A VOLUME PRESERVING DIFFEOMORPHISM WITH ESSENTIAL COEXISTENCE OF ZERO AND NONZERO LYAPUNOV EXPONENTS

HUYI HU, YAKOV PESIN, ANNA TALITSKAYA

Abstract. We show that there exists a C^{∞} volume preserving topologically transitive diffeomorphism of a compact smooth Riemannian manifold which is ergodic (indeed is Bernoulli) on an open and dense subset 𝒢 of not full volume and has zero Lyapunov exponent on the complement of 𝒢.

1. Introduction.

It is shown in [8, 12, 22, 23] that on any manifold ℳ and for any sufficiently large r one has what can be viewed as a discrete version of the classical KAM theory phenomenon in the volume preserving category – there are open sets of volume preserving C^{r} diffeomorphisms of ℳ all of which possess positive volume sets of codimension-1 invariant tori; on each such torus the diffeomorphism is C^{1} conjugate to a Diophantine translation; all of the Lyapunov exponents are zero on the invariant tori. It is expected that the set of invariant tori is surrounded by “chaotic sea”, i.e., outside this set the Lyapunov exponents are nonzero and the system has at most countably many ergodic components. It has since been an open problem to find out to what extend this picture is true.

A first step towards understanding this picture is to establish “essential” coexistence of completely chaotic and regular non-chaotic behavior for the class of volume preserving systems in the spirit of the results mentioned above. To this end in this paper we prove the following result.

Main Theorem. Given α > 0, there exists a compact smooth Riemannian manifold ℳ of dimension 5 and a C^{∞} diffeomorphism P : ℳ → ℳ preserving the Riemannian volume m such that

(1) ∥P − Id ∥_{C^{1}} ≤ α and P is homotopic to Id; (2) P is ergodic on an open and dense subset 𝒢 ⊂ ℳ and m(𝒢) < m(ℳ);

Key words and phrases. Hyperbolic measure, Lyapunov exponents, ergodicity, accessibility.

H. H. was partially supported by NSF grant DMS-0503870; Ya. P. was partially supported by the NSF grant DMS-1101165.

in particular, P is topologically transtive on ℳ; furthermore, P |𝒢 is a Bernoulli diffeomorphism;

(3) the Lyapunov exponents of P are nonzero for almost every x ∈ 𝒢;

(4) the complement 𝒢^{c} = ℳ \ 𝒢 has positive volume, P |𝒢^{c} = Id and the Lyapunov exponents of P on 𝒢^{c} are all zero.

In our example the set 𝒢^{c} is the direct product of a 3-dimensional smooth compact manifold and a Cantor set of positive volume in a two dimensional torus and thus has codimension two. By modifying our construction one can obtain a C^{∞} diffeomorphism P of a compact smooth Riemannian manifold of dimension 4, which is close to the identity map and has nonzero Lyapunov exponents on an open and dense set 𝒢 of positive but not full volume and zero exponents on its complement. The latter is the direct product of a 3dimensional smooth compact manifold and a circle and thus has codimension one and P has countably many ergodic components (see [6]).

Coexistence of elliptic islands and “chaotic sea” is one of the most interesting phenomena in dynamical systems but very few results are known in this direction. We shall briefly describe some results on the topic and we refer the reader to the survey article [7] where more information and references can be found.

Przytycki [19] and Liverani [16] studied a one-parameter family f_{a}, −ε ≤ a ≤ ε, of area preserving diffeomorphisms for which the map f_{0} lies on the boundary of the set of Anosov diffeomorphisms. This example demonstrates a route from uniform hyperbolicity (corresponding to −ε ≤ a < 0) to nonuniform hyperbolicity (corresponding to a = 0) and then to coexistence of regular and chaotic behavior, i.e., the appearance of an elliptic island (for 0 < a ≤ ε). It should be stressed that unlike the constructions in the above mentioned papers in our construction the set of points with nonzero Lyapunov exponents is everywhere dense in the manifold.

An example of a billiard dynamical system – the so-called “mushroom billiards” – with coexistence of “elliptic islands” and “chaotic sea” was constructed by Bunimovich in [3]. However, this case differs substantially from the smooth case due to the presence of singularities.

In [11], Fayad obtained a weaker version of our theorem: only some but not all Lyapunov exponents for P are zero on 𝒢^{c}. Ensuring that all Lyapunov exponents are zero is a substantially more difficult problem and we use completely different techniques than in [11] to make it happen. The matter is that if all Lyapunov exponents in 𝒢^{c} are zero, then a typical trajectory that originates in 𝒢 will spend long time in the vicinity of 𝒢^{c} where contraction and expansion rates are very small. This should be compensated by even longer periods of time that the trajectory should spend away from 𝒢^{c} thus gaining sufficient contraction and expansion and ensuring nonzero Lyapunov exponents.

Let us briefly outline our construction. It starts with a C^{∞} volume preserving diffeomorphism T of a compact smooth 5-dimensional manifold ℳ. The map T is close to and homotopic to the identity and indeed is the identity on an invariant compact subset of positive volume. On its complement 𝒢 the map T is partially hyperbolic with one-dimensional strongly stable, onedimensional strongly unstable subspaces and 3-dimensional center subspace along which dT acts as an isometry and hence has zero Lyapunov exponents. These subspaces are integrable to three transverse one-dimensional strongly stable, one-dimensional strongly unstable and 3-dimensional central invariant foliations of 𝒢. Since this set is open, partial hyperbolicity appears in its weaker pointwise form (see Section 2 for the definition of pointwise partial hyperbolicity).

Pointwise partially hyperbolic maps on compact manifolds were introduced in [5]. They have properties that are pretty much similar to those of uniformly partially hyperbolic systems: 1) strongly stable and unstable subspaces are integrable to continuous strongly stable and unstable foliations that are uniformly transverse to each other; 2) Lyapunov exponents along stable (unstable) subspaces are negative (positive); 3) any sufficiently small perturbation of a pointwise partially hyperbolic map is also pointwise partially hyperbolic. These properties fail to be true if one considers, as we do, pointwise partially hyperbolic maps on open subsets thus providing one of the major obstacles for our construction. To overcome this problem we only consider small perturbations of T that are gentle, i.e., they coincide with T outside a neighborhood of the Cantor set 𝒢^{c}. For those perturbations the above three properties hold. However, the final map P is not a gentle perturbation of T and additional arguments are needed to establish these properties for P.

Our next step is to perturb T gently to a C^{∞} volume preserving diffeomorphism Q, which is concentrated in an open set, which is “far away” from the Cantor set. We arrange this perturbation in such a way that the average Lyapunov exponents of Q in the central direction are positive for points in 𝒢 while the Lyapunov exponents on the complement 𝒢^{c} of 𝒢 are all zero. Our construction of the map Q is built upon some ideas from [21, 9, 14, 2, 10] but requires substantial modifications and new arguments due to nonuniform hyperbolicity of the map T. Note that the restriction Q|𝒢 is not ergodic.

Finally, we perturb Q to a C^{∞} volume preserving diffeomorphism P, which is pointwise partially hyperbolic on 𝒢 and, similarly, to the maps T and Q, possesses three transverse continuous one-dimensional strongly stable, onedimensional strongly unstable and 3-dimensional central invariant foliations.

In doing so we first construct a sequence of small perturbations P_{n} of Q such that each P_{n} coincides with T outside some open invariant subset 𝒰_{n} ⊂ 𝒢 (hence, P_{n} is a gentle perturbation of T) and has the accessibility property on 𝒰_{n} via its strongly stable and unstable foliations (i.e., any two points in 𝒰_{n} can be connected by a path that consists of pieces of strongly stable and unstable manifolds). The sets 𝒰_{n} are nested and exhaust 𝒢 and the sequence P_{n} converges to the desired map P. In constructing the maps P_{n} we use some techniques developed in [9, 14].

At the core of our argument lies the idea that accessibility will be achieved if we show that for a certain point z every point in its local central manifold V^{c}(z) is accessible from z. To this end we define a function from a cube in ℝ^{3} to V^{c}(z) such that every point in the image of the function is accessible from z. We shall show that this function is continuous, which guarantees that it is onto V^{c}(z_{0}), and hence every point in V^{c}(z) is accessible from z.

Although the map P is not a gentle perturbation of T (it coincides with T on the Cantor set only) we shall prove that P has the three properties described above. Furthermore, we show that P has the accessibility property on 𝒢 via its strongly stable and unstable foliations and that the average Lyapunov exponents of P |𝒢 in the central direction remain positive and in fact, central Lyapunov exponents are positive on a subset of positive volume. We then show that P |𝒢 is ergodic and indeed, is a Bernoulli diffeomorphism. To achieve this we extend the argument in [4] to the case of maps that are pointwise partially hyperbolic on open sets. This implies that P has four positive and one negative Lyapunov exponents on 𝒢 while the Lyapunov exponents on the Cantor set 𝒢^{c} are all zero.

In Section 2 we provide some background information and introduce some basic notations. In Section 3 we describe our construction of the map P and prove our result subject to two propositions. In the remaining sections we present the proofs of these propositions and other supporting statements.

2. Preliminaries

See [17, 1] for more details.

Let f be a diffeomorphism of a compact smooth Riemannian manifold ℳ and Λ ⊂ ℳ an f-invariant compact subset. The map f is said to be uniformly partially hyperbolic on Λ if for every x ∈ Λ the tangent space at x admits an invariant splitting

(2.1)  T_{x}ℳ = E^{s}(x) ⊕ E^{c}(x) ⊕ E^{u}(x)

into strongly stable E^{s}(x) = E_{f}^{s}(x), central E^{c}(x) = E_{f}^{c}(x), and strongly unstable E^{u}(x) = E_{f}^{u}(x) subspaces. More precisely, there are numbers 0 < λ < λ′ ≤ 1 ≤ μ′ < μ such that for every x ∈ Λ,

∥dfv∥ ≤ λ ∥v∥, v ∈ E^{s}(x),
λ′ ∥v∥ ≤ ∥dfv∥ ≤ μ′ ∥v∥, v ∈ E^{c}(x),
μ ∥v∥ ≤ ∥dfv∥,  v ∈ E^{u}(x).

Given x ∈ Λ, one can construct a strongly stable local manifold V^{s}(x) = V_{f}^{s}(x) and a strongly unstable local manifold V^{u}(x) = V_{f}^{u}(x) at x. This local manifolds have uniform size, i.e., there are numbers r > 0 and D > 0 such that for every x ∈ Λ there are smooth functions φ^{i} : B^{i}(r) → T_{x}ℳ, i = s or u (here B^{i}(r) ⊂ E^{i}(x) is the ball centered at zero of radius r) such that

φ(0) = 0, dφ(0) = 0, max{∥dφ(a)∥ : a ∈ B^{i}(r)} ≤ A,

and

V^{i}(x) = exp_{x}{(a, φ(a)) : a ∈ B^{i}(r)}.

We define the strongly stable and strongly unstable global manifolds at x by

\begin{aligned}W^{u}(x) & = W_{f}^{u}(x) = ⋃_{n≥0} f^{n}(V^{u}(f^{−n}(x))), \\ W^{s}(x) & = W_{f}^{s}(x) = ⋃_{n≥0} f^{−n}(V^{s}(f^{n}(x))). \end{aligned}

We denote by B(x, r) the ball centered at the point x of radius r. Further, we adopt the following notation: for a smooth submanifold V ⊂ ℳ and a point x ∈ V we denote by B_{V} (x, r) the ball in V centered at x of radius r (with respect to the intrinsic Riemannian metric). We also set

\begin{aligned}B^{s}(x, r) & = B_{f}^{s}(x, r) = B_{V^{s}(x)}(x, r), \\ B^{u}(x, r) & = B_{f}^{u}(x, r) = B_{V^{u}(x)}(x, r). \end{aligned}

In this paper we need a weaker property than uniform partial hyperbolicity. Let 𝒮 ⊂ ℳ be an f-invariant open subset. We say that f is pointwise partially hyperbolic on 𝒮 if for every x ∈ 𝒮 the tangent space at x admits an invariant splitting (2.1) and there are continuous positive functions λ(x) < λ′(x) ≤ 1 ≤ μ′(x) < μ(x), x ∈ 𝒮 such that

∥dfv∥ ≤ λ(x) ∥v∥, v ∈ E^{s}(x),
λ′(x) ∥v∥ ≤ ∥dfv∥ ≤ μ′(x) ∥v∥, v ∈ E^{c}(x),
μ(x) ∥v∥ ≤ ∥dfv∥,  v ∈ E^{u}(x).

We call a partition 𝒫 of 𝒮 a (δ, q)-foliation with smooth leaves or simply a foliation with smooth leaves if there exist continuous functions δ = δ(x) > 0, q = q(x) > 0, and an integer k > 0 such that for each x ∈ 𝒮:

(1) There exists a smooth immersed k-dimensional manifold W (x) containing x for which 𝒫(x) = W (x) where 𝒫(x) is the element of the partition 𝒫 containing x. The manifold W (x) is called the global leaf of the foliation at x; the connected component of the intersection W (x) ∩ B(x, δ(x)) that contains x is called the local leaf at x and is denoted by V (x).

(2) There exists a continuous map ϕ_{x} : B(x, q(x)) → C^{1}(D, ℳ) (where D is the unit ball) such that V (y) is the image of the map ϕ_{x}(y) : D → ℳ for each y ∈ B(x, q(x)); the number q(x) is called the size of V (x).

We say that a foliation with smooth leaves is absolutely continuous if for almost every x ∈ 𝒮 and almost every y ∈ B(x, q(x)) the conditional measure generated on V (y) by volume m (with respect to the partition of B(x, q(x)) by local leaves) is absolutely continuous with respect to the leaf volume m_{V (y)} on V (y).

The strongly stable and unstable global manifolds of a uniformly partially hyperbolic diffeomorphism form two (δ, q)-foliations of Λ with smooth leaves where δ and q are constants. These foliations are absolutely continuous and transverse at every point z ∈ Λ.

Let W_{1} and W_{2} be two foliations of 𝒮 with smooth leaves. Assume that these foliations are transverse at every point z ∈ 𝒮. Let also 𝒮_{1} ⊂ 𝒮 be an open subset. We say that the pair W_{1} and W_{2} has the accessibility property on 𝒮_{1} if any two points z, z′ ∈ 𝒮_{1} are accessible. This means that

(1) There exists a collection of points z_{1}, …, z_{n} ∈ 𝒮 such that x = z_{1}, y = z_{n} and z_{k} ∈ V_{i}(z_{k−1}) for i = 1 or 2 and k = 2, …, n.

(2) The points z_{k−1} and z_{k} can be connected by a smooth curve γ_{k} ⊂

V_{i}(z_{k−1}) in 𝒮 for i = 1 or 2 and k = 2, …, n. ^{1} The collection of such points z_{k} and curves γ_{k} is called the leaf-wise path connecting x and y. In particular, if W_{1} and W_{2} are the strongly stable and unstable foliations, then we say that f has the accessibility property and the leaf-wise path is called the (u, s)_{f}-path or simply (u, s)-path.

It may not be true in general that a diffeomorphism, which is pointwise partially hyperbolic on an open set 𝒮, has strongly stable and unstable local manifolds at every point in 𝒮. However, this is the case for all pointwise partially hyperbolic diffeomorphisms that we construct and in fact, their global strongly stable and unstable manifolds form two transverse foliations with smooth leaves.

More precisely, given a diffeomorphism f that is pointwise partially hyperbolic on an open set 𝒮, we call its small perturbation g in the C^{1} topology gentle if there exists an open set 𝒰 ⊂ 𝒮 such that 𝒰̄ ⊂ 𝒮, 𝒰 is invariant under both f and g and f|𝒰^{c} = g|𝒰^{c}.

1We stress that V_{i}(z_{k−1}) is the local leaf of W_{i} at z_{i}. In particular, the length of the curve γ_{k} (the leg of the path) does not exceed δ(z_{k−1}).

Theorem 2.1. Assume that the strongly stable and unstable subspaces E_{f}^{s} and E_{f}^{u} for f are integrable to continuous strongly stable and unstable foliations W_{f}^{s} and W_{f}^{u} respectively with smooth leaves and that these foliations are transverse. Then for any gentle perturbation g of f that is sufficiently close to f in the C^{1} topology the strongly stable and unstable subspaces E_{g}^{s} and E_{g}^{u} for g are integrable to continuous strongly stable and unstable foliations W_{g}^{s} and W_{g}^{u} respectively with smooth leaves and these foliations are transverse.

The proof of this theorem is based on two simple observations that: (1) a gentle perturbation changes the map f on an invariant subset on which f is uniformly partially hyperbolic and (2) the theorem is true for uniformly partially hyperbolic systems.

Furthermore, we call a diffeomorphism f that is pointwise partially hyperbolic on an open set 𝒮 dynamically coherent if the subbundles E^{cu} = E^{c} ⊕ E^{u}, E^{c}, and E^{cs} = E^{c} ⊕ E^{s} are integrable to continuous foliations with smooth leaves W^{cu}, W^{c} and W^{cs}, called respectively the center-unstable, center and center-stable foliations. Furthermore, the foliations W^{c} and W^{u} are subfoliations of W^{cu}, while W^{c} and W^{s} are subfoliations of W^{cs}.

The following result is an extension of the classical result in [13, 20]. It shows that dynamical coherence is a robust property within the class of gentle perturbations.

Theorem 2.2. Suppose that f is a diffeomorphism that is pointwise partially hyperbolic on an open set 𝒮. Assume that f possesses transverse strongly stable and unstable foliations with smooth leaves. Assume also that the center distribution is integrable to a smooth center foliation W^{c}. Then f is dynamically coherent. Moreover, any diffeomorphism that is close to f in the C^{1} topology and is a gentle perturbation of f is dynamically coherent.

Since both subbundle E^{cu} and E^{cs} vary continuously with the map, so does E^{c} and the corresponding center foliation W^{c}.

We denote by

λ(x, v) = lim sup_{n→∞} \frac{1}{n} log ∥df^{n}v∥

the Lyapunov exponent of a nonzero vector v at x ∈ ℳ and by λ_{i}(x) = λ_{i}(x, f), i = 1, …, dim ℳ, the values of the Lyapunov exponents at x. Note that the functions λ_{i}(x, f) are invariant. We assume that these values are ordered so that

λ_{1}(x, f) ≥ ⋯ ≥ λ_{dim ℳ}(x, f).

We also denote by

(2.2)  L_{k}(f) := ∫_{ℳ} ∑_{i=1}^{k} λ_{i}(x, f) dm(z),

where m is the Riemannian volume. We call this number the k-th average Lyapunov exponent of f.

Consider a volume preserving C^{2} diffeomorphisms f of a compact smooth manifold ℳ that is pointwise partially hyperbolic on an open set 𝒮. We say that f has positive central exponents if there is an invariant set 𝒜 ⊂ 𝒮 of positive volume such that for every x ∈ 𝒜 and every v ∈ E^{c}(x) the Lyapunov exponent λ(x, v) > 0. The following result plays an important role in the proof of our Main Theorem.

Theorem 2.3. Assume that the following conditions hold:

(1) f has strongly stable and unstable (δ, q)-foliations W^{s} and W^{u} where δ = δ(x) and q = q(x) are continuous functions on 𝒮;

(2) the foliations W^{s} and W^{u} are absolutely continuous;

(3) f has the accessibility property via the foliations W^{s} and W^{u}; more precisely, any two points z_{1}, z_{2} ∈ 𝒮 can be connected in 𝒮 via a W^{s} and W^{u} foliations;

(4) f has positive Lyapunov exponents in the strongly unstable directions and negative Lyapunov exponents in the strongly stable directions almost everywhere;

(5) f has positive central exponents. Then f has positive central exponents at almost every point x ∈ 𝒮, f|𝒮 is ergodic and indeed, is a Bernoulli diffeomorphism.

Proof. In the case when f is uniformly partially hyperbolic on the whole manifold ℳ, has positive central exponents and the accessibility property this theorem was proved in [4]. We shall show how to extend the argument presented there to our case.

Note that f is a C^{2} volume preserving diffeomorphism, with nonzero Lyapunov exponents on a set 𝒜 of positive volume. Hence, it has at most countably many ergodic components of positive volume in 𝒜. Each such component contains the set

A(x) = ⋃_{y∈V^{+}(x)} V^{s}(y),

where x is a density point of 𝒜 and V^{+}(x) is a center-unstable local manifold at x. Since the strongly stable foliation W^{s} is continuous, the set A(x) is open in 𝒜 and hence the set 𝒜 itself is open (mod 0). We shall show that the accessibility property of f in 𝒮 and absolute continuity of strongly stable and unstable foliations imply that the trajectory of almost every point in 𝒮 is dense. Clearly, this yields that 𝒜 = 𝒮 (mod 0) and that f|𝒮 is ergodic. Since for each n the map f^{n} satisfies the condition of the theorem, we conclude that f^{n}|𝒮 is ergodic implying that f|𝒮 is a Bernoulli diffeomorphism.

To this end, it suffices to show that if U is an open set then the orbit of almost every point enters U. To see this let us call a point good if it has a neighborhood in which the orbit of almost every point enters U. We wish to show that an arbitrary point p is good. Since f is accessible, there is a (u, s)-path [z_{0}, …, z_{k}] with z_{0} ∈ U and z_{k} = p. We shall show by induction on j that each point z_{j} is good. This is obvious for j = 0. Now suppose that z_{j} is good. Then z_{j} has a neighborhood N such that Orb(x) ∩ U ≠ ∅ for almost every x ∈ N. Let B be the subset of N consisting of points with this property that are also both forward and backward recurrent. It follows from the Poincaré recurrence theorem that B has full volume in N. If x ∈ B, any point y ∈ W^{s}(x) ∪ W^{u}(x) has the property that Orb(y) ∩ U ≠ ∅. The absolute continuity of the foliations W^{s} and W^{u} means that the set

⋃_{x∈B} W^{s}(x) ∪ W^{u}(x)

has full volume in the set

⋃_{x∈N} W^{s}(x) ∪ W^{u}(x).

The latter is a neighborhood of z_{j+1}. Hence z_{j+1} is good. □

3. Construction of the map P: Proof of Main Theorem

We describe a construction of the map P splitting it into several steps.

3.1. Step 1: A Special Flow T^{t}. Let A be an Anosov automorphism of the torus X = 𝕋^{2}. We denote by η_{A} the constant expanding rate of A along the unstable direction.

Consider the special flow T^{t} over A with a constant roof function. The flow acts on the the manifold

𝒩 = {(x, t) : x ∈ X, t ∈ [0, 1]}/ ∼,

where “∼” is the identification (x, 1) = (Ax, 0). We may choose the metric on 𝒩 in such a way that the expansion rate of T^{t} along the one-dimensional strongly unstable direction is tη_{A} at every point (x, t) ∈ 𝒩. For each t ≠ 0 the map T^{t} is uniformly partially hyperbolic with one-dimensional strongly stable E_{T^{t}}^{s}, one-dimensional strongly unstable E_{T^{t}}^{u} and one-dimensional center E_{T^{t}}^{c} subbundles (the latter is the direction of the flow). These subbundles are integrable to smooth strongly stable W_{T^{t}}^{s}, strongly unstable W_{T^{t}}^{u} and center W_{T^{t}}^{c} foliations of 𝒩.

3.2. Step 2: The Original Map T. Set Y = 𝕋^{2} and ℳ = 𝒩×Y. We endow ℳ with the product metric and denote by m its Riemannian volume. We also denote the fiber

(3.1)  𝒩_{y} = 𝒩 × {y}.

For our construction we choose: (A1) A Cantor set C ⊂ Y of positive area whose complement G = Y \ C is

an open connected subset. (A2) An open square G_{0} such that \overline{G}_{0} ⊂ G. (A3) A C^{∞} function κ : Y → ℝ satisfying: (1) κ(y) = 0 if y ∈ C and

κ(y) > 0 if y ∈ G; (2) | grad κ| < 1/4, and (3) κ(y) = κ_{0} for y ∈ U_{1}, where κ_{0} is a constant and U_{1} is a neighborhood of G_{0} whose choice is specified in Subsection 5.1.

The set 𝒢 in the Main Theorem is given by 𝒢 = 𝒩 × G and is open, dense and of positive but not full volume. We let 𝒢^{c} be the complement of 𝒢.

We define a map T : ℳ → ℳ by

T ((x, t), y) = (T^{κ(y)}(x, t), y),

where (x, t) ∈ 𝒩 and y ∈ Y. The proof of the following proposition is immediate.

Proposition 3.1. The map T is a C^{∞} volume preserving diffeomorphism of ℳ with the following properties:

(1) Given δ_{T} > 0, one can chose the function κ such that ∥T −Id ∥_{C^{1}} ≤ δ_{T}. Moreover, T is homotopic to Id.

(2) T preserves the fibers 𝒩_{y}.

(3) T is uniformly partially hyperbolic on any invariant subset 𝒩×A where A ⊂ G is compact. Moreover, T is dynamically coherent with the central foliation W_{T}^{c} = W_{T^{t}}^{c} × Y.

(4) T is pointwise partially hyperbolic on 𝒢 with one-dimensional strongly stable E_{T}^{s} (z), one-dimensional strongly unstable E_{T}^{u}(z) and 3-dimensional center E_{T}^{c} (z) subspaces. The subspaces E_{T}^{s} (z) and E_{T}^{u}(z) are integrable to strongly stable and unstable foliations W_{T}^{s}(z) and W_{T}^{u}(z) with smooth leaves. These foliations are uniformly transverse and their local leaves have uniform size. In addition, these foliations are absolutely continuous.

(5) T |𝒢^{c} = Id and dT_{z} = Id for any z ∈ 𝒢^{c}. In particular, the Lyapunov exponents of T |𝒢^{c} are all zero.

(6) for every z ∈ 𝒢 the Lyapunov exponents of T are as follows:

\begin{aligned}λ_{1}(z, T) = λ^{u}(z, T) & > 0 = λ_{2}(z, T) = λ_{3}(z, T) = λ_{4}(z, T) \\ & > λ_{5}(z, T) = λ^{s}(z, T), \end{aligned}

where λ^{u}(z, T) and λ^{s}(z, T) correspond to the directions E_{T^{t}}^{u} and E_{T^{t}}^{s} respectively and λ_{2}(z, T), λ_{3}(z, T) and λ_{4}(z, T) correspond to the direction of the flow and the Y -direction respectively. Moreover,

L_{1}(T) = L_{2}(T) = L_{3}(T) = L_{4}(T) > 0 and L_{5}(T) = 0,

where each i-th average Lyapunov exponents L_{i}(·) is given by (2.2).

3.3. Step 3: The Perturbation Q. We perturb the map T to a map Q such that it has one negative and four positive average Lyapunov exponents but is not necessarily ergodic. We then perturb Q to a map P which is ergodic on G and has all the desired properties.

Given z ∈ ℳ, we choose a local coordinate system (s, u, t, a, b) such that

(3.2) F^{s}(z) := ∂/∂s = E_{T}^{s} (z), F^{u}(z) := ∂/∂u = E_{T}^{u}(z), F^{t}(z) := ∂/∂t

are the strongly stable, strongly unstable and central (flow) directions of T respectively, and

(3.3)  F^{b}(z) := ∂/∂b, F^{a}(z) := ∂/∂a

are tangent to Y. We shall assume that in these coordinates the square G_{0} has the form

(3.4)  G_{0} = B_{F^{a}}(a_{0}, α_{0}) × B_{F^{b}}(b_{0}, α_{0})

for some (a_{0}, b_{0}) ∈ Y and α_{0} > 0.

The following statement describes some properties of the map Q; its proof is given in Section 4.

Proposition 3.2. Given δ_{Q} > 0, one can construct a C^{∞} volume preserving diffeomorphism Q : ℳ → ℳ which satisfies:

(1) ∥Q − T ∥_{C^{1}} ≤ δ_{Q} and Q is homotopic to Id;

(2) Q = T on the set 𝒩 × (Y \ G_{0}); in particular, Q preserves 𝒩_{y}-fibers if y ∉ G_{0} and is a gentle perturbation of T;

(3) Q is a gentle perturbation of T and satisfies Statements (3)–(5) of Proposition 3.1;

(4) for every z ∈ 𝒢 we have

E_{Q}^{utab}(z) = E_{T}^{utab}(z), det(dQ|E_{Q}^{utab}(z)) = det(dT |E_{T}^{utab}(z)).

(5) L_{1}(Q) < L_{2}(Q) < L_{3}(Q) < L_{4}(Q) = L_{4}(T) and L_{5}(Q) = 0 where L_{i}(·) is given by (2.2).

3.4. Step 4: The Final Perturbation P. Our next step is to perturb the map Q to a map P that is pointwise partially hyperbolic on the open set 𝒢. We shall ensure that P has two transverse strongly stable and unstable foliations W_{P}^{s} and W_{P}^{u} of 𝒢 and satisfies the accessibility property on this set via these foliations. We shall also show that P can be constructed in such a way that the Lyapunov exponents of P on 𝒢^{c} are all zero and that ∫_{ℳ} λ_{i}(z, P) dm > 0 for i = 1, 2, 3, 4.

In order to construct the map P we choose two sequences of open subsets U_{n}, Ũ_{n} ⊂ G, n = 1, 2, … such that (A4) G_{0} ⊂ Ũ_{1}; (A5) Ũ_{n} ⊂ \overline{Ũ}_{n} ⊂ U_{n} ⊂ \overline{U}_{n} ⊂ Ũ_{n+1} ⊂ G and ⋃_{n≥1} U_{n} = G. (A6) Ũ_{n} and U_{n} are connected sets for any n ≥ 1. We set

(3.5)  𝒰_{n} = 𝒩 × U_{n}, 𝒰̃_{n} = 𝒩 × Ũ_{n}.

We will construct a sequence of diffeomorphisms {P_{n}}, whose limit is the desired map P. The following statement is proven in Section 5. Proposition 3.3. Given a number δ_{P} > 0, one can find two sequences of positive numbers {δ_{n}} and {θ_{n}} with δ_{n} ≤ δ_{P} /2^{n} and δ_{n} ≤ d(C, U_{n})^{2} as well as a sequence of C^{∞} volume preserving diffeomorphisms P_{n} : ℳ → ℳ such that for n ≥ 1

(1) ||P_{n} − P_{n−1}||_{C^{n}} < δ_{n} and P_{n} is homotopic to Id;

(2) P_{n}(𝒰_{n}) = 𝒰_{n}, P_{n} = T on ℳ\𝒰_{n}, and P_{n} = P_{n−1} on 𝒰_{n−2}; in particular, P_{n} is a gentle perturbation of T;

(3) P_{n} is a gentle perturbation of T and satisfies Statements (3)–(5) of Proposition 3.1;

(4) for every z ∈ ℳ we have

E_{P_{n}}^{utab}(z) = E_{Q}^{utab}(z), det(dP_{n}|E_{P_{n}}^{utab}(z)) = det(dQ|E_{Q}^{utab}(z));

(5) for all z ∈ 𝒰_{j}, j = 1, …, n and i = u, s, c,

∠(E_{P_{n−1}}^{i}(z), E_{P_{n}}^{i}(z)) ≤ θ_{j}/2^{n−j};

(6) if the number δ_{Q} > 0 (see Proposition 3.2) is sufficiently small, then each map P_{n} is stably accessible in the following sense: let P^{♮} be a C^{2} volume preserving diffeomorphism of ℳ that is a gentle perturbation of T; assume that for all z ∈ 𝒰_{n} and i = u, s, c

∠(E_{P^{♮}}^{i}(z), E_{P_{n}}^{i}(z)) ≤ θ_{n};

then any two points z_{1}, z_{2} ∈ 𝒰̃_{n} are accessible via a (u, s)_{P^{♮}}-path in 𝒢; in particular, P_{n} has the accessibility property on 𝒰̃_{n}.

Statement (1) and (2) of this proposition implies that the limit P = lim_{n→∞} P_{n} exists. We shall show that the map P has all the desired properties.

3.5. Step 5: Proof of the Main Theorem. By Proposition 3.3 (1), we have for any k ≥ 1 and any n > k,

||P_{n} − P_{n−1}||_{C^{k}} ≤ ||P_{n} − P_{n−1}||_{C^{n}} < δ_{P} /2^{n}.

It follows that P_{n} converges to P in the C^{k} topology. Since k is arbitrary, P is a C^{∞} diffeomorphism. Clearly, P preserves volume and ∥P − Id ∥ ≤ δ if δ_{T}, δ_{Q} and δ_{P} are small enough. In addition, since P = P_{n+1} on 𝒰_{n}, by Proposition 3.3 (1), P is homotopic to Id on 𝒰_{n} for any n. The first statement of the Main Theorem follows.

By Proposition 3.3, each diffeomorphism P_{n} is pointwise partially hyperbolic on 𝒰 and uniformly partially hyperbolic on \overline{𝒰}_{n}. By Theorem A.1 in the Appendix, if the sequence δ_{n} decreases sufficiently fast, the limit diffeomorphism P is pointwise partially hyperbolic on 𝒰.

We now claim that the one-dimensional strongly stable E_{P}^{s} and unstable E_{P}^{u} subbundles are integrable to invariant strongly stable W_{P}^{s} and unstable W_{P}^{u} foliations with smooth leaves, which are transverse and absolutely continuous. Recall that the “start-up” map T has strongly stable and unstable local manifolds V_{T}^{s}(z) and V_{T}^{u}(z) respectively at each z ∈ 𝒰. Moreover, these local manifolds are of uniform size, say larger than a certain number 4r > 0.

By Proposition 3.3(3), P_{n}|𝒰_{n}^{c} = T |𝒰_{n}^{c}, and thus V_{P_{n}}^{ω}(z) = V_{T}^{ω}(z) for all z ∈ 𝒢\𝒰_{n}, ω = s, u. On the other hand, each P_{n} is a perturbation of P_{n−1} on the compact set \overline{𝒰}_{n} on which both P_{n} and P_{n−1} are uniformly partially hyperbolic if δ_{n} is sufficiently small. Furthermore, if r_{n} is the size of V_{P_{n}}^{ω}(z) for z ∈ 𝒰_{n}, one can arrange that r_{n}/r_{n−1} ≥ 2^{−1/2^{n}}, and thus by induction we obtain that the size of local manifolds for P_{n}|𝒰_{n} is bigger than r. Therefore, given z ∈ 𝒢, we obtain that the size of V_{P_{n}}^{ω}(z) has a lower bound r > 0, which is independent of z and n.

We can describe the local strongly stable manifold for P_{n} at a point z ∈ 𝒢 in the following way

V_{P_{n}}^{s}(z) = exp_{z}{(v, ψ_{P_{n}}^{s}(v)) : v ∈ B^{s}(0, r_{n})},

where B^{s}(0, r_{n}) ⊂ E_{P_{n}}^{s}(z) is the ball centered at origin of radius r_{n} and ψ_{P_{n}}^{s} : B^{s}(0, r_{n}) → E_{P_{n}}^{cu}(z) is a C^{1} map satisfying:

(1) ψ_{P_{n}}^{s}(0) = 0 and dψ_{P_{n}}^{s}(0) = 0.

(2) If the numbers δ_{n} and θ_{n} decay sufficiently fast then there are r > 0

and Δ > 0 such that r_{n} ≥ r and ∥ψ_{P_{n}}^{s}∥_{C^{1+α}} ≤ Δ for all n ≥ 0. This implies that z ∈ V_{P_{n}}^{s}(z) and T_{z}V_{P_{n}}^{s}(z) = E_{P_{n}}^{s}(z). Furthermore,

(1) P_{n}(V_{P_{n}}^{s}(z)) ⊂ V_{P_{n}}^{s}(P_{n}(z)).

(2) d(P_{n}(z), P_{n}(y)) ≤ λ̃(z)d(z, y) for each y ∈ V_{P_{n}}^{s}(z) and some continuous function λ̃(z) on 𝒢 for which 0 < λ(z) ≤ λ̃(z) < λ′(z) (where λ′(z) is the function in the definition of pointwise partial hyperbolicity).

The sequence of functions ψ_{P_{n}}^{s}(v), ∥v∥ ≤ r is compact in the C^{1} topology and hence, there is a subsequence ψ_{P_{n_{k}}}^{s} that converges to a C^{1} function ψ satisfying ψ(0) = 0, dψ(0) = 0 and ∥ψ∥_{C^{1}} ≤ Δ. Setting

(3.6)  V (z) = exp_{z}{(v, ψ(v)) : v ∈ B^{s}(0, r)}

we have that (1) z ∈ V (z) and T_{z}V (z) = E_{P}^{s} (z);

(2) P (V (z)) ⊂ V (P (z));

(3) d(P (z), P (y)) ≤ λ̃(z)d(z, y) for each y ∈ V (z). This implies that if m_{k} is any subsequence for which ψ_{P_{m_{k}}}^{s} converges in the C^{1} topology to a function ψ̃, then ψ̃ = ψ. Thus the formula (3.6) determines uniquely a local strongly stable manifold through z and the formula W (z) = ∪_{n≥0} P^{−n}(V (P^{n}(z)) defines the global strongly stable manifold through z. These manifolds form a continuous strongly stable foliation with smooth leaves for P. In a similar fashion we can obtain strongly unstable local manifolds and construct a strongly unstable foliation with smooth leaves for P. These two foliations are transverse at every point z ∈ 𝒢.

We shall now show that the Lyapunov exponent λ_{P}^{s} (z) in the direction E_{P}^{s} (z) is negative at almost every point z ∈ 𝒢. Indeed, let Z ⊂ 𝒢 be the set of points at which λ_{P}^{s} (z) = 0. If m(Z) > 0 then

0 = ∫_{Z} λ_{P}^{s} (z) dm = ∫_{Z} lim_{n→∞} \frac{1}{n} log ∏_{i=0}^{n−1} λ_{P} (P^{i}(z))

= lim_{n→∞} \frac{1}{n} ∫_{Z} ∑_{i=0}^{n−1} log λ_{P} (P^{i}(z)) dm(z)

= ∫_{Z} log λ_{P} (z) dm(z) < 0

(recall that λ_{P} (z) is the contraction coefficient along E_{P}^{s} (z)). This contradiction proves our claim. Similarly, one can prove that the Lyapunov exponent λ_{P}^{u} (z) in the direction E_{P}^{u}(z) is positive at almost every point z ∈ 𝒢.

Since P is nonuniformly partially hyperbolic on 𝒢, by Theorem 8.6.1 in[1], we obtain that its strongly stable and unstable foliations are absolutely continuous.

Our next step is to show that the map P has the accessibility property on 𝒢 via its invariant foliations W_{P}^{s} and W_{P}^{u}. Indeed, by Proposition 3.3 (6), for any n > k and any z ∈ 𝒰_{k}, i = s, u, c,

∠(E_{P_{n}}^{i}(z), E_{P_{k}}^{i}(z)) ≤ θ_{k}(1 − \frac{1}{2^{n−k}} ) < θ_{k}.

Taking the limit as n → ∞ yields for i = s, u, c and any z ∈ 𝒰_{k},

(3.7)  ∠(E_{P}^{i} (z), E_{P_{k}}^{i}(z)) ≤ θ_{k}.

Hence, by Proposition 3.3 (6), the map P has the accessibility property on 𝒰̃_{k}. Since k is arbitrary, we obtain that the map P has the accessibility property on 𝒢.

To prove that the map P has nonzero central Lyapunov exponents almost everywhere we let c = L_{4}(Q) − L_{3}(Q) > 0. By semicontinuity of L_{i} with respect to the map, we may take δ_{P} in Proposition 3.3 so small that L_{3}(P) < L_{3}(Q) + c/2. Note that by Proposition 3.3 (4), for all n ≥ 1,

L_{4}(P_{n}) = ∫_{𝒢} log | det(dP_{n}|E_{P_{n}}^{utab}(z))| dm
= ∫_{𝒢} log | det(dQ|E_{Q}^{utab}(z))| dm = L_{4}(Q).

Since P_{n} converges to P in the C^{1} topology, by Proposition 3.3 (4), we have that L_{4}(P_{n}) → L_{4}(P) as n → ∞ and hence L_{4}(P) = L_{4}(Q). It follows that L_{4}(P) − L_{3}(P) ≥ c/2 > 0. Therefore,

∫_{𝒢} λ_{4}(z, P)dm(z) ≥ c/2 > 0.

It follows that there is a subset 𝒜 ⊂ 𝒢 of positive volume such that λ_{4}(z) > 0 for every z ∈ 𝒜. Hence, λ_{2}(z) ≥ λ_{3}(z) ≥ λ_{4}(z) > 0. Thus the map P has positive central exponents at every point in a set of positive volume. Since P is volume preserving, the total sum of the Lyapunov exponents is zero at every point. Therefore, λ_{5}(z, P) < 0 at every point in 𝒜. Since P has the accessibility property and its strongly stable and unstable foliations are absolutely continuous, by Theorem 2.3, we obtain that P has positive central exponents at almost every point in 𝒢, P |𝒢 is ergodic and indeed, is a Bernoulli diffeomorphism.

It follows from Proposition 3.3 (3) and the fact that δ_{n} ≤ d(C, U_{n})^{2}, that P = Id on the set 𝒩 × C and that dP_{z} = Id for all z ∈ 𝒩 × C. In other words, all Lyapunov exponents at every point in the set 𝒩 × C are zero. Since this set has positive volume this completes the proof of the Main Theorem.

4. Construction of the map Q: Proof of Proposition 3.2

We use an approach which is similar to the one in [14] and obtain Q as a result of three consecutive perturbations. First, we perturb the map T to a diffeomorphism S via a gentle perturbation h_{S} so that S = h_{S} ∘ T preserves the fibers 𝒩_{y}, y ∈ G and has two positive average Lyapunov exponents in the E_{T}^{ut} subbundle, i.e, L_{1}(S) < L_{2}(S) (see Lemma 4.1). Next, we perturb S to a diffeomorphism R via a gentle perturbation h_{R} so that R = h_{R} ∘ S has three positive average Lyapunov exponents, i.e., L_{1}(R) < L_{2}(R) < L_{3}(R) (see Lemma 4.2). Finally, we obtain the desired map Q as a perturbation of R via a gentle perturbation h_{Q} so that Q = h_{Q} ∘ R satisfies

L_{1}(Q) < L_{2}(Q) < L_{3}(Q) < L_{4}(Q)

(see Lemma 4.6), or equivalently, ∫_{ℳ} λ_{4}(z, Q) dm(z) > 0.

Given δ > 0 and k = S, R, Q, the perturbations h_{k} are concentrated on pairwise disjoint small open subsets Ω_{k} ⊂ 𝒢_{0} such that ∥h_{k} − Id ∥_{C^{1}} ≤ δ and h_{k} = Id outside Ω_{k}. It follows that Q = T outside Ω_{S} ⋃ Ω_{R} ⋃ Ω_{Q}.

To effect our construction we choose periodic points q, p^{t}, p^{a} and p^{b} of the Anosov automorphism A, which are close to each other and whose orbits are pairwise disjoint. Let V_{A}^{s}(q), V_{A}^{u}(q), V_{A}^{s}(p^{i}) and V_{A}^{u}(p^{i}), i = t, a, b be stable and unstable local manifolds at these periodic points. We may assume that each intersection V_{A}^{u}(q) ∩ V_{A}^{s}(p^{i}) and V_{A}^{u}(p^{i}) ∩ V_{A}^{s}(q) consists of exactly one point, which we denote by [q, p^{i}] and [p^{i}, q] respectively. Consider the closed quadrilateral path with the collection of points q, [q, p^{i}], p^{i}, [p^{i}, q] and q, and let

γ(q) = V_{A}^{u}(q) ∪ V_{A}^{s}(q), γ(p^{i}) = V_{A}^{u}(p^{i}) ∪ V_{A}^{s}(p^{i}).

Given positive numbers ν and σ whose choice will be specified later (see (4.4)), we set for i = t, a, b,

\begin{matrix}Ω^{i}(ν) = ( ⋃_{t∈[0,τ(p^{i})]} B_{𝒩}(T^{t}(p^{i}, 0), ν)) × G, \\ Ω̂^{i}(σ) = ( ⋃_{(x,t)∈(γ(q)×[0,τ(q)])∪(γ(p^{i})×[0,τ(p^{i})])} B_{𝒩}((x, t), σ)) × G, \\ Ω(ν, σ) = ( ⋃_{i=t,a,b} Ω^{i}(ν)) ∪ ( ⋃_{i=t,a,b} Ω̂^{i}(σ)), \end{matrix} ; (4.1)

where τ(q) and τ(p^{i}) are the periods of q and p^{i} and B_{𝒩}((x, t)), r) is the ball in 𝒩 of radius r centered at the point (x, t). Finally, we set

(4.2)  Ω_{0}(ν, σ) = Ω(ν, σ) ∩ 𝒢_{0}

(recall that G_{0} is defined in (A2) and is in the form of (3.4)).

Given δ_{Q} > 0, choose the number θ > 0 according to Sublemma 4.5 below and an integer k_{0} > 0 such that

π/2k_{0} < θ. (4.3)

Now choose positive numbers ν and σ to ensure that the volume of the set Ω_{0}(ν, σ) is so small that

20k_{0}m(Ω_{0}(ν, σ)) < 1. (4.4)

4.1. Construction of the map S.

We obtain the map S as a small perturbation of the map T via a perturbation h_{S}, which is a small rotation in the E_{T}^{ut} subbundle at every point of a small subset of 𝒢_{0} = 𝒩 × G_{0}. This approach is an elaboration of the approach developed in [21, 9] for some uniformly partially hyperbolic systems.

To this end we observe that by the construction of the map T for every z ∈ 𝒢_{0} the expansion rate in the E_{T}^{u}-direction at z, |dT |E_{T}^{u}|, is a constant. We denote this constant by η. Choose a C^{∞} function ψ = ψ(r) : ℝ^{+} → ℝ^{+} such that

(1) ψ(r) = ψ_{0} > 0 if r ∈ [0, 0.9];

(2) ψ(r) > 0 if r ∈ [0, 1) and ψ(r) = 0 if r ≥ 1;

(3) ∥ψ∥_{C^{1}} ≤ 1. Given N_{0} ≥ 20k_{0}, choose a point (x_{0}, t_{0}) ∈ 𝒩 and a number ϵ_{1} > 0 such that

B_{𝒩}((x_{0}, t_{0}), 2ϵ_{1}) ∩ Proj_{𝒩}(Ω_{0}) = ∅,
f^{−kκ_{0}}(B_{𝒩}((x_{0}, t_{0}), 2ϵ_{1})) ∩ B_{𝒩}((x_{0}, t_{0}), 2ϵ_{1}) = ∅, k = 1, …, N_{0},

where Proj_{𝒩} is the projection onto 𝒩, i.e., Proj_{𝒩}(x, t, y) = (x, t) and κ_{0} is defined by (A3) (see Subsection 3.2). Set

(4.5)  Ω_{S} = B_{𝒩}((x_{0}, t_{0}), ϵ_{1}) × G_{0}.

Our choice of ϵ_{1} guarantees that Ω_{S} ∩ Ω_{0} = ∅ and for k = 1, …, N_{0},

(4.6)  T^{−k}(Ω_{S}) ∩ Ω_{S} = ∅.

To define the desired map h_{S} we switch from the coordinate system (s, u, t, a, b) (see (3.2) and (3.3)) in Ω_{S} to the cylindrical coordinate system (r, θ, s, a, b) originated at z_{0} = (x_{0}, t_{0}, a_{0}, b_{0}), where u = r cos θ and t = r sin θ.

Given τ > 0, define the map h_{S} = h_{S,τ} on Ω_{S} as a small rotation in the (u, t)-subspace. More precisely, we set

(4.7) h_{S}(r, θ, s, a, b) = (r, θ + τα_{0}^{2}ϵ_{1}^{2}ψ(\frac{r}{ϵ_{1}} )ψ(\frac{|s|}{ϵ_{1}})ψ(\frac{|b|}{α_{0}})ψ(\frac{|a|}{α_{0}}), s, a, b)

(here α_{0} is defined in (3.4)). We extend the map h_{S} = h_{S,τ} to the whole manifold ℳ by letting it to be the identity outside of Ω_{S}. It is easy to see that h_{S} is a C^{∞} volume preserving diffeomorphism satisfying:

(1) ∥h_{S,τ} − Id ∥_{C^{1}} → 0 as τ → 0;

(2) dh_{S} preserves E_{T}^{ut} bundle;

(3) det(dh_{S}|E_{T}^{ut}(z)) = 1 for any z ∈ ℳ. We define the map S = S_{τ} = T ∘ h_{S,τ} and we set

(4.8)  α_{1} = 0.9α_{0}, G_{1} = B_{F^{a}}(a_{0}, α_{1}) × B_{F^{b}}(b_{0}, α_{1}).

The following statement describes some properties of the map S.

Lemma 4.1. Given δ_{Q} > 0, there exist τ > 0 such that the map S = S_{τ} is a C^{∞} diffeomorphism with the following properties:

(1) ∥S − T ∥_{C^{1}} ≤ δ_{Q} and S is homotopic to Id;

(2) S = T on the sets 𝒩 × (Y \ G_{0}) and Ω_{0}; in particular, S is a gentle perturbation of T;

(3) S satisfies Statements (3)–(5) of Proposition 3.1;

(4) for every z ∈ ℳ,

E_{S}^{ut}(z) = E_{T}^{ut}(z), det(dS|E_{S}^{ut}(z)) = det(dT |E_{T}^{ut}(z));

(5) for any y_{1}, y_{2} ∈ G_{1},

Proj_{𝒩}(S(x, t, y_{1})) = Proj_{𝒩}(S(x, t, y_{2}));

(6) L_{1}(S) < L_{1}(T) and hence,

L_{1}(S) < L_{2}(S) = L_{3}(S) = L_{4}(S) = L_{4}(T), L_{5}(S) = 0;

(7) there exist a number λ_{S} and a set Π_{S} = Proj_{𝒩}(Π_{S}) × G_{1} such that

m(Π_{S}) ≥ 20k_{0}m(Π_{S} ∩ Ω_{S}) > 0,

and for any z ∈ Π_{S} the map S has two positive Lyapunov exponents λ_{1}(z, S) > λ_{2}(z, S) ≥ λ_{S} along the E_{S}^{ut} = E_{T}^{ut} subbundle.

Proof. Statements (1)–(5) follow easily from the construction of the map h_{S}. In particular, S is dynamically coherent in view of Theorem 2.2. It remains to prove Statements (6) and (7).

We prove that there exists τ_{0} > 0 such that for any τ ∈ (0, τ_{0}],

(4.9)  L_{1}(S_{τ}|𝒢_{0}) < L_{1}(T |𝒢_{0}).

Since on the complement of 𝒢_{0} we have S = T, this implies that L_{1}(S) < L_{1}(T).

We outline the proof of (4.9) referring the reader to the proof of Proposition 5.1 in [9] for details (see also [1]). Since E_{S_{τ}}^{u} (z) is one-dimensional, it is easy to see that

L_{1}(S_{τ}|𝒢_{0}) = ∫_{𝒢_{0}} λ_{1}(z, S_{τ}) dm(z) = ∫_{𝒢_{0}} log |dS_{τ}(z)|E_{S_{τ}}^{u} (z)| dm(z).

Since the perturbation h_{S} = h_{S,τ} preserves the E_{T}^{ut} subbundle, we can write

dh_{S,τ}|E_{T}^{ut}(z) = ( \begin{matrix}A(τ, z) & B(τ, z) \\ C(τ, z) & D(τ, z) \end{matrix} ) ,

where

A = A(τ, z) = 1 − τrρ̃_{r} sin θ cos θ − \frac{τ^{2}ρ̃^{2}}{2} − τ^{2}rρ̃ρ̃_{r} cos^{2} θ + O(τ^{3}),
B = B(τ, z) = −τρ̃ − τrρ̃_{r} sin^{2} θ − τ^{2}rρ̃ρ̃_{r} sin θ cos θ + O(τ^{3}),
C = C(τ, z) = τρ̃ + τrρ̃_{r} cos^{2} θ − τ^{2}rρ̃ρ̃_{r} sin θ cos θ + O(τ^{3}),
D = D(τ, z) = 1 + τrρ̃_{r} sin θ cos θ − \frac{τ^{2}ρ̃^{2}}{2} − τ^{2}rρ̃ρ̃_{r} sin^{2} θ + O(τ^{3}),

and

ρ̃(r, s, a, b) = α_{0}^{2}ϵ_{1}^{2}ψ(\frac{r}{ϵ_{1}} )ψ(\frac{|s|}{ϵ_{1}})ψ(\frac{|b|}{α_{0}})ψ(\frac{|a|}{α_{0}}).

Recall that the expanding rate η = η_{A} of dT along E_{z}^{u}(T) is constant for all z ∈ 𝒢_{0}. By the choice of the coordinate systems, we can write

dT |E_{T}^{ut}(z) = ( \begin{matrix}η & 0 \\ 0 & 1 \end{matrix} ) .

Since dS_{τ} = dT ∘ dh_{S,τ}, we have

dS_{τ}(z)|E_{S_{τ}}^{ut} (z) = (\begin{matrix}ηA(τ, z) & ηB(τ, z) \\ C(τ, z) & D(τ, z) \end{matrix}) .

Denote by e_{τ}(z) the unique number such that the vector v_{τ}(z) = (1, e_{τ}(z))^{∗} ∈ E_{S_{τ}}^{u} (z), where ^{∗} denote the transpose of the vector. Repeating the arguments in the proof of Lemma B.7 in [9], one can show that

L_{1}(S_{τ}|𝒢_{0})=∫_{𝒢_{0}} log η dm(z) − ∫_{𝒢_{0}} log[D(τ, z) − ηB(τ, z)e_{τ}(S_{τ0}(z))] dm(z).

Using this formula and applying the same arguments as in the proof of Lemmas B.8 and B.9 in [9] one can show that

\frac{dL_{1}(S_{τ}|𝒢_{0})}{dτ}|_{τ=0} = ∫_{𝒢_{0}} D′_{τ}dm(z) = 0

and

\frac{d^{2}L_{1}(S_{τ}|𝒢_{0})}{dτ^{2}} |_{τ=0} = ∫_{𝒢_{0}} [(D′_{τ})^{2} − D′′_{ττ} + 2ηB′_{τ} \frac{∂e_{τ}(z)}{∂τ}(S_{τ}(z))]_{τ=0} dm(z) < 0.

It follows that there exists τ_{0} > 0 such that (4.9) hold for any τ ∈ (0, τ_{0}]. Therefore, L_{1}(S_{τ}) < L_{1}(T).

Note that for any y ∈ Y the fibers 𝒩_{y} are S_{τ}-invariant and that the subbundles E_{T}^{utab}, E_{T}^{uta} and E_{T}^{ut} are preserved by the perturbation h_{S}. Furthermore, since det(dh_{S,τ}|E_{T}^{ut}(z)) = 1, we have for i = ut, uta, utab,

det(dS_{τ}|E_{T}^{i}) = det(dT |E_{T}^{i}).

Hence, the three smallest Lyapunov exponents remain unchanged and so does the sum of the two largest ones. This implies that L_{i}(S_{τ}) = L_{i}(T) for i = 3, 4, 5 and hence,

L_{1}(S_{τ}) < L_{2}(S_{τ}) = L_{3}(S_{τ}) = L_{4}(S_{τ}) = L_{4}(T)

and L_{5}(S_{τ}) = 0. This proves Statement (6) of the lemma.

To prove Statement (7) we first notice that for any y ∈ G_{1}, the arguments similar to the above ones yield

\frac{dL_{1}(S_{τ}|𝒩_{y})}{dτ}|_{τ=0} = 0, \frac{d^{2}L_{1}(S_{τ}|𝒩_{y})}{dτ^{2}} |_{τ=0} < 0.

It follows that if τ_{0} > 0 is small enough, then L_{1}(S_{τ}|𝒩_{y}) < L_{1}(T |𝒩_{y}) for any τ ∈ (0, τ_{0}]. Let us fix such a τ. There is a subset of 𝒩_{y} on which S_{τ} has two positive Lyapunov exponents λ_{1}(z, S_{τ}) > λ_{2}(z, S_{τ}) > 0. Given λ_{S} > 0, consider the level set Π_{S}(y) = {z ∈ 𝒩_{y} : λ_{2}(z, S_{τ}) ≥ λ_{S}}. If λ_{S} is sufficiently small this set has positive volume. Set Π_{S} = Π_{S}(y) × G_{1}, where the set G_{1} is defined by (4.8). Clearly, Π_{S} is invariant under S_{τ}. Since N_{0} ≥ 20k_{0}, we obtain by (4.4) that 20k_{0}m(Π_{S} ∩ Ω_{S}) ≤ m(Π_{S}). Furthermore, by Statement (5) and definition of Π_{S}, for any z ∈ Π_{S} we have that λ_{2}(z, S_{τ}) ≥ λ_{S} and the lemma follows. □

4.2. Construction of the map R.

We shall obtain the map R as a small perturbation of the map S by a diffeomorphism h_{S}, i.e., R = h_{R} ∘ S. We use some ideas from [2, 10] and construct h_{R} as a composition of rotations in the F^{ta}-subspace along pieces of orbits so that the total rotation is π/2. This allows us to interchange the F^{t}- and F^{a}-directions making the Lyapunov exponents along these directions to be close to each other.

Let us briefly outline the construction. It starts with a choice of the Rokhlin-Halmos tower for S within an invariant set Γ′ of positive volume where at every point the map S has two positive Lyapunov exponents along the E_{T}^{ut}-subspace.

The tower of height 7K + k_{0} consists of disjoint subsets called floors, where K > 0 is a given number and k_{0} is given by(4.3). We then consider a subtower Γ ⊂ Γ′ of height 2K + k_{0}. The number K should be sufficiently large to ensure that the k_{0} floors in the middle of Γ are disjoint from Ω_{S} and Ω_{0} and consist of “good” points z in the sense that every vector v ∈ E_{T}^{ut}-subspace expands by about e^{iλ} times under dS^{i} and contracts by about e^{−iλ} under dS^{−i} for any i ≥ K/2. We then approximate these k_{0} floors by finitely many sets of a special type – in our global coordinate system these sets are cylinders. We obtain the perturbation h_{R} as a composition of finitely many maps where each of these maps rotates the core of the corresponding cylinder by the angle π/2k_{0} in the F^{ta}-subspace at each level so that the total rotation is π/2.

Now consider a “good” orbit, which starts at a point z on the bottom of the subtower Γ, and a vector v ∈ E^{uta}(z). If v is close to the E^{ut}-subspace, then the length of the ut-component of dR^{K}v = dS^{K}v becomes at least about e^{Kλ} times longer than the length of v. Since dS does not contract vectors in the E^{uta}-subspace very much during the remaining k_{0} + K steps, the length of the ut-component stays about the same. If v is close to the E_{T}^{a}-subspace, the length of the a-component of v does not change under the map dR^{K} = dS^{K}. During the next k_{0} iterations the vector dR^{K}v is rotated by π/2 degree into the E^{t}-subspace. During the next K iterations the length of the vector becomes at least about e^{Kλ} times longer. It follows that every vector in E^{uta}(z) expands by about e^{Kλ} times under dR^{2K+k_{0}}. Thus we obtain a set on which R has three positive Lyapunov exponents.

To effect this construction let λ = λ_{S} and Π = Π_{S} be as in Statement (7) of Lemma 4.1. Given K > 0, let

(4.10)

Λ′ = Λ′(K) ={z ∈ Π : log ∥dS^{k}(z, v)∥ − kλ ≥ −0.1kλ,
log ∥dS^{−k}(z, v)∥ + kλ ≤ 0.1kλ
for all v ∈ E_{S}^{ut}(z), ∥v∥ = 1 and all |k| ≥ 0.5K},

and let also

Λ = Λ(K) = ⋂_{i=0}^{k_{0}−1} S^{−i}(Λ′(K)), (4.11)

where k_{0} > 0 is given by (4.3). Note that m(Λ′(K)) → m(Π) as K → ∞ and hence, m(Λ(K)) → m(Π) as K → ∞. Therefore, given a number δ_{Q} > 0, we can choose K so large that

(4.12) Kλ ≥ max{5k_{0}λ, 10 log 2, −10k_{0} log(1 − δ_{Q})}, λm(Π) + 40 log(1 − δ_{Q})m(Π \ Λ) > 0, (4.13) 20m(Π \ Λ) ≤ m(Π). (4.14)

Note that if z ∈ Λ(K) then for n ≥ 0.5K and v ∈ E_{S}^{ut}(z),

∥dS^{n}(z, v)∥ ≥ e^{0.9nλ}∥v∥.

Set

(4.15)  Λ^{∗} = Λ \ ⋃_{i=0}^{k_{0}−1} S^{−i}(Ω_{0} ∪ Ω_{S})

(recall that Ω_{0} and Ω_{S} are given by (4.2) and (4.5) respectively). By Lemma 4.1 (7),

(4.16)  m(Ω_{S} ∩ Π) ≤ m(Π)/20k_{0}.

Furthermore, by choosing the numbers ν and σ in (4.1) appropriately, we may assume that

(4.17)  m(Ω_{0} ∩ Π) ≤ m(Π)/20k_{0}.

Combining (4.14), (4.15), (4.17) and (4.16), we find that

m(Λ^{∗}) ≥ ((1 − 0.05) − 0.05 − 0.05)m(Π) ≥ 0.8m(Π).

By the Rokhlin-Halmos Lemma (see [15]), given K > 0, one can choose a measurable set Γ′ ⊂ Π such that S^{i}(Γ′)∩Γ′ = ∅ for any −K ≤ i ≤ 6K +k_{0} −1, i ≠ 0 and

m( ⋃_{i=−K}^{6K+k_{0}−1} S^{i}(Γ′)) ≥ 0.9m(Π). (4.18)

Set

Γ_{0} = {S^{j}(z) : z ∈ Γ′, 0 ≤ j ≤ 5K − 1, S^{j}(z) ∈ Λ^{∗}, S^{i}(z) ∉ Λ^{∗} for i < j}.

In other words, Γ_{0} is the set of first entries to Λ^{∗} of trajectories {S^{i}(z)}_{i=0}^{5K−1} with z ∈ Γ′. By Lemma 4.1 (5), both sets Λ and Π are of the form

Λ = Proj_{𝒩}(Λ) × G_{1}, Π = Proj_{𝒩}(Π) × G_{1}

and hence so is the set Γ_{0}, i.e., Γ_{0} = Proj_{𝒩}(Γ_{0}) × G_{1}. Let

(4.19)  Γ_{i} = S^{i}(Γ_{0}), Γ = ⋃_{i=−K}^{K+k_{0}−1} Γ_{i}.

Clearly, the sets {Γ_{i}} are pairwise disjoint for i = −K, …, K + k_{0} − 1. We approximate the set Γ_{0} by finitely many disjoint sets Σ_{0j} of the form

Σ_{0j} = B_{F^{u}}(u_{j}, r′_{j}) × B_{F^{s}}(s_{j}, r′′_{j}) × B_{F^{ta}}((t_{j}, a_{j}), r_{j}) × B_{F^{b}}(b_{0}, α_{1}),

where

z_{j} = (u_{j}, s_{j}, t_{j}, a_{j}, b_{j}) ∈ ℳ, r′_{j} ≥ r_{j}, r′′_{j} ≥ r_{j}η^{k_{0}}, j = 1, …, J.

For i = −K, …, K + k_{0} − 1, let

Σ_{ij} = S^{i}(Σ_{0j}), Δ_{i} = ⋃_{j=1}^{J} Σ_{ij}.

We can choose the sets Σ_{0j} in such a way that

Σ_{ij} ∩ Σ_{kl} = ∅

for (i, j) ≠ (k, l), −K ≤ i, k ≤ K + k_{0}, 1 ≤ j, l ≤ J and that

Σ_{ij} ∩ (Ω_{0} ∪ Ω_{S}) = ∅

for 0 ≤ i ≤ k_{0} − 1, 0 ≤ j ≤ J. It follows that for i = 1, …, k_{0}, the set Δ_{i} is an approximation of Γ_{i} and Γ_{i} ∩ (Ω_{0} ∪ Ω_{S}) = ∅. We may assume that for each i = 0, …, k_{0},

m(Γ_{i}△Δ_{i}) ≤ 0.05 max{m(Γ_{i}), m(Δ_{i})}. (4.20)

Note that each set Σ_{ij} is a cylinder in the form described in Sublemma 4.5 below. Applying this sublemma with Δ = Σ_{ij}, we obtain a map ρ_{ij} and a subcylinder Σ′_{ij} ⊂ Σ_{ij} such that ∥ρ_{ij} − Id ∥ ≤ δ_{Q} and

(4.21)  m(Σ′_{ij})/m(Σ_{ij}) ≥ 3/4.

Furthermore, restricted to Σ′_{ij}, the map ρ_{ij} is the rotation by the angle π/2k_{0} along the F^{t} × F^{a}-subspace and is the identity outside Σ_{ij}. In fact, by the construction of the sets Σ′_{ij} (see Sublemma 4.5 below), we can assume that S(Σ′_{ij}) = Σ′_{i+1,j} for i = 0, …, k_{0} − 1. Let

(4.22)  Ω_{R} = ⋃_{i=0}^{k_{0}−1} Δ_{i}, Δ′_{i} = ⋃_{j=1}^{J} Σ′_{ij}.

Hence, by (4.21) and by definition of Δ_{i} and Δ′_{i}, we have

(4.23)  m(Δ′_{i})/m(Δ_{i}) ≥ 3/4.

Then define h_{R} = ρ_{ij} on Σ_{ij}, and h_{R} = Id otherwise. Clearly, h_{R} is a C^{∞} volume preserving diffeomorphism. Moreover, dh_{R} preserves E_{T}^{uta} bundle and det(dh_{R}|E_{T}^{uta}(z)) = 1 for any z ∈ ℳ. We define the map R = h_{R} ∘ S. Some of the properties of R are described in the following lemma.

Lemma 4.2. The following statements hold:

(1) ∥R − T ∥_{C^{1}} ≤ δ_{Q} and R is homotopic to Id;

(2) R = T on the sets 𝒩 × (Y \ G_{0}) and Ω_{0}; in particular, R is a gentle perturbation of T;

(3) R satisfies Statements (3)–(5) of Proposition 3.1;

(4) for any (a, b) ∈ G_{0}, the set 𝒩 × I_{b}, where I_{b} = {(a′, b) : a′ ∈ B_{F^{a}}(a_{0}, α_{0})}, is R-invariant and for y ∉ G_{0} the set 𝒩_{y} is R-invariant;

(5) for every z ∈ ℳ,

E_{R}^{uta}(z) = E_{S}^{uta}(z) = E_{T}^{uta}(z),

det(dR|E_{R}^{uta}(z)) = det(dS|E_{S}^{uta}(z)) = det(dT |E_{T}^{uta}(z));

(6) for α_{2} = 0.9α_{1}, y′ = (a, b′), y_{2} = (a, b′′) ∈ B_{F^{a}}(a_{0}, α_{1}) × B_{F^{b}}(b_{0}, α_{2}) we have Proj_{𝒩×B_{F^{a}}(a_{0},α_{1})}R(x, t, a, b′) = Proj_{𝒩×B_{F^{a}}(a_{0},α_{1})}R(x, t, a, b′′), where Proj_{𝒩×B_{F^{a}}(a_{0},α_{1})} is the projection onto the set 𝒩 × B_{F^{a}}(a_{0}, α_{1}) given by Proj_{𝒩×B_{F^{a}}(a_{0},α_{1})}(x, t, a, b) = (x, t, a);

(7) L_{2}(R) < L_{3}(R) and hence,

L_{1}(R) < L_{2}(R) < L_{3}(R) = L_{4}(R) = L_{4}(T), L_{5}(R) = 0;

(8) there exist a number λ_{R} > 0 and a subset Π_{R} = (Proj_{𝒩×B_{F^{a}}(a_{0},α_{1})}Π_{R}) × B_{F^{b}}(b_{0}, α_{2}) of positive volume such that m(Π_{R}) ≥ 20k_{0}m(Π_{R} ∩ Ω_{i}) for i = R, S, and at any z ∈ Π_{R}, R has three positive Lyapunov exponents λ_{1}(z, R), λ_{2}(z, R), λ_{3}(z, R) ≥ λ_{R} along the E_{R}^{uta} = E_{T}^{uta} subbundle.

Proof. Statements (1)–(6) follows immediately from the construction of h_{R}. In particular, the fact that α_{2} = 0.9α_{1} follows from Statement (4) of Sublemma 4.5.

Now we prove Statements (7) and (8).

Set Δ_{0}^{∗} = Δ′_{0} ∩ Λ, where Δ′_{0} is given by (4.22), and Λ is given by (4.11) (we shall see later that Δ_{0}^{∗} is not empty and indeed has positive volume). Then set

U_{1} = R^{−K}Δ_{0}^{∗},  U_{2} = Δ_{0} \ Δ_{0}^{∗},
U_{3} = R^{k_{0}}((Δ_{0} ∩ Λ) \ Δ_{0}^{∗}), U_{4} = R^{k_{0}}(Δ_{0} \ Λ).

Let U = U_{1} ∪ U_{2} ∪ U_{3} ∪ U_{4} and R̄ = R^{β} : U → U be the first return map where β = β(z) is the first return time of the point z ∈ U to U under R. By Poincarẽs Recurrence Theorem, the map R̄ is defined for almost every z ∈ U. In the proof below, for any z ∈ U, we shall assume that v ∈ E_{R}^{uta}(z) = E_{S}^{uta}(z).

Let ∧^{k}(E_{S}^{uta}(z)) denote the exterior power of E_{S}^{uta}(z) and

∧^{k}(dR|E_{S}^{uta}(z)) : ∧^{k}(E_{S}^{uta}(z)) → ∧^{k}(E_{S}^{uta}(R(z)))

be the exterior power of dR|E_{S}^{uta}(z). It is easy to see that if there exists c ≥ 1 such that ∥dRv∥ ≥ c∥v∥ for any v ∈ E_{S}^{uta}(z), then

(4.24)  \frac{∥∧^{3}(dR|E_{S}^{uta}(z))∥}{∥∧^{2}(dR|E_{S}^{uta}(z))∥} ≥ c.

First we consider the case when z ∈ U_{1}. Then β(z) ≥ 2K + k_{0}. By Sublemma 4.3 below and (4.12),

log ∥dR̄_{z}v∥ ≥ 0.9Kλ − 0.5 log 2 + log ∥v∥ ≥ 0.85Kλ + log ∥v∥.

Hence,

log ∥∧^{3}(dR|E_{S}^{uta}(z))∥ − log ∥∧^{2}(dR|E_{S}^{uta}(z))∥ ≥ 0.85Kλ.

Note that by definition, Γ_{0} ⊂ Λ^{∗}. Since Σ_{ij} ∩ (Ω_{0} ∪ Ω_{S}) = ∅ for 0 ≤ i ≤ k_{0} − 1 and 0 ≤ j ≤ J, we have that Δ′_{0} ∩ Λ = Δ′_{0} ∩ Λ^{∗} ⊃ Δ_{0} ∩ Γ_{0}. Hence, by (4.23) and (4.20),

m(U_{1}) =m(Δ_{0}^{∗}) = m(Δ′_{0} ∩ Λ) ≥ m(Δ′_{0} ∩ Γ_{0}) = m(Δ′_{0}) − m(Δ′_{0} \ Γ_{0})
≥m(Δ′_{0}) − m(Δ_{0} \ Γ_{0}) ≥ \frac{3}{4}m(Δ_{0}) − 0.05m(Δ_{0}) = 0.7m(Δ_{0}).

It follows that

(4.25)  \begin{matrix}∫_{U_{1}}(log ∥∧^{3}(dR̄|E_{S}^{uta}(z))∥ − log ∥∧^{2}(dR̄|E_{S}^{uta}(z))∥) dm(z) \\ ≥ & 0.85Kλ · 0.7m(Δ_{0}). \end{matrix}

Now we consider the case when z ∈ U_{2}. Note that ∥dR − dT ∥_{C^{1}} ≤ δ_{Q} andE_{R}^{uta}(z) = E_{R}^{uta}(z) for all z. Then R̄|U_{2} = R^{k_{0}}|U_{2} and

log ∥dR̄_{z}v∥ ≥ k_{0} log(1 − δ_{Q}) + log ∥v∥.

In addition, by definition of Δ_{0}^{∗} and (4.23),

m(U_{2}) = m(Δ_{0} \ Δ_{0}^{∗}) ≤ m(Δ_{0} \ Δ′_{0}) ≤ \frac{1}{4}m(Δ_{0}).

We conclude that

(4.26) ∫_{U_{2}}(log ∥∧^{3}(dR̄|E_{S}^{uta}(z)) − log ∥∧^{2}(dR̄|E_{S}^{uta}(z))∥) dm(z)

≥k_{0} log(1 − δ_{Q}) · 0.25m(Δ_{0}).

If z ∈ U_{3}, then z ∈ R^{k_{0}}(Λ) ⊂ Λ′, where Λ′ is defined in (4.10), and β(z) > K. Hence, R^{k}(z) = S^{k}(z) for 0 ≤ k ≤ β(z) and

dR̄|E_{S}^{ut}(z) = dS^{β(z)}|E_{S}^{ut}(z).

Therefore if v ∈ E_{S}^{ut}(z), then ∥dR̄_{z}v∥ ≥ 0.9Kλ∥v∥, and if v ∈ E_{S}^{a}(z), then ∥dR̄v∥ = ∥dS^{β(z)}v∥ = ∥v∥. It follows that ∥dR̄_{z}v∥ ≥ ∥v∥ for any v ∈ E_{S}^{uta}(z). Hence, by (4.24) with c = 1, we have

(4.27)  ∫_{U_{3}}(log ∥∧^{3}(dR̄|E_{S}^{uta}(z))∥ − log ∥∧^{2}(dR̄|E_{S}^{uta}(z)∥) dm(z) ≥ 0.

Finally, let us consider the case z ∈ U_{4}. Let β′(z) be the smallest positive integer such that R^{β′(z)}(z) ∈ Λ for some 0 ≤ β′(z) ≤ β(z) and let β′(z) = β(z) if there is no such integer. Denote by

U′_{4} = U_{4} ∩ {z : β(z) − β′(z) ≥ 0.5K}, U′′_{4} = U_{4} ∩ {z : β(z) − β′(z) < 0.5K}.

Since β(z) ≥ K for z ∈ U′′_{4}, we have β(z) ≤ 2β′(z). Note that by (4.10), if n ≥ 0.5K then ∥dS_{z}^{n}v∥ ≥ ∥v∥ for any z ∈ Λ and v ∈ E_{S}^{uta}(z). Also note that R = S on Π \ Ω_{R}. If z ∈ U′_{4} then

∥dR̄_{z}v∥ = ∥dR_{z}^{β(z)}v∥ = ∥dS_{R^{β′(z)}(z)}^{β(z)−β′(z)}(dR_{z}^{β′(z)}v)∥ ≥ ∥dR_{z}^{β′(z)}v)∥.

Hence, by Statement (6) of the lemma,

log ∥dR̄_{z}v∥ ≥ log ∥R_{z}^{β′(z)}(v)∥ ≥ β′(z) log(1 − δ_{Q}) + log ∥v∥.

If z ∈ U′′_{4} then

log ∥dR̄_{z}v∥ ≥ β(z) log(1 − δ_{Q}) + log ∥v∥ ≥ 2β′(z) log(1 − δ_{Q}) + log ∥v∥.

It follows that

∫_{U_{4}}(log ∥∧^{3}(dR̄|E_{S}^{uta}(z))∥ − log ∥∧^{2}(dR̄|E_{S}^{uta}(z)∥) dm(z)
≥2 log(1 − δ_{Q}) ∫_{U_{4}} β′(z) dm(z).

Furthermore, if z ∈ U_{4}, then z, R(z), …, R^{β′(z)−1}(z) ∈ Π\Λ. Hence, we obtain ∫_{U_{4}} β′(z) dm(z) ≤ m(Π \ Λ) and therefore

(4.28) ∫_{U_{4}}(log ∥∧^{3}(dR̄|E_{S}^{uta}(z))∥ − log ∥∧^{2}(dR̄|E_{S}^{uta}(z)∥) dm(z)

≥2 log(1 − δ_{Q})m(Π \ Λ).

Note that the sets R^{K}(U_{1}), R^{−k_{0}}(U_{3}) and R^{−k_{0}}(U_{4}) form a partition of Δ_{0} and hence, by (4.25)–(4.28), we have

∫_{U}(log ∥∧^{3}(dR̄|E_{S}^{uta}(z))∥ − log ∥∧^{2}(dR̄|E_{S}^{uta}(z))∥) dm(z)
≥0.595λKm(Δ_{0}) + 0.25k_{0} log(1 − δ_{Q})m(Δ_{0}) + 2 log(1 − δ_{Q})m(Π \ Λ).

(4.29)

Using (4.12), and then Sublemma 4.4 and (4.13), we conclude that the right hand side of (4.29) is greater than

0.57λKm(Δ_{0}) + 2 log(1 − δ_{Q})m(Π \ Λ)
≥0.0627λm(Π) − 0.05λm(Π) ≥ 0.0127λm(Π) > 0.

Hence,

∫_{U} log ∥∧^{3}(dR̄|E_{S}^{uta}(z))∥ dm(z) > ∫_{U} log ∥∧^{2}(dR̄|E_{S}^{uta}(z))∥ dm(z).

Denote Π′ = ∪_{i=−∞}^{∞}R^{i}(U). Clearly we have

∫_{U} log ∥∧^{3}(dR̄|E_{S}^{uta}(z))∥ dm(z) = ∫_{Π′} log ∥∧^{3}(dR|E_{S}^{uta}(z))∥ dm(z)
= ∫_{Π′}(λ_{1}(z, R) + λ_{2}(z, R) + λ_{3}(z, R)) dm(z)

and

∫_{U} log ∥∧^{2}(dR̄|E_{S}^{uta}(z))∥ dm(z) = ∫_{Π′} log ∥∧^{2}(dR|E_{S}^{uta}(z))∥ dm(z)
= ∫_{Π′}(λ_{1}(z, R) + λ_{2}(z, R)) dm(z).

It follows that L_{3}(R|Π′) > L_{2}(R|Π′), where L_{i} is defined by (2.2). Since R = S outside Π′, we obtain that L_{3}(R) > L_{2}(R). Furthermore, there is an R-invariant subset of Π′ on which R has three positive Lyapunov exponents. Note that the subbundles E_{T}^{utab} and E_{T}^{uta} are preserved by dS and dR and that det(dS|E_{T}^{i}) = det(dT |E_{T}^{i}) for i = uta, utab. Hence, the two smallest Lyapunov exponents remain unchanged, and so does the sum of the three largest ones. This implies that L_{i}(S_{τ}) = L_{i}(T) for i = 4, 5 and hence,

L_{1}(R) < L_{2}(R) < L_{3}(R) = L_{4}(R) = L_{4}(S) = L_{4}(T) and L_{5}(R) = 0.

Statement (7) of the lemma follows.

To prove Statement (8) observe that the above argument applies to the sets

Ũ = U ⋂ 𝒩 × B_{F^{a}}(a_{0}, α_{1}) × B_{F^{b}}(b_{0}, α_{2})

and

Π̃′ = Π′ ⋂ 𝒩 × B_{F^{a}}(a_{0}, α_{1}) × B_{F^{b}}(b_{0}, α_{2}).

Denote by λ_{1}(z, R), λ_{2}(z, R) and λ_{3}(z, R) the positive Lyapunov exponents of z ∈ Π′. Given λ_{R} > 0, consider the level set

Π_{R} = {z ∈ Π̃′ : λ_{1}(z, R), λ_{2}(z, R), λ_{3}(z, R) ≥ λ_{R}}.

If λ_{R} is sufficiently small, this set has positive volume. Note that by (4.12), we have K ≥ 5k_{0}. Furthermore, by definition of sets Γ′, Γ_{0} and Ω_{R}, we have that every piece of an orbit visiting all set S^{i}(Γ′) with −K ≤ i ≤ 6K + k_{0} − 1 consecutively meets Ω_{R} exactly k_{0} times. Moreover, Ω_{R} is contained in the union of these S^{i}(Γ′). Since R preserves volume, we have that

m(Π_{R}) ≥ (7K + k_{0})m(Π_{R} ∩ Ω_{R}) > 20k_{0}m(Π_{R} ∩ Ω_{R}).

Since N_{0} > 20k_{0}, we obtain by (4.6), that m(Π_{R}) ≥ 20k_{0}m(Π_{R} ∩ Ω_{S}). This completes the proof of the lemma. □

4.3. Sublemmas.

We shall prove now the technical sublemmas used in the previous subsection.

Sublemma 4.3. Let z ∈ R^{−K}(Δ_{0}^{∗}). Then for any v ∈ E_{S}^{uta}(z),

∥dR̄_{z}(v)∥ ≥ \frac{\sqrt{2}}{2}∥v∥e^{0.9Kλ}.

Proof. Note that h_{R} = Id on ∪_{i=−K}^{−1}Γ_{i} and hence, R^{K}(z) = S^{K}(z). Since dh_{S} preserves the subbundle E^{ut}(S), we have E_{S}^{ut}(z) = E_{T}^{ut}(z). Write v = v^{ut} + v^{a}, where v^{ut} ∈ E_{T}^{ut}(z) and v^{a} ∈ E_{T}^{a}(z).

We first consider the case ∥v^{a}∥ ≤ \frac{\sqrt{2}}{2}∥v∥. Note that ∥v^{ut}∥ ≥ \frac{\sqrt{2}}{2}∥v∥. Since

dS^{K}v^{ut} ∈ E_{S}^{ut}(S^{K}(z)) and S^{K}(z) ∈ Λ, using (4.10) and (4.11), we find that

∥v^{ut}∥ = ∥dS^{−K}(dS^{K}v^{ut})∥ ≤ ∥dS^{K}v^{ut}∥e^{−0.9Kλ}.

Hence,

∥dR^{K}v∥ = ∥dS^{K}v∥ ≥ ∥dS^{K}v^{ut}∥ ≥ ∥v^{ut}∥e^{0.9Kλ} ≥ \frac{\sqrt{2}}{2}∥v∥e^{0.9Kλ}.

Note that at R^{K}(z), …, R^{K+k_{0}−1}(z) the map dh_{R} is a rotation and that dS|E_{S}^{uta}(R^{i}(z)) = dT |E_{T}^{uta}(R^{i}(z)) is non-contracting for i = K, …, K + k_{0} − 1. Therefore, dR^{k_{0}}|E_{S}^{uta}(R^{K}(z)) is non-contracting. Further, since

{R^{i}(z)}_{i=K+k_{0}}^{β} ∩ Ω_{R} = ∅

and R^{K+k_{0}}(z) ∈ Λ′, we have that the map

dR^{β−(K+k_{0})}|E_{S}^{ut}(R^{K+k_{0}}(z)) = dS^{β−(K+k_{0})}|E_{S}^{ut}(R^{K+k_{0}}(z))

is expanding and the map

dR^{β−(K+k_{0})}|E_{S}^{uta}(R^{K+k_{0}}(z))

is non-contracting. It follows that

\begin{matrix}∥dR̄v∥ = & ∥dR_{R^{K+k_{0}}(z)}^{β−(K+k_{0})}(dR_{z}^{K+k_{0}}v)∥ = ∥dS_{R^{K+k_{0}}(z)}^{β−(K+k_{0})}(dR_{z}^{K+k_{0}}v)∥ \\ ≥ & ∥dR_{z}^{K+k_{0}}v∥ ≥ ∥dR_{z}^{K}v∥ = ∥dS_{z}^{K}v∥ ≥ \frac{\sqrt{2}}{2}∥v∥e^{0.9Kλ}. \end{matrix}

We now consider the case ∥v^{a}∥ ≥ \frac{\sqrt{2}}{2}∥v∥. Note that dS^{K}v^{a} ∈ E_{S}^{a}(S^{K}(z)). By construction of h_{R}, we see that dR_{S^{K}(z)}^{k_{0}} rotates the vector in E_{S}^{ta}(S^{K}(z)) = E_{T}^{ta}(S^{K}(z)) by π/2. It means that

dR^{K+k_{0}}v^{a} = dR^{k_{0}}(dS^{K}v^{a}) ∈ E_{S}^{ut}(R^{K+k_{0}}(z)).

Using the fact that R^{K+k_{0}}(z) ∈ Λ we obtain

\begin{matrix}∥dR̄v^{a}∥ = & ∥dR_{R^{K+k_{0}}(z)}^{β−(K+k_{0})}(R^{K+k_{0}}(z)v^{a}∥ ≥ ∥dR^{K}(dR^{K+k_{0}}v^{a})∥ \\ ≥ & ∥dR^{K+k_{0}}v^{a}∥e^{0.9Kλ} ≥ ∥v^{a}∥e^{0.9Kλ} ≥ \frac{\sqrt{2}}{2}∥v∥e^{0.9Kλ}. \end{matrix}

This implies the desired result. □ Sublemma 4.4. m(Γ_{0}) ≥ 0.12K^{−1}m(Π) and hence, m(Δ_{0}) ≥ 0.11K^{−1}m(Π). Proof. Denote by

Γ̂′ = ⋃_{i=0}^{5K−1} S^{i}(Γ′), \overline{Γ}′ = ⋃_{i=−K}^{6K+k_{0}−1} S^{i}(Γ′)

(recall that Γ′ is given by the Rokhlin-Halmos Lemma in Subsection 4.2). Since K ≥ 5k_{0}, we have that

\frac{m(Γ̂′)}{m(\overline{Γ}′)} = \frac{5K}{7K + k_{0}} ≥ \frac{5K}{7K + 0.2K} ≥ \frac{50}{72}.

By (4.18),

m(Γ̂′) ≥ (50/72) · 0.9m(Π) = 0.625m(Π).

For z ∈ Γ′ denote by O(z) = {Q^{i}(z) : i = 0, …, 5K + k_{0} − 1} the piece of the orbit from 0 to 5K − 1 that start at z. Let

Γ̂′_{1} = {O(z) : z ∈ Γ′, O(z) ∩ Λ ≠ ∅}, Γ̂′_{2} = {O(z) : z ∈ Γ′, O(z) ∩ Λ = ∅}. Clearly {Γ̂′_{1}, Γ̂′_{2}} forms a partition of Γ̂′ and Γ̂′_{2} ⊂ Π \ Λ. Therefore by (4.14),

m(Γ̂′_{1}) =m(Γ̂′) − m(Γ̂′_{2}) ≥ m(Γ̂′) − m(Π \ Λ)
≥0.625m(Π) − 0.025m(Π) = 0.6m(Π).

Note that Γ_{0} consists of exactly one point from each orbit O(z) in Γ̂_{1}. It follows that

m(Γ_{0}) ≥ \frac{m(Γ̂_{1})}{5K} ≥ \frac{0.6m(Π)}{5K} ≥ 0.12K^{−1}m(Π).

By (4.20),

m(Δ_{0}) ≥ m(Γ_{0}) − m(Γ_{0} \ Δ_{0}) ≥ 0.95m(Γ_{0}) ≥ 0.11K^{−1}m(Π).

This is the desired result. □

Sublemma 4.5. For any δ > 0, there is θ_{0} > 0 such that for any θ ∈ [0, θ_{0}], any positive numbers s, s′, s′′, s′′′ satisfying s′, s′′, s′′′ ≥ s and any cylinder Δ ⊂ ℝ^{5} of the form

Δ = Δ_{s,s′,s′′,s′′′} = B_{1}(z_{1}, s′) × B_{2}(z_{2}, s′′) × B_{34}((z_{3}, z_{4}), s) × B_{5}(z_{5}, s′′′)

there exists a set Δ′ ⊂ Δ of the form

Δ′ = Δ′_{s_{0},s′_{0},s′′_{0},s′′′_{0}} = B_{1}(z_{1}, s′_{0}) × B_{2}(z_{2}, s′′_{0}) × B_{34}((z_{3}, z_{4}), s_{0}) × B_{5}(z_{5}, s′′′_{0})

and a C^{∞} map ρ : ℝ^{5} → ℝ^{5} with the following properties:

(1) ρ = r_{θ} on Δ′ where r_{θ} is the rotation

r_{θ}(z_{1}, z_{2}, z_{3}, z_{4}, z_{5}) = (z_{1}, z_{2}, z_{3} cos θ − z_{4} sin θ, z_{3} sin θ + z_{4} cos θ, z_{5});

(2) ρ = Id outside Δ;

(3) m(Δ′)/m(Δ) ≥ 3/4;

(4) s_{0}/s, s′_{0}/s′, s′′_{0}/s′′, s′′′_{0}/s′′′ > 9/10.

(5) ∥ρ − Id ∥_{C^{1}} ≤ δ;

Proof. Due to the particular form of our cylinders there is a number κ ∈ (0, 1/10) such that for any r > 0 and r′, r′′, r′′′ > r we have that

\frac{m(Δ_{r(1−κ),r′(1−κ),r′′(1−κ),r′′′(1−κ)})}{m(Δ_{rr′r′′r′′′})} ≥ 3/4.

Consider a family of C^{∞} functions ζ_{r} = ζ_{r}(s) : ℝ^{+} → ℝ^{+}, for r ≥ 1 such that (a) ζ_{1}(s) = 1 if s ∈ [0, 1 − κ] and ζ_{1}(s) = 0 if s ≥ 1;

(b) ζ_{r}(s) = 1 if s ∈ [0, r − 1) and ζ_{r}(s) = ζ_{1}(s − r + 1) if s ≥ r − 1. Define the map ρ by ρ(z) = r_{θ(τ,s,s′,s′′,s′′′)}(z), where

θ(τ, s, s′, s′′, s′′′) = τζ_{s′/s}(z_{1}/s′)ζ_{s′′/s}(z_{2}/s′′)ζ_{1}(\frac{\sqrt{z_{3}^{2} + z_{4}^{2}}}{s} )ζ_{s′′′/s}(z_{5}/s′′′),

and r_{(s,s′,s′′,s′′′)} is given in Condition (1) of the sublemma. By construction, ρ satisfies Statements 1 and 2. Statement 3 and 4 follows from the choice of the number κ and the definition of ζ_{1} and ζ_{r}. To obtain Statement 5, we first note that if τ = 0 then ρ = Id and that the C^{1} norm of ρ changes smoothly with τ.

It is also easy to check that the C^{1} norm of the rotation is independent of the choice of the size s if s′ = s′′ = s′′′ = s, and the C^{1} norm does not increase if we increase s′, s′′ and s′′′. □

4.4. Construction of the map Q.

We shall obtain the map Q as a small perturbation of the map R by a diffeomorphism h_{Q}, i.e., Q = h_{Q} ∘ R. The construction of h_{Q} is similar to the construction of the map h_{R}: it is a composition of rotations in the F^{ba}-subspace along pieces of orbits so that the total rotation is π/2.

Let λ = λ_{R} and Π = Π_{R} be as in Lemma 4.2 (8). Note that for any z ∈ Π the map R has three positive Lyapunov exponents λ_{1}(z, R), λ_{2}(z, R), and λ_{3}(z, R) ≥ λ along the E_{R}^{uta} = E_{T}^{uta} subbundle. Consider the set

Λ′ = Λ′(K) = {z ∈ Π : log ∥dR_{z}^{k}v∥ − kλ ≥ −0.1kλ,
log ∥dR_{z}^{−k}v∥ + kλ ≤ 0.1kλ,

for all v ∈ E^{uta}(z, R), ∥v∥ = 1, and all |k| ≥ 0.5K}

and define the set Λ and the number K > 0 similar to (4.11)-(4.14). Set

Λ^{∗} = Λ \ ⋃_{i=0}^{k_{0}−1} R^{−i}(Ω_{0} ∪ Ω_{S} ∪ Ω_{R}).

Similar to (4.17), we may assume

m(Ω_{0} ∩ Π) ≤ m(Π)/20k_{0}.

Hence, by the choice of K, and Lemma 4.2 (4), we have

m(Λ^{∗}) ≥ ((1 − 0.05) − 0.05 − 0.05 − 0.05)m(Π) = 0.8m(Π).

We then construct the set Γ′, Γ_{0} in a way similar to the previous subsection and set Γ_{i} = R^{i}(Γ) for −K ≤ i ≤ K + k_{0} − 1. Finally, we approximate Γ_{0} by the sets of the form

Σ_{0j} = B_{F^{u}}(u, t′_{j}) × B_{F^{s}}(s, t′′_{j}) × B_{F^{t}}(t, t′′_{j}) × B_{F^{ab}}((a_{j}, b_{j}), r_{j}),

where r′_{j}, r′′′_{j} ≥ r_{j}, r′′_{j} ≥ r_{j}η^{k_{0}} and set for i = −K, ⋯, K + k_{0} − 1,

Σ_{ij} = R^{i}(Σ_{0j}), Δ_{i} = ⋃_{j=1}^{J} Σ_{ij}.

Define Ω_{Q} = ⋃_{i=0}^{k_{0}−1} Δ_{i}. Applying Sublemma 4.5 to each set Σ_{ij} we obtain a map ρ_{ij} and then set h_{Q} = ρ_{ij} on each Σ_{ij} and h_{Q} = Id otherwise. Finally, define Q = h_{Q} ∘ R.

Lemma 4.6. The map Q satisfies all the properties stated in Proposition 3.2. In particular, L_{1}(Q) < L_{2}(Q) < L_{3}(Q) < L_{4}(Q) = L_{4}(T).

Proof. Statements (1)–(4) of Proposition 3.2 follow from the construction of the map Q. The proof that L_{3}(Q) < L_{4}(Q) is the same as in the proof of Lemma 4.2.

Note that the subbundle E_{T}^{utab} is preserved by both Q and T and that both maps T and Q are volume preserving. Hence the smallest Lyapunov exponents remains unchanged, and so does the sum of the four largest ones. It follows that Q has four positive Lyapunov exponents along the E_{R}^{utab} = E_{T}^{utab} subbundle on a set of positive volume. □

5. Construction of the maps P_{n}: Proof of Proposition 3.3

Recall that the map Q is pointwise partially hyperbolic with one-dimensional strongly stable, one-dimensional strongly unstable and 3-dimensional central subbundles. The strongly stable and unstable subbundles are integrable to (one-dimensional) transverse strongly stable and unstable foliations. The central subbundle corresponds to the flow direction and two directions, F^{a} and F^{b}, in the Y -space and is integrable to a smooth central foliation. However Q does not have the accessibility property: for (a, b) ∉ G_{0} the accessibility class of every point z = (u, s, t, a, b) is the 2-torus (X, t, a, b).

For each n, we construct the map P_{n} to be a sufficiently small gentle perturbation of Q and such that P_{n} has the accessibility property on an invariant open set 𝒰_{n}, and is stably accessible on an open set 𝒰̃_{n} (see (3.5)). These sets are nested and exhaust the set 𝒢, and the sequence of maps P_{n} converges to a map P that is accessible on 𝒢. In our construction we use methods similar to those in [9] and [14], and we obtain each P_{n} as a result of three gentle perturbations h^{t}, h^{a} and h^{b} that ensure accessibility in the flow direction and two directions in Y respectively.

5.1. Construction of sets U_{n}.

In our construction we will heavily exploit the fact that the 2-torus Y has a global coordinate system. This will enable us to define the sets U_{n} in an explicit and specific way, which will serve our goal. At this point we regard the 2-torus Y as the square [0, 8] × [0, 8] whose opposite sides are identified. For each n ≥ 1, consider the partition of Y into squares

Ẑ_{ij}^{(n)} = [\frac{i}{2^{n}} , \frac{i + 1}{2^{n}}] × [\frac{j}{2^{n}} , \frac{j + 1}{2^{n}}], i, j = 0, 1, …, 2^{n+3} − 1.

Without loss of generality we shall assume that the square G_{0}, constructed in Subsection 3.2, is contained in some Ẑ_{i_{0}j_{0}}^{(1)} so that

d(G_{0}, ∂Ẑ_{i_{0}j_{0}}^{(1)}) ≥ 1/2^{4} and d(C, Ẑ_{i_{0}j_{0}}^{(1)}) > 2

(here C is the Cantor set constructed in (A1), see Subsection 3.2). Consider the open squares

\begin{matrix}Z_{ij}^{(n)} = (\frac{i}{2^{n}} − \frac{1}{2^{n+2}} , \frac{i + 1}{2^{n}} + \frac{1}{2^{n+2}} ) × (\frac{j}{2^{n}} − \frac{1}{2^{n+2}} , \frac{j + 1}{2^{n}} − \frac{1}{2^{n+2}} ), \\ Z̃_{ij}^{(n)} = (\frac{i}{2^{n}} − \frac{1}{2^{n+5}} , \frac{i + 1}{2^{n}} + \frac{1}{2^{n+5}} ) × (\frac{j}{2^{n}} − \frac{1}{2^{n+5}} , \frac{j + 1}{2^{n}} − \frac{1}{2^{n+5}} ). \end{matrix}

Clearly, these squares have the same center as Ẑ_{ij}^{(n)} and Ẑ_{ij}^{(n)} ⊂ Z̃_{ij}^{(n)} ⊂ Z_{ij}^{(n)}. For n ≥ 1 consider the set

Y_{n} = {y ∈ Y : d(y, C) ≥ 1/2^{n−2}}.

Since G_{0} ⊂ Y_{1}, we let Y′_{n} be the connected component of Y_{n} that contains G_{0}. Finally, consider the sets

Û_{1} = Ẑ_{i_{0}j_{0}}^{(1)}, U_{1} = Z_{i_{0}j_{0}}^{(1)} and Ũ_{1} = Z̃_{i_{0}j_{0}}^{(1)}

and for n > 1,

Û_{n} = ⋃_{Ẑ_{ij}^{(n)}∩Y′_{n}≠∅} Ẑ_{ij}^{(n)}, U_{n} = ⋃_{Ẑ_{ij}^{(n)}∩Y′_{n}≠∅} Z_{ij}^{(n)}, Ũ_{n} = ⋃_{Ẑ_{ij}^{(n)}∩Y′_{n}≠∅} Z̃_{ij}^{(n)}.

It is clear that the sets U_{n} and Ũ_{n} satisfy Conditions (A4)–(A6) in Subsection 3.4.

Let 𝒵̂_{n} = {Ẑ_{ij}^{(n)} : Ẑ_{ij}^{(n)} ⊂ Û_{n} \Û_{n−1}} and 𝒵_{n} = {Z_{ij}^{(n)} : Ẑ_{ij}^{(n)} ∈ 𝒵̂_{n}}. Relabeling elements of 𝒵_{n} we shall denote them by Z_{1}^{(n)}, …, Z_{k_{n}}^{(n)}, and we shall use the notations Ẑ_{ℓ}^{(n)} and Z̃_{ℓ}^{(n)} for the corresponding squares contained in Z_{ℓ}^{(n)}. Thus we have (see Figure 1)

U_{n} = U_{n−1} ∪ ( ⋃_{Z_{ij}^{(n)}∈𝒵_{n}} Z_{ij}^{(n)}) = U_{n−1} ∪ (⋃_{ℓ=1}^{k_{n}} Z_{ℓ}^{(n)}).

Clearly, Ẑ_{ℓ}^{(n)} ∩ Ẑ_{j}^{(m)} = ∅ if (n, ℓ) ≠ (m, j) and hence, the collection of sets {Ẑ_{ℓ}^{(n)} : n = 1, 2, …, ℓ = 1, …, k_{n}} forms a countable partition of G up to a set of zero volume while the collection of sets {Z_{ℓ}^{(n)} : n = 1, 2, …, ℓ = 1, …, k_{n}} forms a cover of G of multiplicity at most 4.

Note that the requirement d(G_{0}, ∂Ẑ_{1}^{(1)}) ≥ 1/2^{4} yields that G_{0} ∩ Z_{ℓ}^{(n)} = ∅ for any n > 1 and ℓ = 1, …, k_{n}.

PSfrag replacements

The Cantor Set Z_{1}^{(n)}

Z_{2}^{(n)}
Z_{3}^{(n)}
Z_{4}^{(n)}
Z_{5}^{(n)}
Z_{1}^{(n+1)}
Z_{2}^{(n+2)}
Figure 1. Sets U_{n} and U_{n+1}

Lemma 5.1. There is a labeling of the squares {Z_{ℓ}^{(n)}} by integers from 1 to 8 such that for any y ∈ G, the labels of the squares Z_{ℓ}^{(n)} containing y are all different. In particular, Z_{1}^{(1)} can be labelled by 1.

Proof. For each odd number n > 0, we use 1, 2, 3, 4 to label the squares {Z_{ij}^{(n)}} ∈ 𝒵_{n} in such a way that Z_{ij}^{(n)} and Z_{kl}^{(n)} have the same label if i ≡ k(mod 2) and j ≡ l(mod 2). An alternative way of describing this process is that we first label the 4 squares Ẑ_{ij}^{(n)} inside of some Ẑ_{kl}^{(n−1)} by the numbers 1 to 4, and then translate the square Ẑ_{kl}^{(n−1)} to all other such squares. We then let Z_{ij}^{(n)} have the same labeling as Ẑ_{ij}^{(n)}. Clearly, for any y ∈ G, the label of the squares Z_{ij}^{(n)} with Z_{ij}^{(n)} ∋ y are all different. Hence, we obtain a labeling on 𝒵_{n} by restriction.

For even n > 0, we use numbers 5 to 8 to label the squares {Z_{ij}^{(n)}} in a similar way. Since any squares Z_{ij}^{(n)} ∈ 𝒵_{n} and Z_{kl}^{(n+2)} ∈ 𝒵_{n+2} are disjoint, we obtain the desired labeling. □

5.2. Construction of maps P_{n}.

Let q_{j}, j = 1, …, 8 be eight distinct periodic points of the Anosov automorphism A. There is ϵ_{0} > 0 such that B_{X}(A^{l}q_{j}, ϵ_{0}) ∩ B_{X}(A^{l}q_{j′}, ϵ_{0}) = ∅ whenever j ≠ j′ and l = −1, 0, 1. For each q_{j} we choose three distinct periodic points p_{j}^{t}, p_{j}^{a}, p_{j}^{b} ∈ B_{X}(q_{j}, ϵ_{0}/3) for A. We shall assume that q_{1} = q and p_{1}^{i} = p^{i} for i = t, a, b where q and p^{i} are chosen as in the beginning of Section 3.4.

Denote by [q_{j}, p_{j}^{i}] = V^{u}(q_{j}) ∩ V^{s}(p_{j}^{i}), i = t, a, b (where V^{s} and V^{u} are the stable and unstable local manifolds respectively). For i = a, b, t and j = 1, …, 8 consider the closed quadrilateral (u, s)_{A}-path γ_{j}^{i} with the collection of points q_{j}, [q_{j}, p_{j}^{i}], p_{j}^{i}, [p_{j}^{i}, q_{j}], and q_{j}. In the case n = 1, we take γ_{1}^{i}, i = a, b, t as introduced in the beginning of Section 4.

Recall that η = η_{A} is the expanding rate of A along its unstable direction. Clearly, η^{−1} is the contracting rate along the stable direction of A. Recall also that κ is the function in (A2) such that κ = κ_{0} on U_{1} and | grad κ| < 1/4. We have that the expanding rate of T |𝒩_{y} along W_{T}^{u} is ηκ(y) (here 𝒩_{y} is given by (3.1)).

For n ≥ 1 let us choose a rectangle Z_{ℓ}^{(n)} ∈ 𝒵_{n} and assume that it is labelled by a number j. Consider the case n > 1 and let

η_{−} = η_{−}(n, ℓ) = min{ηκ(y) : y ∈ Z_{ℓ}^{(n)}}

and

α_{u}^{i} = α_{u}^{i}(j) = d(p_{j}^{i}, [p_{j}^{i}, q_{j}]), α_{s}^{i} = α_{s}^{i}(j) = d(p_{j}^{i}, [q_{j}, p_{j}^{i}]) (5.1)

ᾰ_{u}^{i} = ᾰ_{u}(n, ℓ) = α_{u}^{i}(j)/η_{−}(n, ℓ), ᾰ_{s}^{i} = ᾰ_{s}(n, ℓ) = α_{s}^{i}(j)/η_{−}(n, ℓ),

where we write ᾰ_{s}(n, ℓ) instead of ᾰ_{s}(j, n, ℓ) since j is determined by n and ℓ (see Figure 2).

Next for i = t, a, b and j = 1, …, 8 we set

Π_{j}^{i} = B_{F^{u}}(p_{j}^{i}, α_{u}^{i}) × B_{F^{s}}(p_{j}^{i}, α_{s}^{i}), Π̆_{j}^{i} = B_{F^{u}}(p_{j}^{i}, ᾰ_{u}^{i}) × B_{F^{s}}(p_{j}^{i}, ᾰ_{s}^{i}).

We shall assume that the points p_{j}^{i} are chosen in such a way that all three rectangles Π_{j}^{i}, i = t, a, b, are pairwise disjoint. Hence, all the 24 rectangles Π_{j}^{i}, i = t, a, b, j = 1, …, 8 are pairwise disjoint.

Finally, we let

(5.2) ϵ_{t} = ϵ_{t}(n, ℓ) = min{κ(y)/2 : y ∈ Z_{ℓ}^{(n)}}, ϵ̆_{t} = ϵ̆_{t}(n, ℓ) = 5ϵ_{t}(n, ℓ)/6.

In the case n = 1, we have Z_{1}^{(1)} = U_{1}. Choose l_{u}^{i} and l_{s}^{i} such that

A^{−l_{u}^{i}}([p_{1}^{i}, q_{1}]) ∈ B_{X}(p_{1}^{i}, ν/2), A^{l_{s}^{i}}([q_{1}, p_{1}^{i}]) ∈ B_{X}(p_{1}^{i}, ν/2),

where ν > 0 is given by (4.1). Then we set

α_{u}^{i} = α_{u}^{i}(1) = d(p_{j}^{i}, A^{−l_{u}^{i}}([p_{j}^{t}, q_{j}])), α_{s}^{i} = α_{s}^{i}(1) = d(p_{j}^{i}, A^{l_{s}^{i}}([q_{j}, p_{j}^{t}]))

with other quantities and sets to be defined in a similar way.

36 HUYI HU, YAKOV PESIN, ANNA TALITSKAYA

PSfrag replacements q_{j}

p_{j}^{t}
p_{j}^{a}
p_{j}^{b}
Π_{j}^{a}
Π_{j}^{b}
Π_{j}^{t}
V^{u}(q_{j})
V^{s}(q_{j})
V^{u}(p_{j}^{t})
V^{u}(p_{j}^{a})
V^{u}(p_{j}^{b})

Figure 2. Quadrilaterals

To effect our construction of the maps P_{n}, in addition to the squares Ẑ_{ij}^{(n)}, Z̃_{ij}^{(n)} and Z_{ij}^{(n)} constructed in the previous subsection, we need to consider for n ≥ 1 the following squares:

\begin{matrix}Z̆_{ij}^{(n)} = (\frac{i}{2^{n}} − \frac{1}{2^{n+3}} , \frac{i + 1}{2^{n}} + \frac{1}{2^{n+3}} ) × (\frac{j}{2^{n}} − \frac{1}{2^{n+3}} , \frac{j + 1}{2^{n}} − \frac{1}{2^{n+3}} ); \\ Z̄_{ij}^{(n)} = (\frac{i}{2^{n}} − \frac{1}{2^{n+4}} , \frac{i + 1}{2^{n}} + \frac{1}{2^{n+4}} ) × (\frac{j}{2^{n}} − \frac{1}{2^{n+4}} , \frac{j + 1}{2^{n}} − \frac{1}{2^{n+4}} ) \end{matrix}

as well as the following intervals:

\begin{aligned}Ĭ_{n} & = J̆_{n} = (−\frac{5}{2^{n+3}} , \frac{5}{2^{n+3}} ) , Ī_{n} = J̄_{n} = (−9/2^{n+4}, 9/2^{n+4}), \\ Î_{n} & = (−\frac{1}{2^{n+1}} , \frac{1}{2^{n+1}} ) , Ĩ_{n} = (−\frac{17}{2^{n+5}} , \frac{17}{2^{n+5}} ) , I_{n} = (−\frac{3}{2^{n+2}} , \frac{3}{2^{n+2}} ) \end{aligned}

and

\begin{aligned}K̆ & = (−\frac{1}{8}, 1 + \frac{1}{8}) , K̄ = (−1/16, 1 + 1/16), \\ K̂ & = (0, 1) , K̃ = (−1/32, 1 + 1/32) , K = (−1/4, 1 + 1/4) . \end{aligned}

We have that

Ẑ_{ij}^{(n)} ⊂ Z̃_{ij}^{(n)} ⊂ Z̄_{ij}^{(n)} ⊂ Z̆_{ij}^{(n)} ⊂ Z_{ij}^{(n)}

with similar relations for I_{n} and J_{n}.

Fix n ≥ 1 and choose C^{∞} functions ϕ^{i} and ψ^{i} on ℝ for i = a, b, t satisfying: · ϕ^{i}(r) = const. for r ∈ (−ᾰ_{u}^{i}, ᾰ_{u}^{i}) and ψ^{i}(r) = const for r ∈ (−ᾰ_{s}^{i}, ᾰ_{s}^{i}); · ϕ^{i}(r) = 0 for |r| ≥ α_{u}^{i}, and ψ^{i}(r) = 0 for |r| ≥ α_{s}^{i}; · ∫_{0}^{±α_{u}^{i}} ϕ^{i}(τ)dτ = 0, and ψ^{i}(x) > 0 for any |x| < α_{s}^{i}; · ∥ϕ^{i}(·)∥_{C^{n}} < 1 and ∥ψ^{i}(·)∥_{C^{n}} < 1.

Further, choose C^{∞} functions ξ_{t} and ξ_{Y} supported on K and I_{n} respectively such that:

· ξ_{t}(r) = const. for r ∈ K̆, and ξ_{Y} (r) = const. for r ∈ Ĭ_{n};

· ξ_{t}(r) > 0 for r ∈ K and ξ_{Y} (r) > 0 for r ∈ I_{n};
· ξ_{t}(r) = 0 for r ∉ K and ξ_{Y} (r) = 0 for r ∉ I_{n};
· ∥ξ_{t}∥_{C^{n}}, ∥ξ_{Y} ∥_{C^{n}} < 1.

Finally, choose C^{∞} functions ζ_{t} and ζ_{Y} supported on (−ϵ_{t}, ϵ_{t}) and I_{n} respectively such that:

· ζ_{t}(r) = const. for r ∈ (−ϵ̆_{t}, ϵ̆_{t}) and ζ_{Y} (r) = const. for r ∈ Ĭ_{n};
· ζ_{t}(r) > 0 for r ∈ (ϵ_{t}, ϵ_{t}) and ζ_{Y} (r) > 0 for r ∈ I_{n};
· ζ_{t}(r) = 0 for r ∉ (ϵ_{t}, ϵ_{t}) and ζ_{Y} (r) = 0 for r ∉ I_{n};
· ∥ζ∥_{C^{n}} < 1.

Let (a_{0}, b_{0}) = (a_{0}(n, ℓ), b_{0}(n, ℓ)) be the center of the square Z_{ℓ}^{(n)}.

In this section we shall use the coordinate system z = (u, s, t, a, b) = (x, t, a, b) introduced in (3.2) and (3.3) with the origin at (p_{j}^{a}, 1/2, a_{0}, b_{0}). In this coordinate system the interval K is in the symmetric form (−3/4, 3/4). Define

Ω^{a} = Ω_{n,ℓ}^{a} = {z = (x, r, â, b̂) : x ∈ Π_{j}^{a}, |r| ≤ ϵ_{t}, (â, b̂) ∈ Z_{ℓ}^{(n)}}

(recall that j labels the square Z_{ℓ}^{(n)}) and for each β > 0 a vector field X^{a} = X_{β,n,ℓ}^{a} by

(5.3) X^{a}(z) = βζ_{Y} (b̂)ζ_{t}(r)ψ^{a}(s)(−ξ′_{Y} (â) ∫_{0}^{u} ϕ^{a}(τ)dτ, 0, 0, ξ_{Y} (â)ϕ^{a}(u), 0),

(here ξ′_{Y} denotes the derivative of ξ_{Y}). The choice of ϵ_{t} guarantees that T (Ω^{a})∩ Ω^{a} = ∅. It is clear that X^{a} is constant on the set

Ω̆^{a} = {z = (x, r, â, b̂) : x ∈ Π̆_{j}^{a}, |r| ≤ ϵ̆_{t}, (â, b̂) ∈ Z̆^{(n)}}.

We define the map h_{n,ℓ}^{a} = h_{β,n,ℓ}^{a} on Ω^{a} to be the time-1 map of the flow generated by X^{a}, and we set h_{n,ℓ}^{a} = Id on the complement of Ω^{a}. It is easy to see that the vector field X^{a} is divergence free, the differential dh_{n,ℓ}^{a} preserves E_{T}^{ua}, and det(dh_{n,ℓ}^{a}|E_{T}^{ua}(z)) = 1.

Then we use the same coordinate system as above but with the origin at (p_{j}^{b}, 1/2, a_{0}, b_{0}). Define

Ω^{b} = Ω_{n,ℓ}^{b} = {z = (x, r, â, b̂) : x ∈ Π_{j}^{b}, |r| ≤ ϵ_{t}, (â, b̂) ∈ Z_{ℓ}^{(n)}}

and for each β > 0 a vector field X^{b} = X_{β,n,ℓ}^{b} by

(5.4) X^{b}(z) = βζ_{Y} (â)ζ_{t}(r)ψ^{b}(s)(−ξ′_{Y} (b̂) ∫_{0}^{u} ϕ^{b}(τ)dτ, 0, 0, 0, ξ_{Y} (b̂)ϕ^{b}(u)).

Let h_{n,ℓ}^{b} = h_{β,n,ℓ}^{b} on Ω^{b} be the time-1 map of the flow generated by X^{b} and let h_{n,ℓ}^{b} = Id on the complement of Ω^{b}. It is clear that X^{b} is divergence free, dh_{n,ℓ}^{b} preserves E_{T}^{ub}, and det(dh_{n,ℓ}^{b}|E_{T}^{ub}(z)) = 1.

Now we use the coordinate system but with the origin at (p_{j}^{t}, 1/2, a_{0}, b_{0}). Define

Ω^{t} = Ω_{n,ℓ}^{t} = {z = (x, r, y) : x ∈ Π_{j}^{t}, r ∈ K, y ∈ Z_{ℓ}^{(n)}}

and for each β > 0 a vector field X^{t} = X_{β,n,ℓ}^{t} by

(5.5) X^{t}(z) = βζ_{Y} (a)ζ_{Y} (b)ψ^{t}(s)(−ξ′_{t}(r) ∫_{0}^{u} ϕ^{t}(τ)dτ, 0, ξ_{t}(r)ϕ^{t}(u), 0, 0).

We define the map h_{n,ℓ}^{t} = h_{β,n,ℓ}^{t} on Ω^{t} to be the time-1 map of the flow generated by X^{t}, and we set h_{n,ℓ}^{t} = Id on the complement of Ω^{t}. Obviously, X^{t} is divergence free, dh_{n,ℓ}^{t} preserves E_{T}^{ut} and det(dh_{n,ℓ}^{t}|E_{T}^{ut}(z)) = 1.

Our construction guarantees that all {Q_{n,ℓ}^{i}} are pairwise disjoint. For n = 1, 2, … define h_{n} = h_{β,n} by

h_{β,n} = h_{β,n,k_{n}}^{b} ∘ h_{β,n,k_{n}}^{a} ∘ h_{β,n,k_{n}}^{t} ∘ ⋯ ∘ h_{β,n,1}^{b} ∘ h_{β,n,1}^{a} ∘ h_{β,n,1}^{t}.

Then we let P_{1} = h_{β_{1},1,1} ∘ Q and define P_{n} inductively by setting P_{n} = h_{β_{n},n} ∘ P_{n−1} for some suitable choice of {β_{n}} which will be determined inductively later.

5.3. Properties of maps P_{n}: Proof of Proposition 3.3.

Statements (2) and (4) of Proposition 3.3 and the fact that the map P_{n} is homotopic to the identity follow directly from the construction.

Note that the unperturbed map T is uniformly partially hyperbolic on each set 𝒰_{n} with smooth 3-dimensional central foliation and is dynamically coherent. Note also that for each n > 0, by choosing β_{n} in (5.3)-(5.5) sufficiently small, we can ensure that ∥h_{n} − Id ∥_{C^{n}} is arbitrarily small. Hence, we can choose a positive sequence {δ′_{n}} such that δ′_{n} ≤ δ′_{1}/2^{n−1} and if h_{n} and P_{n} satisfy

(5.6)  ∥P_{n} − P_{n−1}∥_{C^{n}} ≤ δ′_{n} and ∥h_{n} − Id ∥_{C^{n}} ≤ δ′_{n},

then Statement (3) of the proposition holds. In particular, P_{n} is pointwise partially hyperbolic on an open set 𝒢; it is uniformly partially hyperbolic on 𝒰_{n} with 3-dimensional central foliation and is dynamically coherent. It remains to show how to choose sequences of positive numbers δ_{n} and θ_{n} such that P_{n} also satisfies Statements (5) and (6) of the proposition.

We denote by W_{P_{n}}^{c}(z) the center manifold of P_{n} at the point z ∈ ℳ. Suppose a square Z_{ℓ}^{(n)} is labelled by a number j. Let q_{j} be the periodic point chosen as in the previous subsection and z_{0} = z_{0}(n, ℓ) = (q_{j}, 1/2, a_{0}(n, ℓ), b_{0}(n, ℓ)). We denote by W_{P_{n}}^{c}(z_{0}, K, Z_{ℓ}^{(n)}) the connected component of W_{P_{n}}^{c}(z_{0}) ∩ (X × K × Z_{ℓ}^{(n)}) that contains z_{0}. We shall also use similar notations W_{P_{n}}^{c}(z_{0}, K̆, Z̆_{ℓ}^{(n)}), etc. Note that for all ℓ and n,

W_{P_{n}}^{c}(z_{0}, K, Z_{ℓ}^{(n)}) = W_{Q}^{c}(z_{0}, K, Z_{ℓ}^{(n)}) = W_{T}^{c}(z_{0}, K, Z_{ℓ}^{(n)}).

Recall that γ_{j}^{i} is the quadrilateral (u, s)_{A}-path with the collection of points q_{j}, [q_{j}, p_{j}^{i}], p_{j}^{i}, [p_{j}^{i}, q_{j}], and q_{j} (for i = a, b, t and j = 1, …, 8) introduced in the beginning of Subsection 5.2. In particular, γ_{1}^{i} = γ^{i} is given in the beginning of Section 4.

For any n ≥ 1, ℓ = 1, …, k_{n}, and j = 1, …, 8 such that the label of Z_{ℓ}^{n} is j, we consider a quadrilateral (u, s)_{P_{n}}-path γ̂_{j}^{a} with the initial point z_{1} such that Proj_{X} γ̂_{j}^{a} = γ_{j}^{a}. More precisely, γ̂_{j}^{a} = {z_{1}, …, z_{5}} where

(5.7)  \begin{aligned}z_{2} & = V_{P_{n}}^{u}(z_{1}) ∩ V_{P_{n}}^{sc}(p_{j}^{a}, 1/2, a_{0}, b_{0}), \\ z_{3} & = V_{P_{n}}^{s}(z_{2}) ∩ V_{P_{n}}^{uc}(p_{j}^{a}, 1/2, a_{0}, b_{0}), \\ z_{4} & = V_{P_{n}}^{u}(z_{3}) ∩ V_{P_{n}}^{sc}(z_{1}), \\ z_{5} & = V_{P_{n}}^{s}(z_{4}) ∩ V_{P_{n}}^{uc}(z_{1}). \end{aligned}

This path defines a map Θ = Θ^{a} = Θ_{n,ℓ,P_{n}}^{a}, given by Θ(z_{1}) = z_{5}. Note that z_{4} ∈ V_{P_{n}}^{sc}(z_{1}) and z_{5} ∈ V_{P_{n}}^{s}(z_{4}). Hence, z_{5} ∈ V_{P_{n}}^{sc}(z_{1}). Since also z_{5} ∈ V_{P_{n}}^{uc}(z_{1}), we obtain that z_{5} ∈ V_{P_{n}}^{c}(z_{1}). This implies that Θ maps W_{P_{n}}^{c}(z_{0}, K, Z_{ℓ}^{n}) into itself.

We contract the (u, s)_{P_{n}}-path γ̂_{j}^{a} to a line segment. Namely, let σ : [0, 1] → V_{P_{n}}^{u}(z_{1}) be a parametrization by the arc length of the part of the curve V_{P_{n}}^{s}(z_{1}) from z_{1} to z_{2} so that σ(0) = z_{1} and σ(1) = z_{2}. For each τ ∈ [0, 1], the new path γ̂_{j}^{a}(τ) = {z_{1}(τ), …, z_{5}(τ)} is such that z_{1}(τ) = z_{1}, z_{2}(τ) = σ(τ) and z_{i}(τ) for i = 3, 4, 5 are obtained in the way similar to (5.7). Thus we obtain a map Θ_{τ} = Θ_{τ,n,ℓ,P_{n}}^{a}, given by Θ_{τ}(z_{1}) = z_{5}. It maps W_{P_{n}}^{c}(z_{0}, K, Z_{ℓ}^{n}) into W_{P_{n}}^{c}(z_{0}) and depends continuously on τ ∈ [0, 1].

Clearly, γ̂_{j}^{a}(1) = γ̂_{j}^{a} and hence, Θ_{1,n,ℓ,P_{n}}^{a} = Θ_{n,ℓ,P_{n}}^{a}. Furthermore, the path γ̂_{j}^{a}(0) degenerates to a path on V_{P_{n}}^{s}(z_{1}) that starts from z_{1} = z_{2}, goes to z_{3} = z_{4} and then returns to z_{5} = z_{1}. Hence, Θ_{0} = Id.

We stress that Θ_{n,ℓ,P_{n}}^{a} depends only on h_{n,ℓ}^{a}, since γ̂_{j}^{a} consists of strongly stable and unstable leaves of (q_{j}, 1/2, y) and (p_{j}^{a}, 1/2, y) with y ∈ Z_{ℓ}^{(n)} that are not perturbed by any other perturbations h_{n′,ℓ′}^{a} if (n′, ℓ′) ≠ (n, ℓ). On the other hand, if τ ∈ (0, 1), then Θ_{τ,n,ℓ,P_{n}}^{a} may depend on other perturbations h_{n′,ℓ′}^{a}.

By using the paths γ_{j}^{b} and γ_{j}^{t} respectively, we can define the maps Θ_{τ}^{b} = Θ_{τ,n,ℓ,P_{n}}^{b} and Θ_{τ}^{t} = Θ_{τ,n,ℓ,P_{n}}^{t} for τ ∈ [0, 1] in a similar way. Furthermore, for any gentle perturbation P^{♮} of P_{n} we can also construct the maps Θ_{n,ℓ,P^{♮}}^{a} and Θ_{τ,n,ℓ,P^{♮}}^{a} from W_{P^{♮}}^{c}(z_{0}, K, Z_{ℓ}^{n}) to itself. Clearly, they have properties similar to those of the maps Θ_{n,ℓ,P_{n}}^{a} and Θ_{τ,n,ℓ,P_{n}}^{a}. Note that V_{P^{♮}}^{u}, V_{P^{♮}}^{s} and V_{P^{♮}}^{c} depend continuously on the perturbation P^{♮} as long as P^{♮} is a gentle perturbation of T with P^{♮} = T outside some fixed U_{n} and with ∠(E_{P^{♮}}^{i}(z), E_{T}^{i} (z)) sufficiently small for all z ∈ 𝒰_{n} and i = u, s, c. It follows that Θ_{n,ℓ,P^{♮}}^{i} and Θ_{τ,n,ℓ,P^{♮}}^{i}, for i = u, s, c depend continuously on P^{♮} as well. Since the lengths of all the quadrilateral paths used in the construction of the maps Θ^{i} and Θ_{τ}^{i} are uniformly bounded from above, the continuity is uniform with respect to z.

Given j = 1, … 8 and a point z = (x, t, y), we can find a (u, s)_{T} -path γ_{T} (z) connecting z to the point z′ = (q_{j}, t, y) whose length does not exceed 2d(x, q_{j}) (indeed, such a path can be constructed by using at most three points z, z_{1} and z′). This generates a map Ψ_{T} = Ψ_{T,j} from 𝒢 to {q_{j}} × K × G given by Ψ_{T} (z) = z′.

Furthermore, given a gentle perturbation P^{♮} of T and a point z ∈ Z_{ℓ}^{(n)}, we can find a (u, s)_{P^{♮}}-path γ_{P^{♮}}(z), which is close to γ_{T} (z) and connect z to a point z′ = z′(P^{♮}) ∈ W_{P^{♮}}^{c}(z_{0}(n, ℓ), K, Z_{ℓ}^{(n)}) and we can then define Ψ_{P ♮}(z) = z′(P^{♮}). Again the path can be chosen to consists of at most three point z, z_{1} = z_{1}(P^{♮}) and z′ = z′(P^{♮}), and both z_{1}(P^{♮}) and z′(P^{♮}) depend continuously on P^{♮}. Hence, Ψ_{P^{♮}} depends continuously on P^{♮} as long as P^{♮} is a gentle perturbation of T with P^{♮} = T outside some fixed U_{n} and with ∠(E_{P^{♮}}^{i}(z), E_{T}^{i} (z)) sufficiently small for all z ∈ 𝒰_{n} and i = u, s, c. We stress that the lengths of all the paths used in the construction of the map Ψ are uniformly bounded from above for all z and all gentle perturbations P^{♮}. In particular, the continuity is uniform with respect to z.

Given a set Γ ⊂ ℳ and a gentle perturbation P^{♮} of T, let

(5.8)

𝒜_{P^{♮}}(Γ) = {z ∈ ℳ : there is y ∈ Γ such that
y is accessible to z via a (u, s)_{P^{♮}}-path}.

For n ≥ 1 denote by ϵ_{n} = min{1/2^{n+5}, ϵ̆_{t}(n, ℓ), ℓ = 1, …, k_{n}} where ϵ̆_{t}(n, ℓ) is given by (5.2).

We shall now show how to choose the sequence {δ_{n}}. Recall that U_{1} = Z_{i_{0}j_{0}}^{(1)} = Z_{1}^{(1)} and Ũ_{1} = Z̃_{1}^{(1)}. We can choose a number θ_{0} > 0 such that for any gentle perturbation P^{♮} of T with ∠(E_{P^{♮}}^{i}(z), E_{T}^{i} (z)) ≤ 2θ_{0}(z) for i = s, c, u and z ∈ 𝒰_{1} the maps Ψ_{P^{♮}} and Θ_{τ,1,1,P^{♮}}^{i} are well defined. We also assume that the number δ_{Q} in Proposition 3.2 is so small that the map P_{0} = Q satisfies ∠(E_{P_{0}}^{i}(z), E_{T}^{i} (z)) ≤ θ_{0} and d(Θ_{τ,1,1,P_{0}}^{i}(z), z) ≤ ϵ_{1}/4 for z ∈ 𝒢_{0}, τ ∈ [0, 1] and i = s, c, u.

Now we choose a number θ_{1} such that 0 < θ′_{1} ≤ θ_{0}/2 and if ∠(E_{P^{♮}}^{i}(z), E_{P_{0}}^{i}(z)) ≤ 2θ′_{1} for i = s, c, u and z ∈ 𝒩 × Z_{1}^{(1)}, then

(5.9)  d(Ψ_{P^{♮}}(z), Ψ_{P_{0}}(z)) ≤ 1/2^{8}, z ∈ 𝒩 × Z_{1}^{(1)}.

Finally, we may assume that the number δ′_{1} in (5.6) is chosen so small that if ∥P_{1} − P_{0}∥ ≤ δ′_{1}, then ∠(E_{P_{1}}^{i}(z), E_{P_{0}}^{i}(z)) ≤ θ′_{1} for i = s, c, u and z ∈ 𝒩 × Z_{1}^{(1)}.

Now we set δ_{1} = min{δ′_{1}, δ′′_{1}} and θ_{1} = min{θ′_{1}, θ′′_{1}} where the numbers δ′′_{1} and θ′′_{1} are given by Lemma 5.2 below. For any gentle perturbation P^{♮} of P_{1} with ∠(E_{P^{♮}}^{i}(z), E_{P_{1}}^{i}(z)) ≤ θ′_{1} for i = s, c, u and z ∈ 𝒩 × Z_{1}^{(1)}, we have ∠(E_{P^{♮}}^{i}(z), E_{P_{0}}^{i}(z)) ≤ 2θ′_{1} and therefore (5.9) holds. Since d(Θ_{τ,1,1,P_{0}}^{i}(z), z) ≤ ϵ_{1}/4, we can apply Lemma 5.2 to obtain that d(Θ_{τ,2,ℓ,P_{1}}^{i}(z), z) ≤ ϵ_{2}/4 for all z ∈ W_{P_{1}}^{c}(z_{0}(2, ℓ), K, Z_{ℓ}^{(2)}), i = u, s, c, τ ∈ [0, 1] and ℓ = 1, …, k_{2}. Moreover,

𝒜_{P^{♮}}(z_{0}) ⊃ W^{c}(z_{0}(1, 1), K̄, Z̄_{1}^{(1)}).

Since the distance between the boundaries ∂Z̄_{1}^{(1)} and ∂Z̃_{1}^{(1)} is 1/2^{6}, (5.9) implies that

Ψ_{P^{♮}}(𝒩 × Z̃_{1}^{(1)}) ⊂ W_{P^{♮}}^{c}(z_{0}(1, 1), K̄, Z̄_{1}^{(1)}).

By definition, z and Ψ_{P^{♮}}(z) are (u, s)_{P^{♮}}-accessible and hence, we have that

𝒜_{P^{♮}}(z_{0}(1, 1)) ⊃ 𝒩 × Z̃_{1}^{(1)}.

In particular, for P^{♮} = P_{1}, the inclusion holds and so does (5.9).

Proceeding inductively, we assume that for j = 1, …, n − 1, the maps P_{j}, and the numbers δ_{j} and θ_{j} are chosen such that (5.6) and Statements (5) and (6) of the proposition hold. Moreover, we assume that for all i = u, s, c, τ ∈ [0, 1], ℓ = 1, …, k_{j+1},

(5.10)d(Ψ_{P_{j}}(z), Ψ_{P_{j−1}}(z)) ≤ 1/2^{j+7} for all z ∈ 𝒩 × Z_{ℓ}^{(j)},
(5.11) d(Θ_{τ,j+1,ℓ,P_{j}}^{i}(z), z) ≤ ϵ_{j+1}/4 for all z ∈ W_{P_{j}}^{c}(z_{0}(j + 1, ℓ), K, Z_{ℓ}^{(j+1)}).

Now we choose 0 < θ′_{n} ≤ θ_{n−1}/2 in such a way that for any gentle perturbation P^{♮} of P_{n−1}, if ∠(E_{P^{♮}}^{i}(z), E_{P_{n−1}}^{i}(z)) ≤ 2θ′_{n} for i = u, s, c, z ∈ 𝒩 × Z_{ℓ}^{(n)}, and ℓ = 1, …, k_{n} then

(5.12)  d(Ψ_{P^{♮}}(z), Ψ_{P_{n−1}}(z)) ≤ 1/2^{n+7}

for all z ∈ 𝒩 × Z_{ℓ}^{(n−1)} and ℓ = 1, …, k_{n}. Reducing δ′_{n} in (5.6) further if necessary we may assume that if ∥P_{n}−P_{n−1}∥ ≤ δ′_{n} then ∠(E_{P_{n}}^{i}(z), E_{P_{n−1}}^{i}(z)) ≤ θ′_{n} for i = u, s, c and z ∈ 𝒰_{n}. Then we take δ_{n} = min{δ′_{n}, δ′′_{n}} and θ_{n} = min{θ′_{n}, θ′′_{n}}, where θ′′_{n} and δ′′_{n} are given in Lemma 5.2.

Since 0 < θ_{n} ≤ θ_{n−1}/2, Statement (5) of the proposition holds.

Let P^{♮} be a gentle perturbation of P_{n} such that ∠(E_{P^{♮}}^{i}(z), E_{P_{n}}^{i}(z)) ≤ θ_{n} for i = u, s, c and z ∈ 𝒰_{n}. Then ∠(E_{P^{♮}}^{i}(z), E_{P_{n−1}}^{i}(z)) ≤ 2θ′_{n} ≤ θ_{n−1} for z ∈ 𝒰_{n}. By Statement (6), we get that P^{♮} has the accessibility property on 𝒰̃_{n−1}.

Since P_{n−2} = T on 𝒩 × Z_{ℓ}^{(n)}, applying (5.10) with j = n − 1, we find that d(Ψ_{P_{n−1}}(z), Ψ_{T} (z)) ≤ 1/2^{n+6} for all z ∈ 𝒩 × Z_{ℓ}^{(n−1)} and ℓ = 1, …, k_{n−1}. Therefore by (5.12), we obtain that

d(Ψ_{P^{♮}}(z), Ψ_{T} (z)) ≤ 1/2^{n+6} + 1/2^{n+7} < 1/2^{n+5}.

Applying (5.11) with j = n−1, we conclude that the requirement of Lemma 5.2 below holds. Therefore by the lemma and the fact that d(∂Z̃_{ℓ}^{(n)}, ∂Z̄_{ℓ}^{(n)}) = 1/2^{n+5}, we obtain following the same line of arguments as in the case n = 1 that

𝒜_{P^{♮}}(z_{0}(n, ℓ)) ⊃ 𝒩 × Z̃_{ℓ}^{(n)}

for all ℓ = 1, …, k_{n}. In other words, P^{♮} has the accessibility property on 𝒩 × Z̃_{ℓ}^{(n)} for ℓ = 1, …, k_{n}. By the construction,

𝒰̃_{n} = (𝒰̃_{n−1}) ⋃(⋃_{l=1}^{k_{n}} 𝒩 × Z̃_{ℓ}^{(n)}).

Note that any intersection Z̃_{ℓ}^{(n)} ∩ Z̃_{ℓ′}^{(n)} or Z̃_{ℓ}^{(n)} ∩ Z̃_{ℓ′}^{(n−1)}, if not empty, contains a rectangle of width 1/2^{n+4}. Hence, the intersection of any two sets among 𝒰̃_{n−1} and 𝒩 × Z̃_{ℓ}^{(n)}, ℓ = 1, …, k_{n}, contains a nonempy open set whenever they intersect. Since Ũ_{n} is connected, we obtain accessiblity of P^{♮} on 𝒰̃_{n}.

Applying the above result with P^{♮} = P_{n}, we obtain that P_{n} has the accessiblity property on 𝒰̃_{n}. Moreover, (5.12) for P^{♮} = P_{n} gives (5.10), and (5.13) below gives (5.11) for j = n.

5.4. A technical lemma. We prove here some of our main technical statements.

Lemma 5.2. Suppose for some n > 0, d(Θ_{τ,n,ℓ,P_{n−1}}^{i}(z), z) ≤ ϵ_{n}/4 for all i = u, s, c, τ ∈ [0, 1], z ∈ W_{P_{n−1}}^{c}(z_{0}(n, ℓ), K, Z_{ℓ}^{(n)}), ℓ = 1, …, k_{n}. Then there are δ′′_{n} > 0 and θ′′_{n} > 0 such that if P_{n} satisfies ∥P_{n} − P_{n−1}∥ ≤ δ′′_{n}, then we have

(5.13) d(Θ_{τ,n+1,ℓ,P_{n}}^{i}(z), z) ≤ ϵ_{n+1}/4 as z ∈ W_{P_{n}}^{c}(z_{0}(n + 1, ℓ), K, Z_{ℓ}^{(n+1)}),

for all i = u, s, c, τ ∈ [0, 1], ℓ = 1, …, k_{n+1}; and for any gentle perturbation P^{♮} of P_{n} with

∠(E_{P^{♮}}^{i}(z), E_{P_{n}}^{i}(z)) ≤ θ′′_{n} for all z ∈ 𝒩 × Z_{ℓ}^{(n)}, i = u, s, c,

we have

(5.14)  𝒜_{P^{♮}}(z_{0}(n, ℓ)) ⊃ W_{P^{♮}}^{c}(z_{0}(n, ℓ), K̄, Z̄_{ℓ}^{(n)}) for all ℓ = 1, …, k_{n}.

In particular, (5.14) holds with P^{♮} = P_{n}.

Proof. Take θ′′_{n} ≤ θ_{n−1}/2 such that for any gentle perturbation P^{♮} of P_{n−1}, if

∠(E_{P^{♮}}^{i}, E_{P_{n−1}}^{i}) ≤ 2θ′′_{n} on 𝒩 × Z_{ℓ}^{(n)}, i = u, s, c,

then (5.13) holds with P_{n} = P^{♮}, and

(5.15)  d(Θ_{τ,n,ℓ,P^{♮}}^{i}(z), z) ≤ ϵ_{n}/2 as z ∈ W_{P^{♮}}^{c}(z_{0}(n, ℓ), K, Z_{ℓ}^{(n)}),

for all i = u, s, c, τ ∈ [0, 1], and ℓ = 1, …, k_{n}. (5.15) is possible because of the assumption of the lemma, while (5.13) is possible because on 𝒩 × Z_{ℓ}^{(n+1)}, P_{n−1} = T and therefore d(Θ_{τ,n+1,ℓ,P_{n−1}}^{i}(z), z) = 0. Then we take δ′′_{n} ≤ δ_{n−1}/2 such that if ∥P_{n} − P_{n−1}∥ ≤ δ′′_{n}, then ∠(E_{P_{n}}^{i}, E_{P_{n−1}}^{i}) ≤ θ′′_{n} on 𝒩 × Z_{ℓ}^{(n)} for i = u, s, c. Hence, (5.15) is satisfied with P^{♮} = P_{n}.

Now we only need to prove (5.14) for one square Z_{ℓ}^{(n)}.

Define a continuous function Φ = Φ_{P_{n}}^{(1)} : ℝ → W_{P_{n}}^{c}(z_{0}) by using Θ = Θ_{n,ℓ,P_{n}}^{a} and Θ_{τ} = Θ_{τ,n,ℓ,P_{n}}^{a} such that the image of Φ consists of points accessible to z_{0} = (q_{j}, 1/2, a_{0}, b_{0}). Namely,

(1) Φ(0) = z_{0};

(2) For a positive integer n if Φ(n − 1) = (q_{j}, \frac{1}{2}, a, b_{0}) for some a ∈ I_{n}, then we let Φ(n) = Θ(Φ(n − 1));

(3) For a negative integer −n if Φ(−n + 1) = (q_{j}, \frac{1}{2}, a, b_{0}) for some a ∈ I_{n}, then we let Φ(−n) = Θ^{−1}(Φ(−n + 1)); in other words, Φ(−n) is the terminate point of the quadrilateral (u, s)_{P_{n}}-path γ̂_{j}^{a} with the initial point Φ(−n + 1) such that π_{X} γ̂_{j}^{a} = γ_{j}^{a} with the direction reversed; (4) For any real number n + τ, where n ∈ ℤ and τ ∈ [0, 1) if Φ(n) =

(q_{j}, \frac{1}{2}, a, b_{0}) then we let Φ(n + τ) = Θ_{τ}(Φ(n)). In fact, if we denote by ⌊r⌋ the greatest integer that is less than or equal to r, then we have

Φ_{P_{n}}^{(1)}(r) = Θ_{r−⌊r⌋}^{a} ∘ (Θ^{a})^{⌊r⌋}(z_{0}).

Since, lim_{τ→1} Θ_{τ}^{a} = Θ^{a} and lim_{τ→0} Θ_{τ}^{a} = Id we have that Φ_{P_{n}}^{(1)} is a continuous function of r. Furthermore,

Φ_{P_{n}}^{(1)}(ℝ) ⊂ 𝒜_{P_{n}}(z_{0}(n, ℓ)).

By Lemma 5.3 below,

Φ_{P_{n}}^{(1)}(ℤ) ⊂ {(q_{j}, 1/2, a, b_{0}) : a ∈ I_{n}, } ⊂ W_{P_{n}}^{c}(z_{0}, K, Z_{ℓ}^{(n)}).

Hence, by (5.15) with P^{♮} = P_{n},

Φ_{P_{n}}^{(1)}(ℝ) ⊂ {(q_{j}, t, a, b) : |t − 1/2| < ϵ_{n}/2, a ∈ I_{n}(ϵ_{n}/2), |b − b_{0}| ≤ ϵ_{n}/2},

where I_{n}(ϵ_{n}) denotes the ϵ_{n}-neighborhood of I_{n} in ℝ. It is also clear that

lim_{n→±∞} Φ_{P_{n}}^{(1)}(±n) = (q_{j}, 1/2, a_{0} ∓ \frac{3}{2^{n+2}} , b_{0}),

where the two points on the right hand side is on the boundary of Z_{ℓ}^{(n)}. Hence, we can choose an integer N = N_{n,ℓ}^{a} > 0 such that Φ_{P_{n}}^{(1)}(±N) ∉ π_{Y}^{−1}Z̆_{ℓ}^{(n)}. In other words, Φ_{P_{n}}^{(1)}([−N, N]) forms a continuous curve near the line segment {(q_{j}, 1/2, a, b_{0}) : a ∈ I_{n}} and crosses Z̆_{ℓ}^{(n)} in F^{a} direction.

Now we use the maps Θ = Θ_{n,ℓ,P_{n}}^{b} and Θ_{τ} = Θ_{τ,n,ℓ,P_{n}}^{b} to define a function Φ = Φ_{P_{n}}^{(2)} : ℝ^{2} → W_{P_{n}}^{c}(z_{0}) such that the image of Φ consists of the points accessible to z_{0}. Namely, given r ∈ ℝ, let

(1) Φ(r, 0) = Φ_{P_{n}}^{(1)}(r);

(2) For a positive integer n if Φ(r, n − 1) is defined, we let Φ(r, n) = Θ(Φ(r, n − 1));

(3) For a negative integer −n if Φ(r, −n + 1) is defined, we let Φ(r, −n) = Θ^{−1}(Φ(r, −n + 1));

(4) For any real number n + τ, where n ∈ ℤ and τ ∈ [0, 1) if Φ(r, n) is

defined, we let Φ(r, n + τ) = Θ_{τ}(Φ(r, n)). In other words,

Φ_{P_{n}}^{(2)}(r, r′) = Θ_{r′−⌊r′⌋}^{b} ∘ (Θ^{b})^{⌊r′⌋}(Φ_{P_{n}}^{(1)}(r))

or equivalently,

Φ_{P_{n}}^{(2)}(r, r′) = Θ_{r′−⌊r′⌋}^{b} ∘ (Θ^{b})^{⌊r′⌋} ∘ Θ_{r−⌊r⌋}^{a} ∘ (Θ^{a})^{⌊r⌋}(z_{0}).

It is clear that Φ_{P_{n}}^{(2)} is continuous, and Φ(ℝ^{2}) ⊂ 𝒜(q_{j}, 1/2, a_{0}, b_{0}). Furthermore, for r ∈ ℝ,

Φ(r, ℤ) ⊂ {Φ_{P_{n}}^{(1)}(r) + (0, 0, 0, 0, b) : π_{F^{b}}Φ_{P_{n}}^{(1)}(r) + b ∈ J_{n}}.

Hence, by (5.15) with P^{♮} = P_{n},

Φ(r, ℝ) ⊂ {(q_{j}, t, a, b) : |t − 1/2| ≤ ϵ_{n}, (a, b) ∈ Z_{ℓ}^{(n)}(ϵ_{n})},

where Z_{ℓ}^{(n)}(ϵ_{n}) denotes the ϵ_{n}-neighborhood of Z_{ℓ}^{(n)} in Y. This means that Φ(ℝ^{2}) is contained in the ϵ_{n}-neighborhood of the set {q_{j}} × {1/2} × Z_{ℓ}^{(n)}.

Similarly, for every r ∈ ℝ there exists N(r) = N_{n,ℓ}^{b}(r) such that the set Φ(r, [−N(r), N(r)]) forms a continuous curve near J_{n}(r) and crosses Z̆_{ℓ}^{(n)}. By continuity, we can take N = N_{n,ℓ}^{b} such that Φ(r, [−N, N]) crosses Z̆_{ℓ}^{(n)} for all r ∈ [−N, N]. Moreover, we may assume that N_{n,ℓ}^{a} = N_{n,ℓ}^{b}, since otherwise we may use the larger one instead. By continuity, we obtain that the four curves

Φ(−N, [−N, N]), Φ(N, [−N, N]), Φ([−N, N], −N), Φ([−N, N], N)

form a closed curve. The projection of the curves under Proj_{Y} is outside Z̆_{ℓ}^{(n)}. Hence, by Sublemma 5.4, we get that Proj_{Y} {Φ(r, r′) : r, r′ ∈ [−N, N]} covers Z̆_{ℓ}^{(n)}.

Finally, we use the maps Θ = Θ_{n,ℓ,P_{n}}^{t} and Θ_{τ} = Θ_{τ,n,ℓ,P_{n}}^{t} to define a function Φ = Φ_{P_{n}}^{(3)} : ℝ^{3} → W_{P_{n}}^{c}(z_{0}) such that the image of Φ consists of the points accessible to z_{0}. See Figures 3 and 4. Namely, given r, r′ ∈ ℝ, let

(1) Φ(r, r′, 0) = Φ_{P_{n}}^{(2)}(r, r′);

(2) For a positive integer n if Φ(r, r′, n − 1) is defined, we let Φ(r, r′, n) = Θ(Φ(r, r′, n − 1));

(3) For a negative integer −n if Φ(r, r′, −n+1) is defined, we let Φ(r, r′, −n) = Θ^{−1}(Φ(r, r′, −n + 1));

(4) For any real number n + τ, where n ∈ ℤ and τ ∈ [0, 1) if Φ(r, r′, n) is defined, we let Φ(r, r′, n + τ) = Θ_{τ}(Φ(r, r′, n)).

We have

Φ_{P_{n}}^{(3)}(r, r′, r′′) = Θ_{r′′−⌊r′′⌋}^{t} ∘ (Θ^{t})^{⌊r′′⌋} ∘ Θ_{r′−⌊r′⌋}^{b} ∘ (Θ^{b})^{⌊r′⌋} ∘ Θ_{r−⌊r⌋}^{a} ∘ (Θ^{a})^{⌊r⌋}(z_{0}).

The function Φ_{P_{n}}^{(3)} is continuous and Φ(ℝ^{3}) ⊂ 𝒜(z_{0}).

We also have that there exists N > 0 such that Φ maps the surfaces of the cube [−N, N] × [−N, N] × [−N, N] into outside the corresponding surfaces of W_{P_{n}}^{c}(z_{0}, K̆, Z̆_{ℓ}^{(n)}) and inside the corresponding surfaces of the 2ϵ_{n}-neighborhood PSfrag replacements

t direction b direction (a_{0}, b_{0}) z_{0} N × Y

projection onto Y Z̃_{n}^{l}

p_{j}^{t}

Φ^{(3)}([−N, N] × [−N, N], N)

Figure 3. The action of the function Φ^{(3)} on the central direction of z_{0}

of W_{P_{n}}^{c}(z_{0}, K, Z_{ℓ}^{(n)}). By Sublemma 5.4, {Φ(r, r′, r′′) : r, r′, r′′ ∈ [−N, N]} covers W_{P_{n}}^{c}(z_{0}, K̆, Z̆_{ℓ}^{(n)}), and we obtain that

𝒜(z_{0}) ⊃ W_{P_{n}}^{c}(z_{0}, K̆, Z̆_{ℓ}^{(n)}).

We may reduce δ′′_{n} again if neccesary such that any gentle perturbation P^{♮} of P_{n} satisfying ∥P^{♮} − P_{n}∥ ≤ δ′′_{n} is so close to the unperturbed map P_{n} that the map Θ_{P^{♮}}^{i} = Θ_{n,ℓ,P^{♮}}^{i} and Θ_{τ,P^{♮}}^{i} = Θ_{τ,n,ℓ,P^{♮}}^{i} are well defined on for i = u, s, c and τ ∈ [0, 1], and close to Θ_{P_{n}}^{i} = Θ_{n,ℓ,P_{n}}^{i} and Θ_{τ,P_{n}}^{i} = Θ_{τ,n,ℓ,P_{n}}^{i} respectively. Then we define Φ_{P^{♮}}^{(3)} : ℝ^{3} → W_{P^{♮}}^{c}(z_{0}, K, Z_{ℓ}^{(n)}) by

Φ_{P^{♮}}^{(3)}(r, r′, r′′) = Θ_{{r′′},P^{♮}}^{t} ∘ (Θ_{P^{♮}}^{t})^{⌊r′′⌋} ∘ Θ_{{r′},P^{♮}}^{b} ∘ (Θ_{P^{♮}}^{b})^{⌊r′⌋} ∘ Θ_{{r},P^{♮}}^{a} ∘ (Θ_{P^{♮}}^{a})^{⌊r⌋}(z_{0}),

where {r} = r − ⌊r⌋ denotes the fractional part of r. If θ′′_{n} is small enough and ∠(E_{P_{n}}^{i}(z), E_{P^{♮}}^{i}(z)) ≤ θ′′_{n} for i = u, s, c and all z ∈ 𝒩 × Z_{ℓ}^{(n)}, then Θ_{P^{♮}}^{i} and Θ_{τ,P^{♮}}^{i} are sufficiently close to Θ_{P_{n}}^{i} and Θ_{τ,P_{n}}^{i} respectively for i = a, b, t. Thus we can obtain that

d(Φ_{P^{♮}}^{(3)}(r, r′, r′′), Φ_{P_{n}}^{(3)}(r, r′, r′′)) ≤ 1/2^{n+4},

which is the distance between ∂Z̆_{k}^{(n)} and ∂Z̄_{k}^{(n)}, for r, r′, r′′ ∈ [−N, N]. In other words, Φ_{P^{♮}}^{(3)}(r, r′, r′′) maps the surface of the cube [−N, N]×[−N, N]×[−N, N]

COEXISTENCE OF ZERO AND NONZERO LYAPUNOV EXPONENTS 47

PSfrag replacements

a direction b direction

(a_{0}, b_{0})

Z̃_{n}^{l}
p_{j}^{t}

Φ^{(2)}([−N, N], −N) Φ^{(2)}(N, [−N, N]) Φ^{(2)}(−N, [−N, N]) Φ^{(2)}([−N, N], N)

p_{j}^{a}

p_{j}^{b}
Π_{j}^{a}
Π_{j}^{b}
Π_{j}^{t}
V^{u}(q_{j})
V^{s}(q_{j})
V^{u}(p_{j}^{t})
V^{u}(p_{j}^{a})
V^{u}(p_{j}^{b})

Figure 4. The function Φ

to the surfaces that are close to and outside the corresponding surfaces of W_{P^{♮}}^{c}(z_{0}, K̄, Z̄_{ℓ}^{(n)}). Hence, by Sublemma 5.4, the set

{Φ_{P^{♮}}^{(3)}(r, r′, r′′) : r, r′, r′′ ∈ [−N, N]}

covers W_{P^{♮}}^{c}(z_{0}, K̄, Z̄_{ℓ}^{(n)}) implying that

𝒜_{P^{♮}}(z_{0}(n, ℓ)) ⊃ W_{P^{♮}}^{c}(z_{0}(n, ℓ), K̄, Z̄_{ℓ}^{(n)}).

The desired result follows. □

Sublemma 5.3. For each n > 0, there exists δ′′_{n} > 0 such that if ∥h_{n}−Id ∥_{C^{n}} ≤ δ′′_{n} > 0, then for any a ∈ I_{n},

(1) Θ^{a}((q_{j}, 1/2, a, b_{0})) = (q_{j}, 1/2, a′, b_{0}) with a′ < a;

(2) b ∈ J_{n}, t ∈ (1/2 − ϵ_{t}, 1/2 + ϵ_{t}), Θ^{b}((q, t, a, b)) = (q, t, a, b′) with b′ < b;

(3) b ∈ J_{n}, t ∈ K, Θ^{t}((q, t, a, b)) = (q, t′, a, b) with t′ < t.

Proof. The proof is similar to that of Lemma B.4 in [9].

We prove the first statement. Consider the coordinate system in Ω^{a} with the origin at (p_{j}^{a}, 1/2, a_{0}, b_{0}) which therefore has local coordinates (0, 0, 0, 0, 0). We may assume that the local coordinates of the points q_{j}, [q, p_{j}^{a}] and [p_{j}^{a}, q] are (u_{0}, s_{0}), (0, s_{0}) and (u_{0}, 0) respectively, where u_{0} = α_{u}^{a} and s_{0} = α_{s}^{a} are given by (5.1).

We first consider the case n > 1 and note that the path γ̂_{j}^{a} is contained in the closure of Ω_{n,ℓ}^{a} (see Subsection 5.2) for n > 1 and ℓ = 1, … k_{n}. We have that P_{n}|Ω_{n,ℓ}^{a} = h_{n,ℓ}^{a} ∘ T. Furthermore, since h_{n,ℓ}^{a} = Id on the curve V_{T}^{u}(q_{j}, t, y) for t ∈ K and y ∈ Z_{ℓ}^{(n)}, we have that V_{P_{n}}^{u}(q_{j}, t, y) = V_{T}^{u}(q_{j}, t, y). It follows that if (u_{0}, s_{0}, 0, a_{1}, 0) are the local coordinates of the point z_{1} = (q_{j}, 1/2, a_{1}, b_{0}), then (0, s_{0}, 0, a_{2}, 0) are the local coordinates of the point z_{2} = ([q_{j}, p_{j}^{a}], 1/2, a_{2}, b_{0}) with a_{2} = a_{1}.

Recall that by (5.3), the a-component of the vector field X^{a}(z) is equal to βϕ^{a}(u)ψ^{a}(s)ζ_{t}(t)ζ_{Y} (b)ξ_{Y} (a) and that ϕ^{a}(u), ψ^{a}(s), ζ_{t}(t) and ζ_{Y} (b) are constants for |u| ≤ ᾰ_{u}^{a}, |s| ≤ ᾰ_{s}^{a}, |t| ≤ ϵ̆^{t} and b ∈ J̆_{n} respectively. Recall also that the map h^{a} preserves the s-, t- and b-coordinates. Therefore, if |u| ≤ ᾰ_{u}^{a}, |s| ≤ ᾰ_{s}^{a}, |t| ≤ ϵ̆^{t}, a ∈ I_{n}, and b ∈ J̆_{n}, then

(5.16)  h^{a}(u, s, t, a, b) = (u′, s, t, a + c(a, t), b),

where u′ is close to u provided β is sufficiently small, and c(a, t) > 0 if |t| ≤ ϵ_{t} and c(a, t) = 0 otherwise. Moreover, if |t| ≤ ϵ̆^{t}, then c(a, t) = c(a) is independent of t. Also, if u = 0 then u′ = 0 since in this case ∫_{0}^{u} ϕ^{a}(r)dr = 0 and therefore the u-component of X^{a} is zero. Since α_{s}^{a}/ηκ(a, b) ≤ ᾰ_{s}^{a}, we have that for |s| ≤ α_{s}^{a} and b ∈ J̆_{n},

\begin{matrix}P_{n}(0, s, t, a, b) = h^{a}(T (0, s, t, a, b)) \\ = & (0, s/ηκ(a, b), t + κ(a, b), a + c(a, t), b). \end{matrix}

Note that P_{n} = T near the orbit of (p_{j}^{a}, t, y) and outside Ω^{a}. Hence, under the iterations of P_{n} all points of the set {(0, s, t, a, b) : |s| ≤ α_{s}^{a}} have fixed u- and b-coordinates and the same t- and a-coordinates. Therefore, this set belongs to V^{s}(p_{j}^{a}, t, a, b). Since z_{2} ∈ V^{s}(z_{3}), the fact that(0, s_{0}, 0, a_{2}, 0) are the local coordinates of the point z_{2} yields that (0, 0, 0, a_{3}, 0) are the local coordinates of the point z_{3} = (p_{j}^{a}, 1/2, a_{3}, b_{0}) with a_{3} = a_{2}.

By similar arguments, we can show that if |u| ≤ ᾰ_{u}^{a}, a ∈ I_{n} and b ∈ J̆_{n}, then

(5.17)  \begin{matrix}P_{n}^{−1}(u, 0, t, a, b) = T^{−1}((h^{a})^{−1}(u, 0, t, a, b)) \\ = & (u′′/ηκ(a, b), 0, t − κ(a′, b), a′, b), \end{matrix}

for some u′′ close to u, where by (5.16), a′ satisfies a′+c(a′, t) = a. If we choose δ′′_{n} > 0 small enough, then ∥h_{n}^{a} − Id ∥ ≤ δ′′_{n} implies that u′′ is sufficiently close to u and therefore |u′′|/ηκ(a, b) ≤ ᾰ_{u}^{a}. Hence, under the iterations of P_{n}^{−1} all points of the set {(u, 0, t, a′, b) : |u| ≤ ᾰ_{u}^{a}} have fixed s- and b-coordinates and the same t- and a-coordinates. Therefore, this set belongs to V^{u}(p_{j}^{a}, t, a, b). On the other hand, by the definition of h^{a} and the choice of α_{u}^{a}, we have h^{a} = Id if |u| ≥ α_{u}^{a}. Therefore, since u_{0} = α_{u}^{a},

(5.18)  \begin{matrix}P_{n}^{−1}(u_{0}, 0, t, a′, b) = T^{−1}(u_{0}, 0, t, a′, b) \\ = & (u_{0}/ηκ(a, b), 0, t − κ(a′, b), a′, b). \end{matrix}

Comparing (5.18) with (5.17) we obtain that the point with local coordinates (u_{0}, 0, t, a′, b) is on the strongly unstable local manifold of the point with local coordinates (0, 0, t, a, b), where a′ + c(a′, t) = a and c(a′, t) > 0. So if z_{4} = ([p_{j}^{a}, q_{j}], t, a_{4}, b_{0}) ∈ V^{u}(z_{3}), then z_{4} has local coordinates (u_{0}, 0, 0, a_{4}, b_{0}) with a_{4} < a_{3}.

Since the path on V^{s}(q_{j}) is unperturbed, the fact that z_{4} ∈ V^{s}(z_{5}) yields that the point z_{5} = (q_{j}, 1/2, a_{5}, b_{0}) has local coordinates (u_{0}, s_{0}, 0, a_{5}, b_{0}) with a_{5} = a_{4}. This implies that in the the case n > 1 we have a_{1} = a_{2} = a_{3} > a_{4} = a_{5}.

In the case n = 1 similar arguments can be used with the following modification. To obtain the a-coordinate of the points on V^{s}(p_{j}^{a}, 1/2, a, b_{0}) and V^{u}(p_{j}^{a}, 1/2, a, b_{0}), we need to consider P_{1}^{ℓ^{u}} = h^{a} ∘ T^{ℓ^{u}} and P_{1}^{ℓ^{s}} = h^{a} ∘ T^{ℓ^{s}} respectively, (recall that near the pathes γ_{1}^{a} and on the set Ω^{a}, the map T is unperturbed, and hence Q = T,) and therefore get a_{1} = a_{2} ≥ a_{3} > a_{4} = a_{5}. The assumption κ = κ_{0} on U_{1} guarantees that on these local manifolds the t-coordinates are the same. This implies Statement (1).

In the above arguments, we can actually replace b_{0} by any b ∈ J_{n} and the number 1/2 by any t ∈ (1/2−ϵ_{n}, 1/2+ϵ_{n}) and can still obtain the same results. Therefore, Statement (2) can be proved by switching the roles of a and b.

Statement (3) can be proved in the same way. In particular, since h^{t} preserves a- and b-coordinates and the strongly stable and unstable local manifolds for T at (q_{j}, t, y) and (p_{j}^{t}, t, y) are unperturbed except by h^{t}, the arguments can be carried over on the submanifold 𝒩_{y} (see (3.1)) for each y ∈ Z_{ℓ}^{(n)}. □ Sublemma 5.4. Let Φ : I^{n} → I^{n} be a homeomorphism of the n-dimensional cube I^{n} and ∂_{i}I^{n} be the faces of I^{n}, i = 1, …, 2n. Assume that Φ(∂_{i}I^{n}) ⊂ B(∂_{i}I^{n}, ϵ) \ I^{n} for i = 1, …, 2n, where B(·, ϵ) is the ϵ-neighborhood of the set. Then I^{n} ⊂ Φ(I^{n}).

Proof. This is a variation of a general topology thoerem, which says that in the setting if Φ(∂_{i}I^{n}) ⊂ B(∂_{i}I^{n}, ϵ) for i = 1, …, 2n, then I^{n} \ B(∂_{i}I^{n}, ϵ) ⊂ Φ(I^{n}). □

Appendix A

Let ℳ be a compact smooth Riemannian manifold and 𝒮 ⊂ ℳ an open subset. Let also h be a C^{1} diffeomorphism that is pointwise partially hyperbolic on 𝒮. Further, let 𝒰_{n} ⊂ 𝒮, n ≥ 1 be a sequence of open subsets such that:

(1) 𝒰_{n} ⊂ \overline{𝒰}_{n} ⊂ 𝒰_{n+1} and ⋃ 𝒰_{n} = 𝒮;

(2) each 𝒰_{n} is h-invariant;

(3) h|\overline{𝒰}_{n} is uniformly partially hyperbolic.

The goal of this Appendix is to prove the following statement. Suppose there is a sequence of diffeomorphisms h_{n} such that h_{0} = h, h_{n} = h_{n−1} on ℳ\\overline{𝒰}_{n}. Clearly, \overline{𝒰}_{n} is h_{n}-invariant, and h_{n} = h on ℳ\\overline{𝒰_{n}}.

Theorem A.1. Let h_{n} be a sequence of diffeomorphisms for which h_{0} = h and h_{n} = h_{n−1} on ℳ\\overline{𝒰}_{n}, so that \overline{𝒰}_{n} is h_{n}-invariant and h_{n} = h on ℳ\\overline{𝒰}_{n}. Then there exists a sequence of positive numbers ε_{n} such that if ∥h_{n} − h_{n−1}∥_{C^{1}} ≤ ε_{n}, then

(1) each map h_{n} is uniformly partially hyperbolic on \overline{𝒰}_{n} and hence pointwise partially hyperbolic on 𝒮;

(2) the limit H = lim_{n→∞} h_{n} exists and is a C^{1} pointwise partially hyperbolic diffeomorphism of 𝒮.

We need the following technical statements.

Lemma A.2. Given a sequence of positive numbers {a_{n}}_{n≥1} satisfying

∑_{n=1}^{∞} a_{n} ≤ \frac{1}{4},

we have

∏_{n=1}^{∞}(1 + a_{n}) ≤ 1 + 2 ∑_{n=1}^{∞} a_{n} and ∏_{n=1}^{∞}(1 − a_{n}) ≥ 1 − 2 ∑ a_{n}.

Lemma A.3. Set

M_{n} = sup_{x∈ℳ} ∥d_{x}h_{n}∥ and m_{n} = inf_{x∈ℳ} m(d_{x}h_{n}).

If ε_{n} < m_{0}/2^{n+4}, then M_{n} ≤ 2M_{0} and m_{n} ≥ 0.5m_{0}.

Proof. Note that |M_{n} − M_{n−1}| ≤ ε_{n} and |m_{n} − m_{n−1}| ≤ ε_{n}. Applying Lemma A.2, one can show by induction that

1 − \frac{1}{2^{n+2}} ≤ \frac{M_{n}}{M_{n−1}} , and \frac{m_{n}}{m_{n−1}} ≤ 1 + \frac{1}{2^{n+2}} .

The desired result follows. □

Given two diffeomorphisms f and g with invariant distributions E_{f} and E_{g} on 𝒮 respectively, let

\begin{matrix}Δ_{f,g,E_{f} ,E_{g}}(x) = & max \{|\frac{∥d_{x}g|E_{g}(x)∥}{∥d_{x}f|E_{f}(x)∥} − 1| , |\frac{m(d_{x}g|E_{g}(x))}{m(d_{x}f|E_{f}(x))} − 1|\} , \\ ε_{f,g}(x) = & ∥d_{x}g − d_{x}f∥, θ_{E_{f} ,E_{g}}(x) = ∠(E_{f}(x), E_{g}(x)). \end{matrix} ; (A.19)

Lemma A.4. Assume that

sup_{x∈ℳ} ∥d_{x}f∥ ≤ M := 2M_{0}, inf_{x∈ℳ} m(d_{x}f) ≥ m := 0.5m_{0}.

Then for any x ∈ 𝒮,

Δ_{f,g,E_{f} ,E_{g}}(x) ≤ \frac{1}{m}[ε_{f,g}(x) + CMθ_{E_{f} ,E_{g}}(x)],

where C > 0 is a constant which depends only on the Riemannian metric of ℳ.

Proof. We have that

|∥d_{x}g|E_{g}(x)∥ − ∥d_{x}f|E_{f}(x)∥| ≤ |∥d_{x}g|E_{g}(x)∥ − ∥d_{x}f|E_{g}(x)∥|
+ |∥d_{x}f|E_{g}(x)∥ − ∥d_{x}f|E_{f}(x)∥|
≤ ∥d_{x}g − d_{x}f∥ + ∥d_{x}f∥dist(E_{g}(x), E_{f}(x))
≤ ∥d_{x}g − d_{x}f∥ + C∥d_{x}f∥∠(E_{g}(x), E_{f}(x)),

for some constant C > 0 depending only on the Riemannian metric of ℳ. Dividing both sides of the inequality by ∥d_{x}f|E_{f}(x)∥ and noting that ∥d_{x}f|E_{f}(x)∥ ≥ m(d_{x}f), we obtain that

|\frac{∥d_{x}g|E_{g}(x)∥}{∥d_{x}f|E_{f}(x)∥} − 1| ≤ \frac{∥d_{x}g − d_{x}f∥}{m(d_{x}f)} + C \frac{∥d_{x}f∥}{m(d_{x}f)}∠(E_{g}(x), E_{f}(x))
≤ \frac{1}{m}[ε_{f,g}(x) + CMθ_{E_{f} ,E_{g}}(x)].

Similarly, one can show that |\frac{m(d_{x}g|E_{g}(x))}{m(d_{x}f|E_{f}(x))} − 1| has the same upper bound.

□

Lemma A.5. Suppose that f is uniformly partially hyperbolic on a compact invariant subset Λ ⊂ 𝒮. Pick numbers 0 < λ < λ̃ ≤ 1 ≤ μ̃ < μ such that

λ ≥ λ(f, Λ) = sup_{x∈Λ} ∥d_{x}f^{s}∥, λ̃ ≤ λ̃(f, Λ) = inf_{x∈Λ} m(d_{x}f^{c}),
μ̃ ≥ μ̃(f, Λ) = sup_{x∈Λ} ∥d_{x}f^{c}∥, μ ≤ μ(f, Λ) = inf_{x∈Λ} m(d_{x}f^{u}).

Given Δ > 0, there is ε = ε(Δ, λ, λ̃, μ̃, μ) < \frac{mΔ}{2} such that if ∥g − f∥_{C^{1}} < ε and g = f on 𝒮\Λ, then g|Λ is also uniformly partially hyperbolic and

(A.20)  Δ_{f,g}^{ω}(x) := Δ_{f,g,E_{f}^{ω},E_{g}^{ω}}(x) ≤ Δ, ω = s, c, u, x ∈ Λ.

In particular,

1 − Δ ≤ \frac{λ(g, Λ)}{λ(f, Λ)}, \frac{λ̃(g, Λ)}{λ̃(f, Λ)}, \frac{μ̃(g, Λ)}{μ̃(f, Λ)}, \frac{μ(g, Λ)}{μ(f, Λ)} ≤ 1 + Δ.

Proof. Note that the set of uniformly partially hyperbolic diffeomorphisms is C^{1}-open, and the invariant distributions E_{g}^{ω} depend continuously on g, ω =

s, c, u (see [18]). More precisely, there is ε < \frac{mΔ}{2} depending on Δ, λ, λ̃, μ̃, μ such that if ∥g − f∥_{C^{1}} < ε and g = f on 𝒮\Λ, then g|Λ is uniformly partially hyperbolic with

(A.21)  sup_{x∈Λ} ∠(E_{g}^{ω}(x), E_{f}^{ω}(x)) < \frac{mΔ}{2CM} .

Then by Lemma A.4, it is immediate that Δ_{f,g}^{ω}(x)λΔ. □

We shall now specify how to choose the sequence of numbers ε_{n} in the theorem. First choose four sequences of numbers 0 < λ_{n} < λ̃_{n} ≤ 1 ≤ μ̃_{n} < μ_{n} such that

(1) λ_{n} ≥ λ(h, \overline{𝒰_{n}}), λ̃_{n} ≤ λ̃(h, \overline{𝒰_{n}}), μ̃_{n} ≥ μ̃(h, \overline{𝒰_{n}}), μ_{n} ≤ μ(h, \overline{𝒰_{n}});

(2) λ_{n}, μ̃_{n} are strictly increasing while λ̃_{n}, μ_{n} are strictly decreasing. For all x ∈ 𝒮, let

γ(x) = min \{\frac{min\{1, m(d_{x}h^{c})\}}{∥d_{x}h^{s}∥}, \frac{m(d_{x}h^{u})}{max\{1, ∥d_{x}h^{c}∥\}}\} ,

and choose a strictly decreasing sequence of numbers γ_{n} such that

(A.22)  0 < γ_{n} ≤ inf_{x∈\overline{𝒰_{n}}} \frac{γ(x) − 1}{8}.

Now choose a sequence of positive numbers Δ_{n} such that

(A.23)  max\{\frac{λ̃_{n+1}}{λ̃_{n}} , \frac{μ_{n+1}}{μ_{n}} \} ≤ 1 − Δ_{n} < 1 + Δ_{n} ≤ min\{\frac{λ_{n+1}}{λ_{n}} , \frac{μ̃_{n+1}}{μ̃_{n}} \};

(A.24)  Δ_{n} < \frac{1}{2^{n+2}} , ∑_{k=n}^{∞} Δ_{k} < γ_{n}.

Finally, choose

ε_{n} < \frac{1}{2} min\{\frac{m_{0}}{2^{n+4}} , ε(Δ_{n}, λ_{n}, λ̃_{n}, μ̃_{n}, μ_{n})\},

where ε(Δ, λ, λ̃, μ̃, μ) is given by Lemma A.5.

Proof of Theorem A.1. First we shall show that for every n > 0 the map h_{n} is uniformly partially hyperbolic on \overline{𝒰_{n}}. It is clearly true for h_{0} and we shall use induction assuming that h_{k}|\overline{𝒰_{k}} for k = 1, …, n are uniformly partially hyperbolic. By Lemma A.5 we obtain that

1 − Δ_{k} ≤ \frac{λ(h_{k}, \overline{𝒰_{k}})}{λ(h_{k−1}, \overline{𝒰_{k}})}, \frac{λ̃(h_{k}, \overline{𝒰_{k}})}{λ̃(h_{k−1}, \overline{𝒰_{k}})}, \frac{μ̃(h_{k}, \overline{𝒰_{k}})}{μ̃(h_{k−1}, \overline{𝒰_{k}})}, \frac{μ(h_{k}, \overline{𝒰_{k}})}{μ(h_{k−1}, \overline{𝒰_{k}})} ≤ 1 + Δ_{k}.

Note that

λ(h_{k}, \overline{𝒰_{k+1}}) ≤ max\{λ(h, \overline{𝒰_{k+1}}), λ(h_{k}, \overline{𝒰_{k}})\}
≤ max\{λ_{k+1}, λ(h_{k}, \overline{𝒰_{k}})\}
≤ max\{λ_{k+1}, λ(h_{k−1}, \overline{𝒰_{k}})(1 + Δ_{k})\}.

The fact that λ(h_{0}, \overline{𝒰_{1}}) ≤ λ_{1} and the choice of Δ_{n} in (A.23) guarantee that

λ′_{n} := λ(h_{n}, \overline{𝒰_{n+1}}) ≤ λ_{n+1}.

Similarly, we have

λ̃′_{n} := λ̃(h_{n}, \overline{𝒰_{n+1}}) ≥ λ̃_{n+1}, μ̃′_{n} := μ̃(h_{n}, \overline{𝒰_{n+1}}) ≤ μ̃_{n+1},
μ′_{n} := μ(h_{n}, \overline{𝒰_{n+1}}) ≥ μ_{n+1}.

It follows that

ε_{n} ≤ ε(Δ_{n}, λ_{n}, λ̃_{n}, μ̃_{n}, μ_{n}) ≤ ε(Δ_{n}, λ′_{n}, λ̃′_{n}, μ̃′_{n}, μ′_{n}).

Since ∥h_{n+1} − h_{n}∥_{C^{1}} ≤ ε_{n}, by Lemma A.5 we obtain that h_{n+1}|\overline{𝒰_{n+1}} is uniformly partially hyperbolic.

Next we shall show that H = lim_{n→∞} h_{n} exists and is indeed pointwise partially

hyperbolic on 𝒮. Since ε_{n} < m_{0}/2^{n+4}, the sequence of maps h_{n} is a Cauchy sequence and hence it converges in the C^{1} topology. Moreover, as shown in (A.21), given x ∈ 𝒰_{k} and n > k, we have

∠(E_{h_{n}}^{ω}(x), E_{h_{n−1}}^{ω}(x)) < \frac{mΔ_{n}}{2CM} ≤ \frac{m}{2^{n+3}CM} , ω = s, c, u.

Hence, the sequence of subspaces E_{h_{n}}^{ω}(x) is a Cauchy sequence, and thus there is a limit

E_{H}^{ω}(x) = lim_{n→∞} E_{h_{n}}^{ω}(x).

Fix n > 0. We now wish to estimate Δ_{H,h}^{ω}(x) for x ∈ 𝒰_{n}\\overline{𝒰_{n−1}} (see (A.19) and (A.20)). We have

Δ_{h_{k},h_{k−1}}^{ω}(x) \begin{cases}= 0, & k < n, \\ ≤ Δ_{k}, & k ≥ n. \end{cases}

Note that

\frac{∥d_{x}h_{l}^{ω}∥}{∥d_{x}h^{ω}∥} = ∏_{k=1}^{l} \frac{∥d_{x}h_{k}^{ω}∥}{∥d_{x}h_{k−1}^{ω}∥}, \frac{m(d_{x}h_{l}^{ω})}{m(d_{x}h^{ω})} = ∏_{k=1}^{l} \frac{m(d_{x}h_{k}^{ω})}{m(d_{x}h_{k−1}^{ω})},

and by (A.24), ∑ Δ_{k} < 1/4. It follows from Lemma A.2 that

Δ_{h_{l},h}^{ω}(x) ≤ ∏_{k=1}^{l}(1 + Δ_{h_{k},h_{k−1}}^{ω}(x)) − 1 ≤ ∏_{k=n}^{∞}(1 + Δ_{k}) − 1 ≤ 2 ∑_{k=n}^{∞} Δ_{k}.

Letting l → ∞, we find that

Δ_{H,h}^{ω}(x) ≤ 2 ∑_{k=n}^{∞} Δ_{k}, ω = s, c, u.

Therefore by (A.22),

\frac{∥d_{x}H^{s}∥}{min\{1, m(d_{x}H^{c})\}} ≤ \frac{1 + 2 ∑_{k=n}^{∞} Δ_{k}}{1 − 2 ∑_{k=n}^{∞} Δ_{k}} \frac{∥d_{x}h^{s}∥}{min\{1, m(d_{x}h^{c})\}}

< (1 + 8γ_{n})\frac{∥d_{x}h^{s}∥}{min\{1, m(d_{x}h^{c})\}}

≤ γ(x)\frac{∥d_{x}h^{s}∥}{min\{1, m(d_{x}h^{c})\}} < 1.

Similarly, one can show m(d_{x}H^{u}) > max{1, ∥d_{x}H^{c}∥}. It follows that H is pointwise partially hyperbolic on 𝒮. □

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H. Hu, Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

E-mail address: hu@math.msu.edu

Ya. Pesin, Department of Mathematics,Pennsylvania State University, University Park, PA 16802, USA

E-mail address: pesin@math.psu.edu

56 HUYI HU, YAKOV PESIN, ANNA TALITSKAYA

A. Talitskaya

E-mail address: anjuta@math.northwestern.edu