{j=1}^{u(x,g)} E_{ij}(x),

where E_{ij}(x) = E_{p}(x, f^{n}g), if nλ_{i}(x, f) + λ_{j}(x, g) = λ_{p}(x, f^{n}g) for some 1 ≤ p ≤ u(x, f^{n}g), and otherwise E_{ij}(x) = {0}. The Ledrappier-Young’s formula relating entropy, exponents and dimensions is

h_{μ}(x, f) = ∑_{i=1}^{u(x,f)} λ_{i}(x, f)γ_{i}(x, f),

where γ_{i}(x, f) denotes a notion of fractional dimension defined as follows.

Let w^{(i)}(x, f) = \{y ∈ M : lim sup_{n→∞} \frac{1}{n} log d(f^{−n}x, f^{−n}y) ≤ −λ_{i}(x, f)\} for each i with

λ_{i}(x, f) > 0, which is a C^{2} immersed submanifold of M with dimension ∑_{s≤i} dim E_{s}(x, f), and

η^{(i)} be a partition subordinating to {w^{(i)}(x, f)}. Denote by B^{(i)}(x, ϵ) the ball in w^{(i)}(x, f) centered at x of radius ϵ in the distance induced by the Riemannian structure on w^{(i)}(x, f). For each i = 1, ⋯, u(x, f) with λ_{i}(x, f) > 0, define

δ_{i}(x, f) = lim_{ϵ→0} \frac{log μ(B^{(i)}(x, ϵ)|η^{(i)}(x))}{log ϵ},

where the limits in the right hand side exist μ− a.e. x ∈ M and are independent of the choice of η (See [LY], §7). And then let

γ_{i}(x, f) = δ_{i}(x, f) − δ_{i−1}(x, f), i = 1, ⋯, u(x, f),

where we regard δ_{0}(x, f) = 0, and δ_{u(x,f)}(x, f) as any fixed constant between δ_{u(x,f)−1}(x, f) and dim M if λ_{u(x,f)}(x, f) = 0. γ_{i}(x, g) and γ_{i}(x, f^{n}g) are defined similarly.

For any 1 ≤ i ≤ u(x, f), let p_{i} be the smallest number such that λ_{p_{i}}(x, f^{n}g) ≥ nλ_{i}(x, f), and let p_{0} = 0. It means that if p_{i} ≤ p < p_{i+1}, then λ_{p}(x, f^{n}g) = nλ_{i}(x, f) + λ_{j}(x, g) for some 1 ≤ j ≤ u(x, g). Thus w^{(i)}(x, f) = w^{(p_{i})}(x, f^{n}g) and δ_{i}(x, f) = δ_{p_{i}}(x, f^{n}g). So

γ_{i}(x, f) = δ_{p_{i}}(x, f^{n}g) − δ_{p_{i−1}}(x, f^{n}g) = ∑_{p=p_{i−1}+1}^{p_{i}} γ_{p}(x, f^{n}g)

except for i = u(x, f) as λ_{u(x,f)}(x, f) = 0. Put γ_{ij}(x) = γ_{p}(x, f^{n}g) if E_{ij}(x) = E_{p}(x, f^{n}g),

and γ_{ij}(x) = 0 if E_{ij}(x) = {0}. Above formula shows that γ_{i}(x, f) = ∑_{j=1}^{u(x,g)} γ_{ij}(x). Now the

Ledrappier-Young’s formula for diffeomorphism f^{n}g can be expressed as

h_{μ}(x, f^{n}g) = ∑_{p=1}^{u(x,f^{n}g)} λ_{p}(x, f^{n}g)γ_{p}(x, f^{n}g) = ∑_{i=1}^{u(x,f)} ∑_{j=1}^{u(x,g)}(nλ_{i}(x, f) + λ_{j}(x, g))γ_{ij}(x).

Therefore

h_{μ}(x, f^{n}g) ≥ ∑_{i=1}^{u(x,f)} ∑_{j=1}^{u(x,g)} nλ_{i}(x, f)γ_{ij}(x) = ∑_{i=1}^{u(x,f)} nλ_{i}(x, f)γ_{i}(x) = nh_{μ}(f).

Similarly we have N_{g} > 0, such that ∀ n > N_{g}, h_{μ}(fg^{n}) ≥ h_{μ}(g^{n}). Then N = max{N_{f}, N_{g}} is a required number.

The Proof of Theorem B (Second Part).

Notice for any n > 0, E^{u}(x, f^{n}g) = E^{u}(x, fg^{n}), μ − a.e. because of the supposition of the theorem. By Proposition 9.1 and Lemma 9.2, if n is large sufficiently, then

\begin{aligned}h_{μ}(fg) & = \frac{1}{n + 1}h_{μ}(f^{n}g · fg^{n}) = \frac{1}{n + 1}[h_{μ}(f^{n}g) + h_{μ}(fg^{n})] \\ & ≥ \frac{1}{n + 1}[h_{μ}(f^{n}) + h_{μ}(g^{n})] = \frac{n}{n + 1}[h_{μ}(f) + h_{μ}(g)]. \end{aligned}

Since n is arbitrary, h_{μ}(fg) = h_{μ}(f) + h_{μ}(g).

§10. Topological Entropy

By applying Theorem B, we can get similar result for topological entropies of commuting diffeomorphisms. We will use the relationship between topological entropies and measuretheoretic entropies, i.e. h(f) = max{h_{ν}(f) : ν ∈ M(M, f)}. Next proposition is in fact a generalization of Theorem C.

Proposition 10.1. Suppose M is a compact C^{∞} Riemannian manifold without boundary, f and g are commuting diffeomorphisms in Diff^{2}(M). If ∀ α > 0, ∃ 0 ≤ β ≤ α, such that the set

V_{β} = {ν ∈ M(M, fg) : h_{ν}(fg) ≥ h(fg) − β}

is compact in weak ^{∗} topology, then

h(fg) ≤ h(f) + h(g).

Proof. Since entropy map ν → h_{ν}(fg) is affine, V_{β} is a convex set. So V_{β} is a nonempty convex set which is compact in weak ^{∗} topology. By Lemma 6.1, f^{∗}V_{β} ⊂ V_{β}. So f^{∗} has a fixed point in V_{β}, i.e. V_{β} ∩ M(M, f) ≠ ∅.

Take μ ∈ V_{β} ∩ M(M, f), then μ ∈ M(M, f, fg). By Corollary 1.5, μ ∈ M(M, f, g). Hence h_{μ}(fg) ≤ h_{μ}(f) + h_{μ}(g) ≤ h(f) + h(g). Also, μ ∈ V_{β}, so h(fg) ≤ h_{μ}(fg) + β ≤ h(f) + h(g) + β. Since 0 ≤ β ≤ α and α is arbitrary, our result follows.

Now we consider two important cases, i.e. the entropy map of fg is upper semicontinuous and the set of measures with maximal entropy of fg is a finite dimensional simplex.

Corollary 10.2. Suppose the entropy map of fg is upper semi-continuous, i.e. ∀ν_{0} ∈ M(M, fg) and β > 0, there exists a neighborhood U of ν_{0} in M(M, fg) such that ν ∈ U implies h_{ν}(fg) ≤ h_{ν_{0}}(fg) + β. Then h(fg) ≤ h(f) + h(g).

Proof. Since entropy map ν → h_{ν}(fg) is upper semi-continuous, ∀ α ≥ 0, V_{α}, the preimage of [h(fg) − α, h(fg)] under the entropy map of fg, is compact.

Corollary 10.3. If the set of measures with maximal entropy of fg is a finite dimensional simplex, then h(fg) ≤ h(f) + h(g).

Proof. This is because the set V_{0} = {ν ∈ M(M, fg) : h_{ν}(fg) = h(fg)} is compact.

The Proof of Theorem C.

i) and ii) are from Corollary 10.2 plus [N] and [W] (Chapter 8) respectively, and iii) is Corollary 10.3.

We end this paper by a counterexample of two commuting homeomorphisms f and g with zero entropies on a smooth manifold M = S^{1} × S^{2} whose composition has positive entropy.

Example. Let S^{1} = {θ ∈ [0, 2π] : {0} = {2π} }. Take a homeomorphism α : S^{1} → S^{1}, such that α(0) = 0 and α(θ) ≤ θ, ∀ θ ∈ (0, 2π). Hence α has unique fixed point θ = 0.

Let S^{2} = {(r, τ) ∈ [0, 2] × [0, 2π] : {(r, 0)} = {(r, 2π)}, ∀r ∈ [0, 2]; {(r, τ)} = {(r, 0)}, ∀τ ∈ [0, 2π], r = 0, 2}. Take D = {(r, τ) ∈ S^{2} : 0 ≤ r ≤ 1}. Define a homeomorphism ψ : S^{2} → S^{2}, such that ψ(D) = D, restricted on D, ψ|_{D} has positive entropy, and ψ|_{S^{2}\D} = id|_{S^{2}\D}.

Define a continuous map β̃ : (0, 2π) × [0, 2] → [0, 2], such that ∀θ ∈ (0, 2π), β̃(θ, ·) = β̃_{θ}(·) : [0, 2] → [0, 2] is a homeomorphism, and for any r ∈ [0, 1], β̃_{θ}(r) ≤ θr, if θ ∈ [0, π]; β̃_{θ}(r) ≤ (2π − θ)r, if θ ∈ [π, 2π]. Then we use β̃_{θ} define a family of homeomorphisms β_{θ} on S^{2} by putting

β_{θ}(r, τ) = (β̃_{θ}(r), τ), ∀ (r, τ) ∈ S^{2}.
Now we take M = S^{1} × S^{2}. Define f, g : M → M by
f(θ, v) = (α(θ), β_{α(θ)}ψβ_{θ}^{−1}(v) ),
g(θ, v) = (α^{−1}(θ), β_{α^{−1}(θ)}β_{θ}^{−1}(v) ),

for any θ ∈ S^{1}\{0}, v = (r, τ) ∈ S^{2}, and

f(0, v) = (0, v) = g(0, v), ∀v ∈ S^{2}.

Clearly, both f and g are homeomorphisms under a suitable choice of β̃_{θ}.

For θ ∈ S^{1}\{0}, v ∈ S^{2},

fg(θ, v) = f(α^{−1}(θ), β_{α^{−1}(θ)}β_{θ}^{−1}(v) )
= (αα^{−1}(θ), β_{αα^{−1}(θ)}ψβ_{α^{−1}(θ)}^{−1}β_{α^{−1}(θ)}β_{θ}^{−1}(v) ) = (θ, β_{θ}ψβ_{θ}^{−1}(v) ),
gf(θ, v) = g(α(θ), β_{α(θ)}ψβ_{θ}^{−1}(v) )
= (α^{−1}α(θ), β_{α^{−1}α(θ)}β_{α(θ)}^{−1}β_{α(θ)}ψβ_{θ}^{−1}(v) ) = (θ, β_{θ}ψβ_{θ}^{−1}(v) ),

and

fg(0, v) = (0, v) = gf(0, v), ∀ v ∈ S^{2}.

So fg = gf holds on M.

The nonwandering set for f and g are Ω(f) = {(0, v) : v ∈ S^{2}} = Ω(g), and f|_{Ω(f)} = id = g|_{Ω(g)}. So h(f) = 0 = h(g). But for any θ ≠ 0, restrict to the set {(θ, v) : v ∈ S^{2}}, fg is conjugate to ψ. Therefore h(fg) ≥ h(ψ) ≥ 0.

Since the support of any f− or g−invariant measure μ must be contained in the set {(0, v) : v ∈ S^{2}}, h_{μ}(fg) should equal to zero if μ ∈ M(M, f, g). It means that the example does not violate Theorem B, though f, g are not diffeomorphisms.

Acknowledgment. I am very thankful to my adviser L.-S. Young for her helpful discussions and important suggestions.

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