A VOLUME PRESERVING FLOW WITH ESSENTIAL COEXISTENCE OF ZERO AND NONZERO LYAPUNOV

EXPONENTS

JIANYU CHEN, HUYI HU, AND YAKOV PESIN

Abstract. We demonstrate essential coexistence of hyperbolic and non-hyperbolic behavior in the continuous-time case by constructing a smooth volume preserving flow on a 5-dimensional compact smooth manifold that has nonzero Lyapunov exponents almost everywhere on an open and dense subset of positive but not full volume and is ergodic on this subset while having zero Lyapunov exponents on its complement. The latter is a union of 3dimensional invariant submanifolds and on each of these submanifolds the flow is linear with Diophantine frequency vector.

1. Introduction

The goal of this paper is to extend the main result in [10] to dynamical systems with continuous time thus demonstrating coexistence of regular and chaotic dynamics in an “essential” way.

Theorem 1.1 (Main Theorem). There exists a compact smooth Riemannian manifold M of dimension 5 and a C^{∞} flow h^{t} : M → M such that

(1) h^{t} preserves the Riemannian volume m on M;

(2) h^{t} (t ≠ 0) has nonzero Lyapunov exponents (except for the exponent in the flow direction) almost everywhere on an open, dense and connected subset U ⊂ M; moreover, h^{t}|U is an ergodic flow;

(3) the complement U^{c} has positive volume and is a union of 3dimensional invariant submanifolds; h^{t} is a non-identity linear flow with Diophantine frequency vector on each invariant submanifold and h^{t} has zero Lyapunov exponents on U^{c}.

We stress that each of the 3-dimensional invariant submanifolds is in turn a union of 2-dimensional invariant tori on which h^{t} is a linear flow with Diophantine frequency vector (see Section 3 for details). This fact makes our construction nontrivial.^{1}

Key words and phrases. Pointwise partial hyperbolicity, Lyapunov exponents, ergodicity, accessibility.

J. Chen and Ya. Pesin are partially supported by NSF grant DMS-1101165.

We emphasize the requirement that the open set U is everywhere dense.^{2} Donnay [8] constructed an example of a surface on which the geodesic flow exhibits the coexistence phenomenon. It is obtained by inserting a light-bulb cap into a negatively curved surface. In this example the set of geodesics, which are trapped in the cap, is invariant, has positive volume and almost every point in this set has zero Lyapunov exponents. Since it has non-empty interior, the stochastic sea in this example, i.e., the analog of the set U in our case, is not dense.

While this paper deals only with dynamical systems with continuous time, it is worth mentioned that in the discrete-time case essential coexistence of chaotic and regular behavior have been demonstrated in various situations by Przytycki and Liverani for area preserving diffeomorphisms and by Bunimovich for billiards; see the paper [5], which surveys recent result on essential coexistence, and references therein.

We split the proof of Theorem 1.1 into several steps. In Section 2 we present some background information and basic notations from the theory of partial hyperbolicity and in particular, introduce the notion of pointwise partial hyperbolicity on open sets for flows. In Section 3 we construct the manifold M, the open set U and introduce the “start-up” flow f^{t} that satisfies the statements (3) of the theorem. In Section 4 we construct a volume preserving flow g^{t}, which is a small perturbation of f^{t} and does not affect the action of f^{t} on the set U^{c}. The flow g^{t} has nonzero Lyapunov exponents on a subset of positive volume in U. Finally, in Section 5 we construct the desired flow h^{t} as a small perturbation of the flow g^{t}. The proof of the Main Theorem is given in Section 6.

In our construction of the flows g^{t} and h^{t} we use the perturbation techniques developed in [10] for the case of diffeomorphisms (these techniques originated in [15], [7] and [6]). However, there is a crucial difference between the discrete-time and continuous-time cases. To effect our construction we ought to make perturbations of the vector fields that generate the required flows but we want these perturbations to produce similar effects on the time-1 maps of the flows as in [10]. This is made possible due to the crucial fact that the “start-up” flow f^{t} has a global cross-section and in our construction we ensure that both perturbation flows g^{t} and h^{t} preserve this cross-section. We achieve this by using specific formulae for perturbations of the vector fields. At the core of our construction lies a new concept of pointwise partially hyperbolic flows on open sets. In Section 2 we introduce such flows and we study their ergodicity.

1Indeed, consider the C^{∞} volume preserving diffeomorphism P constructed in [10]. It has nonzero Lyapunov exponents almost everywhere on an open, dense and connected subset U ⊂ M and P |U is ergodic. Furthermore, the complement U^{c} has positive volume, P |U^{c} is the identity map and has zero Lyapunov exponents. A special flow P^{t} over P has nonzero Lyapunov exponents (except for the exponent in the flow direction) almost everywhere on an open, dense and connected subset U × [0, 1]/ ∼ (where ∼ means that the points (x, 1) and (P x, 0) are identified) of not full volume, however, P^{t} is a periodic flow on its complement U^{c} × [0, 1]/ ∼.

2To some extent this justifies to view our example as of “KAM-type” despite the fact that the flow h^{t} is not close to a completely integrable one.

2. Preliminaries

See [2,10,13] for more details. Consider a diffeomorphism f acting on a compact smooth Riemannian manifold M. It is called uniformly partially hyperbolic on a compact invariant subset Λ ⊂ M if

(1) 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 stable E^{s}(x) = E_{f}^{s}(x), central E^{c}(x) = E_{f}^{c}(x) and unstable E^{u}(x) = E_{f}^{u}(x) subspaces;

(2) there are numbers 0 < λ < λ̃ ≤ 1 ≤ μ̃ < μ such that for every t = 1, 2, …

∥df^{t}v∥ ≤ λ^{t}∥v∥, v ∈ E^{s}(x),
(2.2)  λ̃^{t}∥v∥ ≤ ∥df^{t}v∥ ≤ μ̃^{t}∥v∥, v ∈ E^{c}(x),
μ^{t}∥v∥ ≤ ∥df^{t}v∥,  v ∈ E^{u}(x).

In this paper we need a weaker property than uniform partial hyperbolicity. Let S ⊂ M be an invariant open subset. We say that a diffeomorphism 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 functions 0 < λ(x) < λ̃(x) ≤ 1 ≤ μ̃(x) < μ(x) such that (2.2) holds with constants λ, λ̃, μ̃ and μ replaced with these functions. Pointwise partially hyperbolic diffeomorphisms on compact manifolds were introduced in [4] where their ergodic properties were studied. If a diffeomorphism is pointwise partially hyperbolic on an open subset of a compact manifold then it could fail to have “nice” properties and in particular, could be not ergodic (see the discussion below).

We now consider a smooth flow f^{t} on M which is generated by the vector field X_{f}(x) = \frac{d}{dt}f^{t}(x)|_{t=0}. We say that the flow is uniformly partially hyperbolic on a compact invariant subset Λ ⊂ M if for every x ∈ M the tangent space at x admits an invariant splitting (2.1) into stable E^{s}(x) = E_{f}^{s}(x), central E^{c}(x) = E_{f}^{c}(x) and unstable E^{u}(x) = E_{f}^{u}(x) subspaces such that the vector field X_{f}(x) is contained in the central subspace E^{c}(x) and there are numbers 0 < λ < λ̃ ≤ 1 ≤ μ̃ < μ such that (2.2) holds for all t ∈ [0, 1]. Note that if a flow f^{t} is uniformly partially hyperbolic, then for every t ≠ 0 the time-t map is uniformly partially hyperbolic with the same invariant splitting.

Given an invariant open subset S ⊂ M we call a flow f^{t} pointwise partially hyperbolic on S if its time-1 map f^{1} is pointwise partially hyperbolic on S.

Given δ > 0, we say that a flow g^{t} is (C^{1}, δ)-close to f^{t} on an invariant set Λ if X_{g} = X_{f} outside Λ and ∥X_{g} − X_{f}∥ ≤ δ. Uniformly partially hyperbolic flows form an open set in the C^{1} topology (see Lemma B.8 in the Appendix B).

Given a an open subset S ⊂ M, we call a partition P of S a (δ, q)-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 on the local leaf V (y), generated by the volume m on M and the partition of B(x, q(x)) by the local leaves, is absolutely continuous with respect to the leaf volume m_{V (y)} on V (y). ^{3}

Let W_{1} and W_{2} be two foliations of S with smooth leaves that are transversal to each other 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 via a (u, s)- path in S, that is

3The leaf volume m_{V (y)} is generated by the restriction of the Riemannian metric on M to the smooth submanifod V (y).

(1) there exists a collection of points z_{1}, …, z_{n} ∈ S such that z = z_{1}, z′ = z_{n} and z_{k} ∈ W_{i}(z_{k−1}) for i = 1, 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. ^{4}

The collection of the leaf-wise paths γ_{k} is called a (u, s)-path and is denoted by [z_{1}, …, z_{n}].

For a uniformly partially hyperbolic flow f^{t} one can construct stable and unstable local manifolds of uniform size at every point in Λ. This may not be true for a flow that is pointwise partially hyperbolic on an open set S. However, all pointwise partially hyperbolic flows that we consider in this paper will have global stable and unstable transverse foliations with smooth leaves. We denote these foliations by W^{s} = W_{f}^{s} and W^{u} = W_{f}^{u} respectively.

More precisely, let f^{t} be a flow that is pointwise partially hyperbolic on an open set S and let g^{t} be a sufficiently small perturbation of f^{t} in the C^{1} topology.

Definition 2.1. We call the perturbation g^{t} gentle if there exists an open set U ⊂ S such that Ū ⊂ S, U is invariant under both f^{t} and g^{t} and f^{t}|U^{c} = g^{t}|U^{c}.

Theorem 2.2. Assume that the strongly stable and unstable subspaces E_{f^{t}}^{s} and E_{f^{t}}^{u} for f^{t} are integrable to continuous strongly stable and unstable foliations W_{f^{t}}^{s} and W_{f^{t}}^{u} respectively with smooth leaves and that these foliations are transverse. Then for any sufficiently small gentle perturbation g^{t} in the C^{1} topology the strongly stable and unstable subspaces E_{g^{t}}^{s} and E_{g^{t}}^{u} for g^{t} are integrable to continuous strongly stable and unstable foliations W_{g^{t}}^{s} and W_{g^{t}}^{u} respectively with smooth leaves and these foliations are transverse.

We call a flow f^{t} 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 [9, 14] to the case of flows that are pointwise partially hyperbolic on an open subset. It shows that dynamical coherence is a robust property within the class of gentle perturbations. The proof of this result is a simple modification of the argument in [9].

4We 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 2.3. Suppose that f^{t} is a flow that is pointwise partially hyperbolic on an open set S. Assume that f^{t} 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^{t} is dynamically coherent. Moreover, any flow that is close to f^{t} in the C^{1} topology and is a gentle perturbation of f^{t} is dynamically coherent.

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

Given a smooth flow f^{t}, we denote by

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

the Lyapunov exponent of a nonzero vector v at x ∈ M and by λ_{i}(x) = λ_{i}(x, f^{t}), i = 1, …, dim M, the values of the Lyapunov exponents at x in the decreasing order. We also denote by

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

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

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

Theorem 2.4. Assume that the following conditions hold:

(1) f^{t} 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^{t} has the accessibility property via the foliations W^{s} and W^{u} on S;

(4) f^{t} has positive central exponents;

(5) The Lyapunov exponents in the stable subspace E^{s}(x) are all negative and the Lyapunov exponents in the unstable subspace E^{u}(x) are all positive for almost every x.

Then f^{t} has positive central exponents at almost every point x ∈ S and f^{t}|S is an ergodic flow.

Proof. Following Theorem 2.2 in [10], we first show that f^{t} has positive central exponents at almost every point x ∈ S and f^{t}|S is an ergodic flow.

There is a set A ⊂ S of positive measure such that the flow f^{t}|A has nonzero Lyapunov exponents except along the flow direction. Hence, it has at most countably many ergodic components of positive measure in A (see [2]). Each such component contains the set

A(x) = ⋃_{y∈V^{cu}(x)} V^{s}(y),

where x is the density point of A and V^{cu}(x) is a center-unstable local manifold at x. Since the strong stable foliation W^{s} is continuous, the set A(x) is open (mod 0) in M^{5} and hence, the set A is itself open (mod 0). We will show that the trajectory of almost every point in S is dense, which yields that A = S (mod 0) and that f^{t}|S is ergodic.

The proof of this claim for partially hyperbolic diffeomorphisms is given in [1] and can be extended to our case literally. We present the argument here for the reader’s convenience. We call a point p good for a given open set U if p has a neighborhood in which the orbit of almost every point enters U. It suffices to show that an arbitrary point p is good. Since f^{t} is accessible, there is a (u, s)-path [z_{0}, …, z_{k}] with z_{0} ∈ U and z_{k} = p. We will 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 measure 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 measure 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. □

5That is there is an open set V ⊂ M such that A(x) = V (mod 0) with respect to the volume m.

3. Construction of the “start-up” flow f^{t}

Let A be an Anosov automorphism of the 2-torus X = 𝕋^{2} with expanding rate η > 1 along the unstable direction. Consider the suspension flow S^{t} on the suspension manifold N = X × ℝ/ ∼, with the identification (x, τ + 1) ∼ (Ax, τ). The action of the suspension flow on N is exactly S^{t}(x, τ) = (x, τ + t). See Appendix A for more details of the geometric structure of N.

Given α ∈ 𝕋^{2}, let T_{α}^{t} : N → N be a linear flow defined by (x, τ) ↦ (x + tα, τ). It preserves each level set X × {τ}.

Set Y = 𝕋^{2} and M = N × Y. To effect our construction we choose: (A1) a Cantor set C ⊂ Y of positive area whose complement U =

Y \C is a non-empty open and connected set; (A2) an open square U_{0} such that \overline{U_{0}} ⊂ U; (A3) an open neighborhood U_{1} of \overline{U_{0}} such that \overline{U_{1}} ⊂ U, whose choice

will be specified in Section 5.2.1. We also choose a C^{∞} function κ : Y → ℝ such that

(κ1) κ(y) > 0 for y ∈ U and κ(y) = 0 for y ∈ C; (κ2) κ(y) = 1 for y ∈ U_{1}; (κ3) ∥κ∥_{C^{1}} ≤ 1

(the existence of such a function κ follows from the specific construction of the Cantor set and the choice of the set U_{1} in Section 5.1) and a C^{∞} map α : Y → ℝ^{2} such that

(α1) α(y) = 0 for y ∈ U_{1};

(α2) α(y) = α_{0} > 0 for all y ∈ C where α_{0} is a Diophantine vector; (α3) sup_{y∈Y} ∥α(y)∥ ≤ ᾱ, where ᾱ is a positive number determined

in Section 5.2. We now set U = N × U and U^{c} = N × C and define the flow f^{t} on

M by the formula

(3.1)  f^{t}((x, τ), y) = ((x + tα(y), τ + tκ(y)), y)

where (x, τ) ∈ N and y ∈ Y. The following proposition describes the properties of the flow f^{t} and its proof follows immediately from the definitions.

Proposition 3.1. The following statements hold:

(1) f^{t} is a C^{∞} volume preserving flow;

(2) f^{t} preserves each fiber N × {y}, on which f^{t} is the composition of the scaled suspension flow S^{tκ(y)} and the linear flow T_{α(y)}^{t}; in particular, f^{t} is exactly the suspension flow S^{t} on N × {y} for y ∈ \overline{U_{1}};

(3) f^{t} is pointwise partially hyperbolic on U with one-dimensional stable E_{f}^{s}, one-dimensional unstable E_{f}^{u} and 3-dimensional center E_{f}^{c} subbundles; E_{f}^{s} and E_{f}^{u} are integrable to strongly stable and unstable foliations W_{f}^{s} and W_{f}^{u} with smooth leaves, which are absolutely continuous, uniformly transversal and have local leaves of uniform size;

(4) f^{t} is uniformly partially hyperbolic on N × A where A ⊂ U is a subset, and hence f^{t} is dynamically coherent with the central foliation W_{f}^{c} = W_{S^{t}}^{c} × Y;

(5) f^{t} preserves every 2-dimensional torus X × {τ} × {y} (τ ∈ [0, 1], y ∈ C are fixed) and acts on it as a linear flow with a Diophantine frequency vector; moreover, f^{t}|U^{c} has all zero Lyapunov exponents on U^{c};

(6) for every z = ((x, τ), y) ∈ M the Lyapunov exponents of f^{t} are as follows:

\begin{aligned}λ_{1}(z, f^{t}) = κ(y) log η ≥ 0 & = λ_{2}(z, f^{t}) = λ_{3}(z, f^{t}) = λ_{4}(z, f^{t}) \\ & ≥ λ_{5}(z, f^{t}) = −κ(y) log η; \end{aligned}

moreover, if z ∈ U, then λ_{1}(z, f^{t}) = λ^{u}(z, f^{t}) > 0 is the Lyapunov exponent in the E_{f}^{u}(z) subspace, λ_{5}(z, f^{t}) = λ^{s}(z, f^{t}) < 0 is the Lyapunov exponent in the E_{f}^{s}(z) subspace, and λ_{2}(z, f^{t}), λ_{3}(z, f^{t}) and λ_{4}(z, f^{t}) are Lyapunov exponents in the flow direction and two directions in Y respectively.

4. Removing zero exponents

In this section we will construct a gentle perturbation g^{t} of the original flow f^{t} with positive central Lyapunov exponents on a set of positive volume but not necessarily ergodic. Then we will perturb g^{t} to the desired flow h^{t} of the main theorem in Section 5.

Given z ∈ M, there is a local Cartesian coordinate system (u, s, τ, a, b) (see Appendix A) such that

F^{u}(z) := \frac{∂}{∂u} = E_{f}^{u}(z), F^{s}(z) := \frac{∂}{∂s} = E_{f}^{s}(z), F^{τ}(z) := \frac{∂}{∂τ} = E_{f}^{τ}(z)

are the unstable, stable and flow directions of f^{t} respectively, and

F^{a}(z) := \frac{∂}{∂a} = E_{f}^{a}(z), F^{b}(z) := \frac{∂}{∂b} = E_{f}^{b}(z)

are the other two central directions tangent to Y.

The following statement describes properties of the flow g^{t}.

Proposition 4.1. Given δ_{g} > 0, there is a C^{∞} volume preserving flow g^{t} on M such that the following hold.

(1) g^{t} is (C^{1}, δ_{g})-close to f^{t}, i.e., ∥X_{f} − X_{g}∥ ≤ δ_{g}, where X_{f} and X_{g} are the vector fields of the flows f^{t} and g^{t} respectively.

(2) g^{t} = f^{t} outside N × U_{0}, and hence g^{t} is a gentle perturbation of f^{t} and satisfies Statements (3)-(5) of Proposition 3.1.

(3) g^{t} preserves the subbundles E_{f}^{ω}, ω = uab, uabτ; moreover,

(4.1)  det(dg^{t}|E_{f}^{ω}(z)) = det(df^{t}|E_{f}^{ω}(z)), for all z ∈ M.

(4) The average Lyapunov exponents of g^{t} satisfy

(4.2)  L_{5}(g^{t}) = 0 < L_{1}(g^{t}) < L_{2}(g^{t}) < L_{3}(g^{t}) = L_{4}(g^{t}).

To prove this proposition, we extend the approach in [10] to the case of flows and obtain the flow g^{t} as a result of two consecutive perturbations. First, we perturb the start-up flow f^{t} to a flow g̃^{t} by adding a rotational vector field X̃_{R} to the vector field X_{f}.^{6} This produces two positive average Lyapunov exponents for the flow g̃^{t} in the E_{f}^{ua} subbundle, i.e., L_{1}(g̃^{t}) < L_{2}(g̃^{t}). Next, we perturb g̃^{t} to the desired flow g^{t} by adding another rotational vector field X_{R} to the vector field Xg̃ for the flow g̃^{t}. As a result the flow g^{t} has three positive average Lyapunov exponents in the E_{f}^{uab} subbundle, i.e., L_{1}(g^{t}) < L_{2}(g^{t}) < L_{3}(g^{t}).

The vector fields X̃_{R} and X_{R} are chosen to be supported on disjoint open subsets Ω̃_{R} and Ω_{R} of N × U_{0} respectively such that X̃_{R} = 0 outside Ω̃_{R}, X_{R} = 0 outside Ω_{R} and ∥X̃_{R}∥_{C^{1}}, ∥X_{R}∥_{C^{1}} < δ_{g}/2. Since N ×U_{0} is invariant under f^{t}, we have that g^{t} = g̃^{t} = f^{t} outside N ×U_{0}.

Our construction utilizes the following crucial feature of the flow f^{t}: the set

(4.3)  Π_{0} = X × {0} × U_{0}

is a global cross-section of f^{t}|N × U_{0}, and the time-1 map restricted to Π_{0} is exactly the Poincaré return map of f^{t} to Π_{0}. Furthermore, we make the construction of vector fields X̃_{R} and X_{R} in such a way that Π_{0} is also a global cross-section for both flows g̃^{t} and g^{t} with the time-1 maps to be the Poincaré return map to Π_{0}. This fact allows us to apply arguments similar to those in [10] to our flow case by focusing on the time-1 maps.

6That is the flow generated by X̃_{R} is concentrated in a small neighborhood of a point in M where it acts as a small rotation around this point; see (4.6).

COEXISTENCE OF ZERO AND NONZERO LYAPUNOV EXPONENTS 11

4.1. Construction of the flow g̃^{t}. In this section we construct the flow g̃^{t} by perturbing the vector field X_{f} inside the set N × U_{0}.

To effect our construction we choose distinct periodic points q, p^{a}, p^{b} and p^{τ} of the Anosov automorphism A of X, which are close to each other. Let V_{A}^{s}(q), V_{A}^{u}(q), V_{A}^{s}(p^{i}) and V_{A}^{u}(p^{i}), i = a, b, τ be the 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 a single point, which we denote by [q, p^{i}] and [p^{i}, q] respectively. Let γ^{i} denote 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}).

Choose ν > 0 and set for i = a, b, τ,

\begin{matrix}Ω_{0}^{i}(ν) = & ( ⋃_{t∈[0,ι(p^{i})]} B_{N} (f^{t}(p^{i}, 0), ν)) × U_{0}, \\ Ω̂_{0}^{i}(ν) = & ( ⋃_{(x,τ)∈(γ(q)×[0,ι(q)])∪(γ(p^{i})×[0,ι(p^{i})])} B_{N} ((x, τ), ν)) × U_{0}, \\ Ω_{0} = & Ω_{0}(ν) = ( ⋃_{i=a,b,τ} Ω^{i}(ν)) ∪ ( ⋃_{i=a,b,τ} Ω̂^{i}(ν)), \end{matrix} ; (4.4)

where ι(q) and ι(p^{i}) are the periods of q and p^{i} respectively, and B_{N} ((x, τ), r) is the ball in N of radius r centered at the point (x, τ) ∈ N. We choose a sufficiently small number ν to ensure that

m(Proj_{Π_{0}}Ω_{0}) ≤ 0.05m(Π_{0}),

where Π_{0} is given by (4.3), and Proj_{Π_{0}} : N × U_{0} → Π_{0} is the natural projection onto Π_{0} given by the formula

Proj_{Π_{0}}((x, τ), y) = ((x, 0), y).

To construct the vector field X̃_{R} we choose a C^{∞} function ψ : ℝ → [0, 1] such that

(1) ψ = 1 on (−0.9, 0.9);

(2) ψ > 0 on (−1, 1) and ψ = 0 outside (−1, 1);

(3) ∥ψ∥_{C^{1}} ≤ 10. Observe that N × U_{0} is invariant under the flow f^{t} and that

f^{t}((x, τ), y) = ((x, τ + t), y)

for ((x, τ), y) ∈ N × U_{0}. In other words, f^{t}|N × U_{0} is the product of the suspension flow on N and the identity map on U_{0}. It follows that Π_{0} is a global cross-section for the flow f^{t}|N × U_{0}, and the time-1 map restricted to Π_{0} is f^{1} = A × Id and is exactly the Poincaré return map of f^{t} to Π_{0}. Given a subset Π ⊂ Π_{0}, we call a set Π × [τ_{1}, τ_{2}] ⊂ Π_{0} × ℝ/ ∼= N × U_{0} a tube if

f^{t}(Π × {τ_{1}}) ∩ (Π × {τ_{1}}) = ∅ for all t ∈ [0, τ_{2} − τ_{1}].

It is easy to check that Π × [τ_{1}, τ_{2}] is a tube if and only if the sets Π, f^{1}(Π), f^{2}(Π), …, f^{l}(Π) are pairwise disjoint, where l = ⌊τ_{2} − τ_{1}⌋. Choose a non-periodic point z_{0} = (x_{0}, 0, y_{0}) ∈ Π_{0}\Proj_{Π_{0}}Ω_{0}, where x_{0} is a non-periodic point of the Anosov automorphism A, y_{0} is the center of the square U_{0} and Ω_{0} is the set given by (4.4).

In what follows we will use the local Cartesian coordinate system in a neighborhood of Π_{0} originated at z_{0} given by (u, s, t, a, b) where (u, s) are the coordinates in X along the stable and unstable directions of the hyperbolic diffeomorphism A, t is the coordinate along the time direction and (a, b) are coordinates in Y. In this coordinate system a point z ∈ Π_{0} is given as z = (u, s, 0, a, b). We will also use the ua-cylindrical coordinates (r, θ, τ, s, b), where u = r cos θ, a = r sin θ. Given ε > 0, one can choose a ua-cylinder B ⊂ Π_{0} centered at z_{0} of size ε, i.e.,

B = {(r, θ, 0, s, b) : r ≤ ε, |s| ≤ ε, |b| ≤ ε}.

Given a sufficiently large N_{0} ≥ 20k_{0} (the number k_{0} is defined below in Lemma 4.5), we can choose ε so small that f^{i}(B) ∩ B = ∅ for i = 1, …, N_{0}. Consider the tube

(4.5)  Ω̃_{R} = B × [0, 1/2].

Since z_{0} ∉ Proj_{Π_{0}}Ω_{0}, we can further reduce ε to ensure that B ∩ Proj_{Π_{0}}(Ω_{0}) = ∅. Hence, Ω̃_{R} ∩ Ω_{0} = ∅.

Given β > 0, define a C^{∞} rotational vector field X̃_{R} = X̃_{R,β} on M as follows:

(4.6)  X̃_{R,β}(z) = \begin{cases}βψ̃(z)\frac{∂}{∂θ}, & z ∈ Ω̃_{R}, \\ 0, & z ∈ M\Ω̃_{R}, \end{cases}

where

ψ̃(z) = ψ̃(r, θ, τ, s, b) = ψ (\frac{r^{2}}{ε^{2}} ) ψ (\frac{s}{ε}) ψ (\frac{b}{ε}) ψ (\frac{τ − (1/4)}{1/4}) .

It is easy to see that ∥ψ̃\frac{∂}{∂θ}∥ ≤ c where c > 0 is a constant, which is

independent of ε. Hence, ∥X̃_{R,β}∥ → 0 as β → 0. Furthermore, X̃_{R,β} is divergence free. Let g̃^{t} = g̃_{β}^{t} be the flow generated by the vector field

Xg̃ = X_{g̃,β} = X_{f} + X̃_{R,β}.

Proposition 4.2. There exists β > 0 such that g̃^{t} = g̃_{β}^{t} is a C^{∞} volume preserving flow with the following properties:

(1) g̃^{t} is (C^{1}, δ_{g}/2)-close to f^{t}, i.e., ∥X_{f} − Xg̃∥_{C^{1}} ≤ δ_{g}/2, where X_{f} and Xg̃ are the vector fields corresponding to flows f^{t} and g̃^{t} respectively;

(2) g̃^{t} = f^{t} outside N × U_{0}, and hence g̃^{t} is a gentle perturbation of f^{t} and satisfies Statements (3)-(5) of Proposition 3.1;

(3) g̃^{t} preserves the subbundles E_{f}^{ω}, ω = ua, uab, uabτ; moreover,

(4.7)  det(dg̃^{t}|E_{f}^{ω}(z)) = det(df^{t}|E_{f}^{ω}(z)) for all z ∈ M;

(4) the average Lyapunov exponents of g̃^{t} satisfy

(4.8)  L_{5}(g̃^{t}) = 0 < L_{1}(g̃^{t}) < L_{2}(g̃^{t}) = L_{3}(g̃^{t}) = L_{4}(g̃^{t});

(5) Π_{0} is a global cross-section of the flow g̃^{t}|N × U_{0}, and the time-1 map g̃^{1} is the Poincaré return map of g^{t} to Π_{0}; furthermore, there exist λ > 0 and a g̃^{1}-invariant subset Π ⊂ Π_{0} such that

m(Π) ≥ 20k_{0}m(Π ∩ B) > 0

and for any z ∈ Π the flow g̃^{t} has two positive Lyapunov exponents λ_{1}(z, g̃^{t}) > λ_{2}(z, g̃^{t}) ≥ λ along the E_{f}^{ua} subbundle.

Proof. Statements (1) and (2) are easy corollaries of the construction of the flow g̃^{t}. To prove Statement (3) we will first show that dg̃^{t} preserves the subbundles E_{f}^{ua}. It suffices to check that for any smooth vector field X ∈ E_{f}^{ua} and any z ∈ Ω̃_{R}, the Lie bracket [Xg̃(z), X (z)] ∈ E_{f}^{ua}(z). Indeed, we have

Xg̃(z) = \frac{∂}{∂τ} + βψ̃(z)\frac{∂}{∂θ} = \frac{∂}{∂τ} + βψ̃(z) (−a\frac{∂}{∂u} + u\frac{∂}{∂a}) ,

and the direct calculation yields

[Xg̃, \frac{∂}{∂ω} ] = β (\frac{∂(aψ̃)}{∂ω} \frac{∂}{∂u} − \frac{∂(uψ̃)}{∂ω} \frac{∂}{∂a}) ∈ E_{f}^{ua}, ω = u, a.

Since dg̃^{t} preserves the subbundle E_{f}^{ua}, it also preserves the subbundles E_{f}^{uab} and E_{f}^{uabτ}. Next, consider the variational differential equations

\frac{d}{dt}f^{t} = DX_{f}df^{t}, \frac{d}{dt}g̃^{t} = DXg̃dg̃^{t}.

The determinants along E_{f}^{ω} with ω = ua, uab, uabτ satisfy

\frac{d}{dt} det(df^{t}|E_{f}^{ω}) = div(X_{f}|E_{f}^{ω}) det(df^{t}|E_{f}^{ω}),

\frac{d}{dt} det(dg̃^{t}|E_{f}^{ω}) = div(Xg̃|E_{f}^{ω}) det(dg̃^{t}|E_{f}^{ω}).

(4.9)

Direct calculations show that X̃_{R} = Xg̃ − X_{f} is divergence free along E_{f}^{ω} and thus div(X_{f}|E_{f}^{ω}) = div(Xg̃|E_{f}^{ω}). Therefore, using (4.9) and the fact that det(dg^{0}|E_{f}^{ω}) = det(df^{0}|E_{f}^{ω}) = 1 we find that

det(dg̃^{t}|E_{f}^{ω}(z)) = det(df^{t}|E_{f}^{ω}(z))

and Statement (3) follows.

It remains to prove Statements (4) and (5). We need the following lemma showing that Π_{0} is a global cross-section for the flow g̃^{t}|N × U_{0}.

Lemma 4.3. Given z ∈ B, the r-, s-, b- and τ- coordinates of g̃^{t}(z) and f^{t}(z) are the same for t ∈ [0, 1/2]. Consequently, g̃^{\frac{1}{2}} (B) = f^{\frac{1}{2}} (B) and Π_{0} is a global cross-section for the flow g̃^{t}|N × U_{0}.

Proof of the lemma. Let us compare the orbit segments of g̃^{t}(z) and f^{t}(z) for t ∈ [0, 1/2]. Note that for any smooth function φ and any vector field X we have that

\frac{d}{dt}φ(F^{t}(z)) = L_{X} φ|_{F^{t}(z)},

where L_{X} (·) is the Lie derivative and F^{t} is the flow that is generated by X. This implies that for ω = r, s, b,

\frac{d}{dt}τ(g̃^{t}(z)) = L_{Xg̃}τ = \frac{∂τ}{∂τ} + βψ̃(g̃^{t}(z))\frac{∂τ}{∂θ} = 1 = L_{X_{f}} τ = \frac{d}{dt}τ(f^{t}(z)),

\frac{d}{dt}ω(g̃^{t}(z)) = L_{Xg̃}ω = \frac{∂ω}{∂τ} + βψ̃(g̃^{t}(z))\frac{∂ω}{∂θ} = 0 = L_{X_{f}} ω = \frac{d}{dt}ω(f^{t}(z)).

Under the same initial condition at t = 0, we get that the r-, s-, band τ-coordinates of g̃^{t}(z) and f^{t}(z) are the same. Since B has the cylindrical structure, we obtain that g̃^{t}(z) ∈ B × τ(f^{t}(z)) = B × {t}. In particular, g̃^{\frac{1}{2}} (B) = f^{\frac{1}{2}} (B) = B × \{\frac{1}{2}\}. Since Xg̃ = X_{f} outside Ω̃_{R} = B × [0, 1/2], we have that g̃^{1}(Π_{0}) = f^{1}(Π_{0}) = Π_{0}. In other words, Π_{0} is also a global cross-section for g̃^{t}|N × U_{0}. □

It follows from the lemma that Π_{0} is a global cross-section for the flow g̃^{t}|N × U_{0} and the time-1 map g̃^{1} restricted to Π_{0} is the Poincaré return map of g̃^{t} on Π_{0}. Therefore, (4.8) is equivalent to

(4.10)  L_{4}(G̃) = 0 < L_{1}(G̃) < L_{2}(G̃) = L_{3}(G̃),

where G̃ = G̃_{β} = g̃^{1}|Π_{0}. In fact, by (4.7), we have that L_{k}(G̃) = L_{k}(f^{1}|Π_{0}) for k = 2, 3, 4, and thus we only need to show that

(4.11)  L_{1}(G̃) < L_{1}(f^{1}|Π_{0}).

To this end we will use the following lemma.

Lemma 4.4. For all z ∈ Π_{0} the derivative of G̃ = g̃^{1}|Π_{0} along E_{f}^{ua} has the form

(4.12)  dG̃_{β}(z)|E_{f}^{ua}(z) = (\begin{matrix}ηA(β, z) & ηB(β, z) \\ C(β, z) & D(β, z) \end{matrix}) ,

where

A = A(β, z) = 1 − βrσ_{r} sin θ cos θ − \frac{β^{2}σ^{2}}{2} − β^{2}rσσ_{r} cos^{2} θ + O(β^{3}),
B = B(β, z) = −βσ − βrσ_{r} sin^{2} θ − β^{2}rσσ_{r} sin θ cos θ + O(β^{3}),
C = C(β, z) = βσ + βrσ_{r} cos^{2} θ − β^{2}rσσ_{r} sin θ cos θ + O(β^{3}),
D = D(β, z) = 1 + βrσ_{r} sin θ cos θ − \frac{β^{2}σ^{2}}{2} − β^{2}rσσ_{r} sin^{2} θ + O(β^{3}).

Proof of the lemma. The desired relation (4.12) is apparent for z ∈ Π_{0}\B since G̃ = f^{1} and σ = 0 on Π_{0}\B. Given z = (r, θ, 0, s, b) ∈ B in the ua-cylindrical coordinate, by Lemma 4.3, we have that g̃^{t}(z) = (r, θ + θ(t), t, s, b) where θ(t) = β ∫_{0}^{t} ψ̃(g̃^{τ}(z))dτ for 0 ≤ t ≤ 1/2. In particular, the coordinate of g̃^{\frac{1}{2}} (z) is (r, θ + βσ, 1/2, s, b), where

σ = σ(r, s, b) = \frac{1}{4}ψ (\frac{r^{2}}{ε^{2}} ) ψ (\frac{b}{ε}) ψ (\frac{s}{ε}) ∫_{−1}^{1} ψ(u)du.

Back in the Cartesian coordinate system (u, a, τ, s, b), we obtain that

g̃^{\frac{1}{2}} (z) = (u_{1}, a_{1}, 1/2, s, b)
:= (u cos(βσ) − a sin(βσ), u sin(βσ) + a cos(βσ), 1/2, s, b),

and hence,

G̃(z) = g̃^{1}(z) = f^{\frac{1}{2}} g̃^{\frac{1}{2}} (z) = (u_{1}, a_{1}, 1, s, b) = (ηu_{1}, a_{1}, 0, η^{−1}s, b).

The last equality follows from (A.1) and the fact that g̃^{1}(z_{0}) = f^{1}(z_{0}), where z_{0} is the center of B. Since G̃ preserves the E_{f}^{ua} subbundle, we have that

dG̃_{β}(z)|E_{f}^{ua}(z) = (\begin{matrix}ηA(β, z) & ηB(β, z) \\ C(β, z) & D(β, z) \end{matrix}) ,

where

A(β, z) = \frac{∂u_{1}}{∂u} , B(β, z) = \frac{∂u_{1}}{∂a} , C(β, z) = \frac{∂a_{1}}{∂u} , D(β, z) = \frac{∂a_{1}}{∂a} .

Then

A = \frac{∂u_{1}}{∂u} = cos(βσ) + [−u sin(βσ) − a cos(βσ)]βσ_{u}

= cos(βσ) − βr sin(θ + βσ)σ_{r} cos θ

= 1 − βrσ_{r} sin θ cos θ − \frac{β^{2}σ^{2}}{2} − β^{2}rσσ_{r} cos^{2} θ + O(β^{3}).

Similarly, we can obtain the formulae for B, C and D. □

Lemma 4.4 allows us to follow the line of argument in the proof of Lemma 4.1 in [10] to establish (4.11). For the reader’s convenience we outline the argument here.

Denote by e_{β}(z) the unique number such that the vector v_{β}(z) = (1, e_{β}(z))^{t} ∈ Eu ; G̃_{β}(z) for all z ∈ Π_{0}. One can show that

L_{β} = L_{1}(G̃_{β}) = ∫_{Π_{0}} log η dm(z)
− ∫_{Π_{0}} log[D(β, z) − ηB(β, z)e_{β}(G̃_{β}(z))]dm(z).

Note that L_{0} = L_{1}(f^{1}|Π_{0}), and we will show that

(4.13)  \frac{dL_{β}}{dβ} |_{β=0} = 0, \frac{d^{2}L_{β}}{dβ^{2}} |_{β=0} < 0,

which immediately implies that (4.11) holds for all sufficiently small β > 0.

To show (4.13) observe that

\frac{dL_{β}}{dβ} |_{β=0} = − ∫_{Π_{0}} D_{β}|_{β=0} dm(z) = 0

thus proving the first relation in (4.13). To prove the second relation note that

\frac{d^{2}L_{β}}{dβ^{2}} |_{β=0} = ∫_{Π_{0}}[(D_{β})^{2} − D_{ββ} + 2ηB_{β} \frac{∂}{∂β} (e_{β}(G̃_{β}(z)))]_{β=0} dm(z).

This integral can be written as

(4.14) \begin{aligned}∫_{Π_{0}}[(D_{β}(0, z))^{2} & − D_{ββ}(0, z) + 2ηB_{β}(0, z)C_{β}(0, z)] dm(z) \\ & + ∫_{Π_{0}} ∑_{i=1}^{∞} \frac{1}{η^{i}} 2B_{β}(0, z)C_{β}(0, f^{−i}(z)) dm(z). \end{aligned}

The first term in (4.14) is bounded from above by

−(1 − ε) ∫_{Π_{0}} σ^{2}dm(z) − \frac{1}{8} ∫_{Π_{0}} r^{2}σ_{r}^{2}dm(z).

To estimate the second term in (4.14) note that

∫_{Π_{0}} 2B_{β}(0, z)C_{β}(0, f^{−i}(z))dm(z) ≤ 4 ∫_{Π_{0}}(σ^{2} + r^{2}σ_{r}^{2}) dm(z)

and that B_{β}(0, z)C_{β}(0, f^{−i}(z)) = 0 for all z ∈ Π_{0}\B and all i. Moreover, B_{β}(0, z)C_{β}(0, f^{−i}(z)) = 0 for every z ∈ B and i = 1, …, N_{0} − 1 since f^{i}(B) ∩ B = ∅. This allows us to take N_{0} > 0 large enough to ensure that the second term is bounded by

\frac{1}{10} ∫_{B}(σ^{2} + r^{2}σ_{r}^{2}) dm(z).

Hence,

\frac{d^{2}L_{β}}{dβ^{2}} |_{β=0} ≤ −(\frac{9}{10} − ε) ∫_{Π_{0}} σ^{2}dm(z) − \frac{1}{40} ∫_{Π_{0}} r^{2}σ_{r}^{2}dm(z) < 0.

This completes the proof of the inequality (4.11) thus guaranteeing that for any sufficiently small λ > 0 the level set

Π = {z ∈ Π_{0} : λ_{1}(z, G̃) ≥ λ_{2}(z, G̃) > λ}

has positive volume. It is also invariant under G̃. Since f^{i}(B) ∩ B = ∅ for i = 1, …, N_{0}, we obtain that the sets g̃^{i}(Π ∩ B) = Π ∩ g̃^{i}(B) = Π ∩ f^{i}(B) corresponding to different i are pairwise disjoint subsets of Π. This implies that

m(Π) ≥ N_{0}m(Π ∩ B) ≥ 20k_{0}m(Π ∩ B) > 0

thus completing the proof of Proposition 4.2. □

4.2. Construction of the flow g^{t}. We perturb the flow g̃^{t} to a flow g^{t} by adding a vector field X_{R} to the vector field Xg̃. We obtain X_{R} as a sum of rotational vector fields in the ab-direction along several pairwise disjoint tubes so that the total rotation along an orbit that passes through these tubes is π/2. This ensures positive Lyapunov exponents along the E_{f}^{uab} subbundle for the flow g^{t}.

Note that there is M_{0} > 0 such that for any flow F^{t} that is sufficiently C^{1}-close to the flow f^{t}

(4.15)  ∥F^{1} − f^{1}∥_{C^{1}} ≤ M_{0}∥X_{F} − X_{f}∥_{C^{1}}.

According to Lemma B.5, M_{0} depends only on the Riemannian metric and the start-up flow f^{t}.

Let the number λ > 0 and the subset Π ⊂ Π_{0} be as in Statement (5) of Proposition 4.2. Given K > 0, let

(4.16) Λ′ = Λ′(K) = \{z ∈ Π : |\frac{1}{k} log ∥dg̃^{k}(z, v)∥ − λ| ≤ 0.1λ,

for all v ∈ E_{f}^{ua}(z), ∥v∥ = 1 and all |k| ≥ 0.5K}

and

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

where k_{0} > 0 is a number which will be defined later (see Lemma 4.5). Since m(Λ′(K)) → m(Π) as K → ∞ and hence, m(Λ(K)) → m(Π) as K → ∞, one can choose K so large that

(4.18) Kλ ≥ max{5k_{0}λ, 10 log 2, −10k_{0} log(1 − M_{0}δ_{g})}, λm(Π) + 40 log(1 − M_{0}δ_{g})m(Π\Λ) > 0, (4.19) 20m(Π\Λ) ≤ m(Π). (4.20)

Set

(4.21)  Λ^{∗} = Λ\ ⋃_{i=0}^{k_{0}−1} g̃^{−i}(Proj_{Π_{0}}(Ω_{0} ∪ Ω̃_{R})),

where Ω_{0} and Ω̃_{R} are given by (4.4) and (4.5) respectively. If the number ν is chosen small enough, Statement (5) of Proposition 4.2, allows us to assume that

m(Proj_{Π_{0}}Ω_{0} ∩ Π) ≤ m(Π)/20k_{0},

m(Proj_{Π_{0}} Ω̃_{R} ∩ Π) = m(B ∩ Π) ≤ m(Π)/20k_{0}.

(4.22)

Combining (4.20), (4.21) and (4.22), we find that

(4.23)  m(Λ^{∗}) ≥ 0.8m(Π).

By Statement (5) of Proposition 4.2, the set Π is invariant under the time-1 map of the flow g̃^{t}.

We will approximate the set Π by constructing an appropriate Rokhlin-Halmos tower (see [11]) for the map g̃^{1}. More precisely, we choose a measurable subset Γ′ ⊂ Π such that the sets g̃^{i}(Γ′) are pairwise disjoint for −K ≤ i ≤ 6K + k_{0} − 1 and

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

Consider the set Γ_{0} of first entries of orbits {g̃^{i}(z)}_{i=0}^{5K−1} (with z ∈ Γ′) to the set Λ^{∗}. More precisely, set

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

and let

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

Clearly, the sets {Γ_{i}} are pairwise disjoint for −K ≤ i ≤ K + k_{0} − 1. We then approximate Γ_{0} by finitely many disjoint ab-cylinders B_{0j} of the form

B_{0j} = B^{u}(z_{j}, r′_{j}) × B^{s}(z_{j}, r′′_{j}) × B^{ab}(z_{j}, r_{j})
= {(u_{j}, s_{j}, a_{j}, b_{j}) : |u_{j}| ≤ r′_{j}, |s_{j}| ≤ r′′_{j}, a_{j}^{2} + b_{j}^{2} ≤ r_{j}^{2}}
= {(u_{j}, s_{j}, ρ_{j}, φ_{j}) : |u_{j}| ≤ r′_{j}, |s_{j}| ≤ r′′_{j}, ρ_{j} ≤ r_{j}},

where r′_{j}, r′′_{j}, r_{j} > 0 for j = 1, …, J and z_{j} = (u_{j}, s_{j}, a_{j}, b_{j}) = (u_{j}, s_{j}, ρ_{j}, φ_{j}) ∈ Π_{0} is the center of B_{0j}. For i = −K, …, K + k_{0} − 1 set

(4.26)  B_{ij} = g̃^{i}(B_{0j}), Δ_{i} = ⋃_{j=1}^{J} B_{ij}.

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

(1) B_{ij} ∩ B_{kl} = ∅ for (i, j) ≠ (k, l) with −K ≤ i, k ≤ K + k_{0} − 1 and 1 ≤ j, l ≤ J;

(2) for each i = 0, 1, …, k_{0}

(4.27)  m(Γ_{i}△Δ_{i}) ≤ 0.05 max{m(Γ_{i}), m(Δ_{i})};

(3) B_{ij} ∩ Proj_{Π_{0}}(Ω_{0} ∪ Ω̃_{R}) = ∅ for 0 ≤ i ≤ k_{0} − 1, 1 ≤ j ≤ J. The last property implies that the set B_{ij} = g̃^{i}(B_{0j}) = f^{i}(B_{0j}) lies in a neighborhood around f^{i}(z_{j}) and hence is still an ab-cylinder if the numbers r_{j}, r′_{j}, r′′_{j} are chosen small enough.

We need the following lemma.

Lemma 4.5. Given δ > 0, there is θ_{0} = θ_{0}(δ) > 0 such that for any θ ∈ [0, θ_{0}] and any tube T = C × [0, 1/2], where C ⊂ Π_{0} is an ab-cylinder of the form

C = B^{u}(z, r′) × B^{s}(z, r′′) × B^{ab}(z, r),

there exist a subtube T′ = C′ × [1/40, 19/40] ⊂ T, where C′ ⊂ C is a cylinder of the form

C′ = B^{u}(z, r′_{0}) × B^{s}(z, r′′_{0}) × B^{ab}(z, r_{0}),

and a C^{∞} vector field X = X_{T,θ} on M such that (1) X is a rotation vector field with speed θ in the ab-plane, i.e.,

X (z) = X (u, s, a, b, τ) = θ(−b\frac{∂}{∂a} + a\frac{∂}{∂b}), z ∈ T′;

(2) X = 0 outside T;

(3) m(C′)/m(C) ≥ 0.75;

(4) r_{0}/r, r′_{0}/r′, r′′_{0}/r′′ ≥ 0.9;

(5) ∥X ∥_{C^{1}} < δ. Moreover, let k_{0} > 0 be such that

(4.28)  θ̄:= \frac{2π}{k_{0} ∫_{−1}^{1} ψ(t)dt} < θ_{0}(δ_{g}/2),

where ψ(t) is the function in Section 4.1 and δ_{g} is given by Proposition 4.1. Then ∥X_{T,θ̄}∥ ≤ δ_{g}/2.

Proof of the lemma. Given 0 < α < 1, we define a subcylinder

C_{α} = B^{u}(z, αr′) × B^{s}(z, αr′′) × B^{ab}(z, αr) ⊂ C.

By (A.2), the volume of C_{α} and C is induced by the flat metric du^{2} + ds^{2} + da^{2} + db^{2}, and hence the ratio m(C_{α})/m(C) depends only on α but not on the cylinder C. It follows that m(C_{α})/m(C) → 1 as α → 1.

Fix α > 0.9 such that m(C_{α})/m(C) > 0.75, and set C′ = C_{α}. Let us choose a C^{∞} function ξ : ℝ → [0, 1] satisfying:

(1) ξ = 1 on (−α, α);

(2) ξ > 0 on (−1, 1) and ξ = 0 outside (−1, 1);

(3) ∥ξ∥_{C^{1}} ≤ \frac{2}{1−α}. We introduce the ab-cylindrical coordinate (u, s, ρ, φ), and define a C^{∞} rotational vector field X = X_{T,θ} by the formula

(4.29)  X_{T,θ}(z) = \begin{cases}θξ̃(z)\frac{∂}{∂φ}, & z ∈ T, \\ 0, & z ∈ M\T, \end{cases}

where

ξ̃(z) = ξ̃(u, s, ρ, ξ, τ) = ξ(\frac{u}{r′})ξ(\frac{s}{r′′})ξ(\frac{ρ}{r})ψ(\frac{τ − 1/4}{1/4})

and ψ is the smooth function in Section 4.1. Note that ∥ξ̃\frac{∂}{∂φ}∥ ≤ c

where c > 0 depends only on α but not on the choice of the cylinder C. Thus for any δ > 0, there is θ_{0} = θ_{0}(δ) > 0 such that ∥X ∥_{C^{1}} < δ for any θ ∈ [0, θ_{0}]. □

Consider the tubes T_{ij} = B_{ij} × [0, 1/2]. Applying Lemma 4.5 with T = T_{ij}, we obtain a vector field X_{R,ij} = X_{T_{ij},θ̄} such that ∥X_{R,ij}∥ ≤ δ_{g}/2, where θ̄ is given by (4.28). Moreover, there is a sub-cylinder B′_{ij} ⊂ B_{ij} such that m(B′_{ij})/m(B_{ij}) ≥ 0.75. Furthermore, by Lemma 4.5, we may assume that g̃^{i}(B′_{0j}) = B′_{ij} for i = 1, …, k_{0}. Finally, let

(4.30)  Δ′_{i} = ⋃_{j=1}^{J} B′_{ij}, Ω_{R} = ⋃_{i=0}^{k_{0}−1} ⋃_{j=1}^{J} T_{ij},

and define the vector field X_{R} by

(4.31)  X_{R} = ∑_{i=0}^{k_{0}−1} ∑_{j=1}^{J} X_{R,ij}.

We obtain a new flow g^{t} generated by the vector field X_{g} = Xg̃ + X_{R}. Clearly, g^{t} is a C^{∞} volume preserving flow since X_{R} is divergence free. We will show that the flow g^{t} has all the desired properties stated in Proposition 4.1.

Proof of Proposition 4.1. Statements (1) and (2) follow immediately from the construction of the flow g^{t} and Statement (3) can be proved in the same way as Statement (3) of Proposition 4.2.

We will prove Statement (4). We need the following statement whose proof is very similar to the proof of Lemma 4.3.

Lemma 4.6. Given z ∈ B_{ij}, the ρ_{j}-, u_{j}-, s_{j}- and τ- coordinates of g^{t}(z) and f^{t}(z) are the same for t ∈ [0, 1/2]. Consequently, g^{\frac{1}{2}} (B_{ij}) = f^{\frac{1}{2}} (B_{ij}) and hence Π_{0} is a global cross-section for the flow g^{t}|N × U_{0}.

By the lemma, the time-1 map g^{1} restricted to Π_{0} is the Poincaré return map of g^{t} on Π_{0}. Therefore, (4.2) is equivalent to

(4.32)  L_{4}(G) = 0 < L_{1}(G) < L_{2}(G) < L_{3}(G),

where G = g^{1}|Π_{0}. In fact, by (4.1) and (4.7) we have for k = 3, 4 that

L_{k}(G) = L_{k}(f^{1}|Π_{0}) = L_{k}(g̃^{1}|Π_{0}) = L_{k}(G̃).

Hence, we only need to show that

(4.33)  L_{2}(G) < L_{3}(G).

We follow the argument in Section 4.2 in [10] and give a sketch of the proof of (4.33).

Set Δ_{0}^{∗} = Δ′_{0} ∩ Λ, where Δ′_{0} and Λ are given by (4.30) and (4.17) respectively, and

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

Consider the first return map \overline{G} = G^{R} on the set

U = U_{1} ∪ U_{2} ∪ U_{3} ∪ U_{4} ⊂ Π_{0},

where R = R(z) is the first return time of z ∈ U to U under G. Note that the flow g^{t} preserves the E_{f}^{uab}-subbundle, and so does G.

We intend to show that (4.34)

∫_{U} (log ∥ ∧^{3} (d\overline{G}|E_{f}^{uab}(z))∥ − log ∥ ∧^{2} (d\overline{G}|E_{f}^{uab}(z))∥) dm(z) > 0,

where

∧^{k}(d\overline{G}|E_{f}^{uab}(z)) : ∧^{k}(E_{f}^{uab}(z)) → ∧^{k}(E_{f}^{uab}(z))

is the k-th exterior power of d\overline{G}|E_{f}^{uab}(z). Indeed, assuming that (4.34) holds, consider the G-invariant set

Π′ = ⋃_{i=−∞}^{∞} G^{i}(U) ⊂ Π_{0}.

For k = 2, 3 we have that

∫_{U} log ∥ ∧^{k} (d\overline{G}|E_{f}^{uab}(z))∥dm(z) = ∫_{Π′} log ∥ ∧^{k} (dG|E_{f}^{uab}(z))∥dm(z)

= ∫_{Π′} ∑_{i=1}^{k} λ_{i}(z, G)dm(z) = L_{k}(G|Π′)

and hence, (4.34) implies that L_{2}(G|Π′) < L_{3}(G|Π′). Since G = G̃ outside Π′, we obtain that L_{2}(G) < L_{3}(G).

To show (4.34) we split the integral over U into four integrals I_{1}, I_{2}, I_{3} and I_{4} over the domains U_{1}, U_{2}, U_{3} and U_{4} respectively, and we obtain lower bounds for each of them. Namely, we will show that

(4.35) I_{1} ≥ 0.85Kλ · 0.7m(Δ_{0}), I_{2} ≥k_{0} log(1 − M_{0}δ_{g}) · 0.25m(Δ_{0}), I_{3} ≥ 0,  I_{4} ≥2 log(1 − M_{0}δ_{g})m(Π\Λ).

The lower bounds for I_{2}, I_{3} and I_{4} can be obtained using arguments in the proof of Lemma 4.2 in [10]. However, the proof of the lower bound for I_{1} in our continuous-time case requires substantial changes and we will present it here. We need the following lemma.

Lemma 4.7. Let z ∈ U_{1} = G^{−K}(Δ_{0}^{∗}). Then for any v ∈ E_{f}^{uab}(z), we have

(4.36)  ∥d_{z}\overline{G}(v)∥ ≥ \frac{\sqrt{2}}{2}∥v∥e^{0.9Kλ}

Proof of the lemma. Note that for any z ∈ G^{−K}(Δ_{0}^{∗}), the first return time R(z) is at least 2K + k_{0}. Set

z_{1} = G^{K}(z), z_{2} = G^{k_{0}}(z_{1}) = G^{K+k_{0}}(z), z_{3} = \overline{G}(z) = G^{R(z)}(z).

Since the orbit segments {g^{t}(z)}_{0≤t≤K} from z to z_{1} and {g^{t}(z_{2})}_{0≤t≤R(z)−K−k_{0}} from z_{2} to z_{3} are outside the set Ω_{R}, we have that

z_{1} = G^{K}(z) = G̃^{K}(z), z_{3} = G^{R(z)−K−k_{0}}(z_{2}) = G̃^{R(z)−K−k_{0}}(z_{2}).

On the other hand, since z_{1} ∈ Δ_{0}^{∗} = Δ′_{0} ∩ Λ, we can assume that z ∈ B′_{0j} for some j, and by our construction, we have G^{i}(z_{1}) ∈ B′_{ij} for i = 1, …, k_{0} − 1, and every cylinder B′_{ij} is inside a local coordinate neighborhood of its center. Therefore, we write z_{1} = (u, s, a, b, 0) ∈ B′_{0j}, and apply the similar arguments as in the proof of Lemma 4.4, we have that

G(z_{1}) = f^{\frac{1}{2}} g^{\frac{1}{2}} (z_{1}) = f^{\frac{1}{2}} (u, s, a cos ϕ − b sin ϕ, a sin ϕ + b cos ϕ, 1/2)
= (u, s, a cos ϕ − b sin ϕ, a sin ϕ + b cos ϕ, 1)
= (ηu, η^{−1}s, a cos ϕ − b sin ϕ, a sin ϕ + b cos ϕ, 0),

where

ϕ = \frac{1}{4}θ̄∫_{−1}^{1} ψ(t)dt = \frac{π}{2k_{0}} .

Repeating this calculation for G^{1}(z_{1}), G^{2}(z_{1}), …, G^{k_{0}−1}(z_{1}) and observing that k_{0}ϕ = \frac{π}{2}, we obtain that

z_{2} = G^{k_{0}}(z_{1}) = (η^{k_{0}}u, η^{−k_{0}}s, a cos(k_{0}ϕ)
− b sin(k_{0}ϕ), a sin(k_{0}ϕ) + b cos(k_{0}ϕ), 0)
= (η^{k_{0}}u, η^{−k_{0}}s, −b, a, 0).

This formula means that d_{z_{1}}G^{k_{0}} is non-contracting along the E_{f}^{uab} subbundle and rotates the vector in E_{f}^{ab} by the angle π/2.

To obtain (4.36), we write v = v^{ua}+v^{b} ∈ E_{f}^{ua}(z)⊕E_{f}^{b}(z) and consider the following two cases:

(1) if ∥v^{b}∥ ≤ \frac{\sqrt{2}}{2}∥v∥, since d_{z}G^{K} = d_{z}G̃^{K} and z ∈ G^{−K}Δ′_{0} ⊂

G̃^{−K}Λ′, by (4.16) and (4.17), we find that

∥d_{z}G^{K}v∥ = ∥d_{z}G̃^{K}v∥ ≥ ∥d_{z}G̃^{K}v^{ua}∥ ≥ ∥v^{ua}∥e^{0.9Kλ} ≥ \frac{\sqrt{2}}{2}∥v∥e^{0.9Kλ},

and hence

∥d_{z}\overline{G}v∥ = ∥d_{z^{2}} G̃^{R(z)−K−k_{0}}d_{z_{1}}G^{k_{0}}d_{z}G̃^{K}v∥ ≥ ∥d_{z}G̃^{K}v∥ ≥ \frac{\sqrt{2}}{2}∥v∥e^{0.9Kλ}.

(2) if ∥v^{b}∥ ≥ \frac{\sqrt{2}}{2}∥v∥, since d_{z_{1}}G^{k_{0}} rotates the vector in E_{f}^{ab} by the angle π/2, we have

d_{z}G^{K+k_{0}}v^{b} = d_{z_{1}}G^{k_{0}}(d_{z}G̃^{K}v^{b}) ∈ E_{f}^{ua}(z_{2}).

Since z_{2} ∈ Λ′, by (4.16) we obtain

∥d_{z}\overline{G}v∥ ≥ ∥d_{z}\overline{G}v^{b}∥ = ∥d_{z_{2}} G̃^{R(z)−K−k_{0}}d_{z}G^{K+k_{0}}v^{b}∥
≥ ∥d_{z}G^{K+k_{0}}v^{b}∥e^{0.9Kλ} ≥ \frac{\sqrt{2}}{2}∥v∥e^{0.9Kλ}.
□

By the lemma and (4.18), we find that

log ∥d_{z}\overline{G}(v)∥ ≥ 0.9Kλ − 0.5 log 2 + log ∥v∥ ≥ 0.85Kλ + log ∥v∥,

for any z ∈ U_{1} and v ∈ E_{f}^{uab}(z). Hence,

log ∥ ∧^{3} (d\overline{G}|E_{f}^{uab}(z))∥ − log ∥ ∧^{2} (d\overline{G}|E_{f}^{uab}(z))∥ ≥ 0.85Kλ.

On the other hand, it is proved in [10] that m(U_{1}) ≥ 0.7m(Δ_{0}). Therefore,

I_{1} = ∫_{U_{1}} (log ∥ ∧^{3} (d\overline{G}|E_{f}^{uab}(z))∥ − log ∥ ∧^{2} (d\overline{G}|E_{f}^{uab}(z))∥) dm(z)
≥ 0.85Kλ · 0.7m(Δ_{0}).

It follows from (4.35) that

∫U (log ∥ ∧^{3} (d\overline{G}|E_{f}^{uab}(z))∥ − log ∥ ∧^{2} (d\overline{G}|E_{f}^{uab}(z))∥)
≥ 0.595λKm(Δ_{0}) + 0.25k_{0} log(1 − M_{0}δ_{g})m(Δ_{0})
+ 2 log(1 − M_{0}δ_{g})m(Π\Λ)
≥ 0.57λKm(Δ_{0}) + 2 log(1 − M_{0}δ_{g})m(Π\Λ)
≥ 0.0627λm(Π) − 0.05λm(Π) = 0.0127λm(Π) > 0.

The last two inequalities follow from (4.18), (4.19) and Sublemma 4.4 in [10] that states that m(Δ_{0}) ≥ 0.11K^{−1}m(Π). This completes the proof of Statement (4) of Proposition 4.1. □

5. Accessibility

Notice that the flow g^{t} has positive central exponents on a set of positive volume but is not necessarily ergodic. We will perturb g^{t} to a flow h^{t} that is pointwise partially hyperbolic on the open set U and still has positive central exponents. Furthermore, we will ensure that the flow h^{t} possesses two transversal strongly stable and unstable foliations W_{h}^{s} and W_{h}^{u} of U and satisfies the accessibility property on U via these two foliations. In view of Theorem 2.4, h^{t} is indeed the desired flow in our Main Theorem.

We will follow the arguments in [10] and make some necessary modifications for the flow case. We choose two sequences of open subsets U_{n}, Ũ_{n} ⊂ U, n = 1, 2, … such that (A4) U_{0} ⊂ Ũ_{1}; (A5) Ũ_{n} ⊂ \overline{Ũ_{n}} ⊂ U_{n} ⊂ \overline{U_{n}} ⊂ U and ⋃_{n≥1} U_{n} = U; (A6) Ũ_{n} and U_{n} are connected sets for any n ≥ 1. Set

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

We will construct a sequence of flows {h_{n}^{t}}_{n≥0}, whose limit is the desired flow h^{t}. The goal of this section is to prove the following statement.

Proposition 5.1. Given δ_{h} > 0, one can find a sequence of positive numbers {δ_{n}} with δ_{n} ≤ min{δ_{h}/2^{n}, d(C, U_{n})^{2}} as well as a sequence of C^{∞} divergence free vector fields X_{n} on M, generating a sequence of volume preserving flows h_{n}^{t}, such that for n ≥ 0

(1) X_{0} = X_{g}, and hence h_{0}^{t} = g^{t};

(2) ∥X_{n+1} − X_{n}∥_{C^{n+1}} ≤ δ_{n};

(3) X_{n} = X_{f} on M\U_{n}, and X_{n} = X_{n−1} on U_{n−2}; in particular, each flow h_{n}^{t} is a gentle perturbation of f^{t} and hence satisfies Statements (3)-(5) of Proposition 3.1;

(4) for every z ∈ M, we have

E_{h_{n}}^{uabτ}(z) = E_{g}^{uabτ}(z), det(dh_{n}^{t}|E_{h_{n}}^{uabτ}(z)) = det(dg^{t}|E_{g}^{uabτ}(z));

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

∠(E_{h_{n}}^{ω}(z), E_{h_{n−1}}^{ω}(z)) ≤ δ_{j}/2^{n−j};

(6) if the number δ_{g} in Proposition 4.1 is sufficiently small, then each flow h_{n}^{t} is stably accessible in the following sense: Let a flow h̃^{t} be a gentle perturbation of the flow f^{t}, and assume that

∠(Eω ; h̃(z), E_{h_{n}}^{ω}(z)) ≤ δ_{n} for all z ∈ U_{n} and ω = u, s, c. Then any

two points z_{1}, z_{2} ∈ Ũ_{n} are accessible via a (u, s)h̃^{t}-path in U. In particular, h_{n}^{t} has the accessibility property in Ũ_{n}.

Statements (1)–(3) imply that the limit vector field X_{h} = lim_{n→∞} X_{n} exists. Moreover,

∥X_{n} − X_{k}∥_{C^{k+1}} ≤ ∑_{j=k}^{n−1} ∥X_{j+1} − X_{j}∥_{C^{j+1}} ≤ ∑_{j=k}^{n−1} δ_{j} ≤ δ_{h}/2^{k−1}

for any n ≥ k ≥ 0. It follows that X_{n} converges to X_{h} uniformly in the C^{k+1} topology. Since k is arbitrary, X_{h} is a C^{∞} vector field. In the following section we will show that the flow h^{t} generated by X_{h} has all the desired properties.

5.1. Construction of the sets U_{n} and Ũ_{n}. We view 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 will assume that the square U_{0}, constructed in Section 3, is contained in some Ẑ_{i_{0}j_{0}}^{(1)} so that

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

where C is the Cantor set constructed in Section 3. Consider the open squares

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}} ) .

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 U_{0} ⊂ Y_{1}, we let Y′_{n} be the connected component of Y_{n} that contains U_{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).

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 will denote them by Z_{1}^{(n)}, …, Z_{k_{n}}^{(n)} , and we will use the notations Ẑ_{l}^{(n)} and Z̃_{l}^{(n)} for the corresponding squares contained in Z_{l}^{(n)}. Thus we have

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

Clearly the collection of sets {Ẑ_{l}^{(n)} : n = 1, 2, …, l = 1, …, k_{n}} forms a countable partition of U 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 U of multiplicity at most 4. The following lemma is proved in [10].

Lemma 5.2. There is a labeling of the squares {Z_{l}^{(n)}} by integers from 1 to 8 such that for any y ∈ U, the labels of the squares Z_{l}^{(n)} containing y are all different. In particular, Z_{1}^{(1)} can be labeled by 1.

5.2. Construction of the vector fields X_{n}. The construction is similar to the one in Section 5.2 in [10], with a slightly modification on the choice of the collection of periodic points. We need the following preparations before we construct the vector fields X_{n}.

Let q_{j}, j = 1, …, 8 be eight periodic points of the Anosov automorphism A whose orbits are pairwise disjoint. There is ϵ_{0} > 0 such that

B_{X}(A^{i}q_{j}, ϵ_{0}) ∩ B_{X}(A^{i}q_{j′}, ϵ_{0}) = ∅

whenever j ≠ j′ and i = −1, 0, 1. For each q_{j} we choose three periodic points p_{j}^{i} ∈ B_{X}(A^{i}q_{j}, ϵ_{0}/3) for A, i = a, b, τ, whose orbits are pairwise disjoint. Denote by [q_{j}, p_{j}^{i}] = V_{A}^{u}(q_{j}) ∩ V_{A}^{s}(p_{j}^{i}), i = a, b, τ, where V_{A}^{s} and V_{A}^{u} are the stable and unstable local manifolds respectively. For i = a, b, τ 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}. Without loss of generality, we will assume that q_{1} = q, p_{1}^{i} = p^{i} and γ_{1}^{i} = γ^{i} for i = a, b, τ where q, p^{i} and γ^{i} are chosen as in the beginning of Section 4.1.

For j = 1, …, 8 and i = a, b, τ, we have

A^{ι(q_{j})}(q_{j}) = q_{j}, A^{ι(p_{j}^{i})}(p_{j}^{i}) = p_{j}^{i},

where ι(q_{j}) and ι(p_{j}^{i}) are periods of q_{j} and p_{j}^{i} respectively. There exists ᾱ(j, i) > 0 such that for any α ∈ Y = 𝕋^{2} with ∥α∥ ≤ ᾱ(j, i), the Anosov affine map A + α has a ι(q_{j})-periodic point q_{j}(α) close to q_{j} and a ι(p_{j}^{i})-periodic point p_{j}^{i}(α) close to p_{j}^{i}. Moreover, we can choose the number ᾱ (in Condition (α3) at the beginning of Section 3) to be less than min{ᾱ(j, i) : j = 1, …, 8, i = a, b, τ} such that any two points from the set of periodic points

{q_{j}(α), p_{j}^{i}(α) : j = 1, …, 8, i = a, b, τ, ∥α∥ ≤ ᾱ}

are disjoint.

Given n ≥ 1 and l = 1, …, k_{n}, let j be the label of Z_{l}^{(n)} in Lemma 5.2, and y_{0}(n, l) = (a_{0}(n, l), b_{0}(n, l)) the center of Z_{l}^{(n)}. We take the points associated to Z_{l}^{(n)} as follows:

(5.2)  q(n, l) = q_{j}(α(y_{0}(n, l))), p^{i}(n, l) = p_{j}^{i}(α(y_{0}(n, l))),

where i = a, b, τ. Recall that η is the expanding rate of A along its unstable direction, and the function κ : Y → ℝ is given in the beginning of Section 3. For n ≥ 1 let us choose a square Z_{l}^{(n)} ∈ Z_{n}. In the case n > 1, we write for simplicity q = q(n, l) and p^{i} = p^{i}(n, l) and we let η_{−}(n, l) = min{η^{κ(y)} : y ∈ Z_{l}^{(n)}}. Define the numbers

(5.3)  \begin{aligned}α_{u}^{i} & = α_{u}^{i}(n, l) = d(p^{i}, [p^{i}, q]), \\ α_{s}^{i} & = α_{s}^{i}(n, l) = d(p^{i}, [q, p^{i}]), \\ ᾰ_{u}^{i} & = ᾰ_{u}^{i}(n, l) = α_{u}^{i}(n, l)/η_{−}(n, l), \\ ᾰ_{s}^{i} & = ᾰ_{s}^{i}(n, l) = α_{s}^{i}(n, l)/η_{−}(n, l) \end{aligned}

and the rectangles in X

\begin{aligned}Π^{i}(n, l) & = B_{F^{u}}(p^{i}, α_{u}^{i}) × B_{F^{s}}(p^{i}, α_{s}^{i}), \\ Π̆^{i}(n, l) & = B_{F^{u}}(p^{i}, ᾰ_{u}^{i}) × B_{F^{s}}(p^{i}, ᾰ_{s}^{i}). \end{aligned}

We will assume that the rectangles Π^{i}(n, l), n ≥ 1, l = 1, …, k_{n} and i = a, b, τ are pairwise disjoint if the number ᾱ is chosen sufficiently small. Finally, we let

(5.4)  \begin{aligned}ϵ_{τ} & = ϵ_{τ}(n, l) = min\{κ(y)/2 : y ∈ Z_{l}^{(n)}\}, \\ ϵ̆_{τ} & = ϵ̆_{τ}(n, l) = 5ϵ_{τ}(n, l)/6. \end{aligned}

In the case n = 1, we have Z_{1}^{(1)} = U_{1} and q(1, 1) = q_{1}, p^{i}(1, 1) = p_{1}^{i} since the function α = 0 on 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 is given in (4.4). Then we set

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

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

In addition to the squares Ẑ_{ij}^{(n)}, Z̃_{ij}^{(n)} and Z_{ij}^{(n)} constructed in the previous subsection, we need to consider the following squares:

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}} )

as well as the following intervals:

I_{n} = J_{n} = (−\frac{3}{2^{n+2}} , \frac{3}{2^{n+2}} ) , Ĭ_{n} = J̆_{n} = (−\frac{5}{2^{n+3}} , \frac{5}{2^{n+3}} ) ,

and

K = (−1/4, 1 + 1/4), K̆ = (−1/8, 1 + 1/8), K̄ = (−1/16, 1 + 1/16).

Note that we have that

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

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

Fix n ≥ 1 and l = 1, …, k_{n}, and write α_{ω}^{i} = α_{ω}^{i}(n, l), ᾰ_{ω}^{i} = ᾰ_{ω}^{i}(n, l) for i = a, b, τ, ω = u, s, and ϵ_{τ} = ϵ_{τ}(n, l), ϵ̆_{τ} = ϵ̆_{τ}(n, l). We choose functions as follows:

(1) ϕ^{i} and ψ^{i} are C^{∞} functions on ℝ such that · ϕ^{i} =const. on (−ᾰ_{u}^{i}, ᾰ_{u}^{i}) and ψ^{i} =const. on (−ᾰ_{s}^{i}, ᾰ_{s}^{i}); · ϕ^{i}(r) = 0 for |r| ≥ α_{u}^{i}, ψ^{i}(r) = 0 for |r| ≥ α_{s}^{i}; · ∫_{0}^{±α_{u}^{i}} ϕ^{i}(r)dr = 0, and ψ^{i}(r) > 0 for any |r| < α_{s}^{i}; · ∥ϕ^{i}∥_{C^{n}}, ∥ψ^{i}∥_{C^{n}} ≤ 1.

(2) ξ_{τ} and ξ_{Y} are C^{∞} functions supported on K and I_{n} respectively such that · ξ_{τ} =const. on K̆, and ξ_{Y} =const. on Ĭ_{n}; · ξ_{τ}(r) > 0 for r ∈ K, and ξ_{Y} (r) > 0 for r ∈ I_{n}; · ξ_{τ}(r) = 0 for r ∉ K, and ξ_{Y} (r) = 0 for r ∉ I_{n}; · ∥ξ_{τ}∥_{C^{n}}, ∥ξ_{Y} ∥_{C^{n}} ≤ 1.

(3) ζ_{τ} and ζ_{Y} are C^{∞} functions supported on (−ϵ_{τ}, ϵ_{τ}) and I_{n} respectively such that · ζ_{τ} =const. on (−ϵ̆_{τ}, ϵ̆_{τ}), and ζ_{Y} =const. on Ĭ_{n}; · ζ_{τ}(r) > 0 for r ∈ (−ϵ_{τ}, ϵ_{τ}), and ζ_{Y} (r) > 0 for r ∈ I_{n}; · ζ_{τ}(r) = 0 for r ∉ (−ϵ_{τ}, ϵ_{τ}), and ζ_{Y} (r) = 0 for r ∉ I_{n}; · ∥ζ_{τ}∥_{C^{n}}, ∥ζ_{Y} ∥_{C^{n}} ≤ 1.

Now we are ready to construct the sequence of vector fields X_{n}. Given n ≥ 1, l = 1, …, k_{n} and i = a, b, τ, take the Cartesian coordinate system z = (u, s, τ, a, b) = (x, τ, a, b) with the origin at (p^{i}(n, l), 1/2, y_{0}(n, l)). In this coordinate system the interval K is in the symmetric form (−3/4, 3/4). Take the boxes for i = a, b

Ω^{i} = Ω_{n,l}^{i} = {(x, τ, y) : x ∈ Π^{i}(n, l), |τ| ≤ ϵ_{τ}(n, l), y ∈ Z_{l}^{(n)}},

and

Ω^{τ} = Ω_{n,l}^{τ} = {(x, τ, y) : x ∈ Π^{τ}(n, l), τ ∈ K, y ∈ Z_{l}^{(n)}}.

By the construction of the rectangles Π^{i}(n, l), we have that Ω^{i}(n, l) ∩ Ω^{i′}(n′, l′) = ∅ if (i, n, l) ≠ (i′, n′, l′). Similarly, we can choose Ω̆^{i}, i = a, b, τ by taking Π̆^{i}, ϵ̆_{τ}, K̆ and Z̆_{l}^{(n)}. Next we define three divergence free vector fields

X^{a} = X_{n,l}^{a} = ζ_{Y} (b)ζ_{τ}(τ)ψ^{a}(s)(−ξ′_{Y} (a) ∫_{0}^{u} ϕ^{a}(r)dr, 0, 0, ξ_{Y} (a)ϕ^{a}(u), 0),

X^{b} = X_{n,l}^{b} = ζ_{Y} (a)ζ_{τ}(τ)ψ^{b}(s)(−ξ′_{Y} (b) ∫_{0}^{u} ϕ^{b}(r)dr, 0, 0, 0, ξ_{Y} (b)ϕ^{b}(u)),

X^{τ} = X_{n,l}^{τ} = ζ_{Y} (a)ζ_{Y} (b)ψ^{τ}(s)(−ξ′_{τ}(τ) ∫_{0}^{u} ϕ^{τ}(r)dr, 0, ξ_{τ}(τ)ϕ^{τ}(u), 0, 0).

Clearly each vector field X_{n,l}^{i} vanishes outside the corresponding box Ω_{n,l}^{i}, and it is constant on the smaller box Ω̆_{n,l}^{i}. Finally, we set

(5.5)  X̂_{n} = ∑_{l=1}^{k_{n}}(X_{n,l}^{a} + X_{n,l}^{b} + X_{n,l}^{τ}), X_{n} = X_{g} + ∑_{k=1}^{n} β_{k}X̂_{k},

where the sequence of small positive numbers {β_{n}} is determined inductively to ensure Statements (2) and (5) of Proposition 5.1. Let h_{n}^{t} be the flow on M generated by the vector fields X_{n}.

5.3. Proof of Proposition 5.1. Statements (1)-(4) follow directly from our construction. It remains to show how to choose the sequence of positive numbers δ_{n} such that h_{n}^{t} satisfies Statements (5) and (6) of the proposition. Note that these two statements only concern those invariant subbundles E^{ω} and foliations W^{ω}, ω = u, s, c, cs, cu, which are the same for the flow and its time-1 map. Therefore, the choice of δ_{n} and related arguments are similar to the diffeomorphism case in [10]. We will outline the proof here.

For any gentle perturbation h_{♮}^{t} of f^{t} (see Definition 2.1), we denote by W_{h_{♮}}^{c}(z) the center manifold of h_{♮}^{t} at the point z ∈ M. Given

a square Z_{l}^{(n)} with the center y_{0}(n, l), let q(n, l), p^{i}(n, l), i = a, b, τ be the associated periodic points given by (5.2), and z_{0} = z_{0}(n, l) = (q(n, l), 1/2, y_{0}(n, l)). We denote by W_{h_{♮}}^{c}(z_{0}, K, Z_{l}^{(n)}) the connected

component of W_{h_{♮}}^{c}(z_{0}) ∩ (X × K × Z_{l}^{(n)}) that contains z_{0}. We will

also use similar notations W_{h_{♮}}^{c}(z_{0}, K̆, Z̆_{l}^{(n)}), etc.

Next we will introduce two important families of maps Θ and Ψ for a gentle perturbation h_{♮}^{t} of f^{t}.

Fix n ≥ 1 and l = 1, …, k_{n}, we take the collection of points q = q(n, l), p^{i} = p^{i}(n, l), i = a, b, τ. Consider a quadrilateral (u, s)_{h_{♮}^{t}}-path γ̂^{i} = {z_{1}, …, z_{5}} with initial point z_{1} defined by

(5.6)  \begin{aligned}z_{2} & = V_{h_{♮}}^{u}(z_{1}) ∩ V_{h_{♮}}^{cs}(p^{i}, 1/2, a_{0}, b_{0}), \\ z_{3} & = V_{h_{♮}}^{s}(z_{2}) ∩ V_{h_{♮}}^{cu}(p^{i}, 1/2, a_{0}, b_{0}), \\ z_{4} & = V_{h_{♮}}^{u}(z_{3}) ∩ V_{h_{♮}}^{cs}(z_{1}), \\ z_{5} & = V_{h_{♮}}^{s}(z_{4}) ∩ V_{h_{♮}}^{cu}(z_{1}). \end{aligned}

This path defines a map Θ^{i} = Θ_{n,l,h_{♮}}^{i} given by Θ^{i}(z_{1}) = z_{5}. It is

easy to see that z_{5} ∈ V_{h_{♮}}^{c}(z_{1}), and Θ^{i} maps W_{h_{♮}}^{c}(z_{0}, K, Z_{l}^{(n)}) into itself. Reparameterizing the curve on V_{h_{♮}}^{u}(z_{1}) from z_{1} to z_{2} by σ : [0, 1] → V_{h_{♮}}^{u}(z_{1}) so that σ(0) = z_{1} and σ(1) = z_{2}, we obtain a parameterized family of quadrilaterals γ̂^{i}(ϑ) = {z_{1}(ϑ), …, z_{5}(ϑ)}, ϑ ∈ [0, 1], where z_{1}(ϑ) = z_{1}, z_{2}(ϑ) = σ(ϑ), and z_{k}(ϑ), k = 3, 4, 5 are obtained in the way similar to (5.6). Then we define Θ_{ϑ}^{i} = Θ_{ϑ,n,l,h_{♮}}^{i} given by Θ_{ϑ}^{i}(z_{1}) = z_{5}(ϑ).

Clearly Θ_{0}^{i} = Id, Θ_{1}^{i} = Θ^{i}, Θ_{ϑ}^{i} maps W_{h_{♮}}^{c}(z_{0}, K, Z_{l}^{(n)}) into W_{h_{♮}}^{c}(z_{0}) and depends continuously on ϑ ∈ [0, 1].

On the other hand, given z = ((x, τ), y) ∈ U, there is a (u, s)_{f^{t}}-path γ_{f}(z) connecting z to z′ = ((q, τ), y) whose length does not exceed 2d(x, q). This generates a map Ψ_{f} = Ψ_{f,n,l} from U to {q} × K × G given by Ψ_{f}(z) = z′. Furthermore, given a gentle perturbation h_{♮}^{t} of f^{t} and a point z ∈ Z_{l}^{(n)}, we can find a (u, s)_{h_{♮}}-path γ_{h_{♮}}(z), which is close to γ_{f}(z) and connects z to a point z′ = z′(h_{♮}^{t}) ∈ W_{h_{♮}}^{c}(z_{0}, K, Z_{l}^{(n)}). We can then define Ψ_{h_{♮}} = Ψ_{h_{♮},n,l} by Ψ_{h_{♮}}(z) = z′(h_{♮}^{t}).

Note that the maps Ψ_{h_{♮},n,l}, Θ_{n,l,h_{♮}}^{i} and Θ_{ϑ,n,l,h_{♮}}^{i}, i = a, b, τ depend continuously on h_{♮}^{t} as long as h_{♮}^{t} is a gentle perturbation of f^{t} with h_{♮}^{t} = f^{t} outside some fixed U_{n} and with ∠(E_{h_{♮}}^{ω}(z), E_{f}^{ω}(z)) sufficiently small for all z ∈ U_{n} and ω = u, s, c. Moreover, the continuity is uniform with respect to z.

Given a set Γ ⊂ M and a gentle perturbation h_{♮}^{t} of f^{t}, set

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

For n ≥ 1 denote by ϵ_{n} = min{1/2^{n+5}, ϵ̆_{τ}(n, l), l = 1, …, k_{n}}, where ϵ̆_{τ}(n, l) is defined by (5.4).

We now briefly describe how to choose the sequence {δ_{n}}. See [10] for more details. Recall that U_{1} = Z_{1}^{(1)}, Ũ_{1} = Z̃_{1}^{(1)}, and U_{1} = N × U_{1}, Ũ_{1} = N × Ũ_{1}. Choose θ_{0} > 0 such that the families of maps Ψ_{h_{♮}} and Θ_{h_{♮}}^{i} are well-defined for any gentle perturbation h_{♮}^{t} of f^{t} with ∠(E_{h_{♮}}^{ω}(z), E_{f}^{ω}(z)) ≤ 2θ_{0} for ω = u, s, c. We assume that the number δ_{g} in Proposition 4.1 is so small that the flow h_{0}^{t} = g^{t} satisfies ∠(E_{h_{0}}^{ω}(z), E_{f}^{ω}(z)) ≤ θ_{0} and d(Θ_{ϑ,1,1,h_{0}}^{i}(z), z) ≤ ϵ_{1}/4 for z ∈ N × U_{0}, ϑ ∈ [0, 1] and i = a, b, τ.

Now choose θ′_{1} with 0 < θ′_{1} ≤ θ_{0}/2 such that d(Ψ_{h_{♮}}(z), Ψ_{h_{0}}(z)) ≤ 1/2^{8} if ∠(E_{h_{♮}}^{ω}(z), E_{h_{0}}^{ω}(z)) ≤ 2θ′_{1} for all z ∈ N × Z_{1}^{(1)}. Also choose δ′_{1} > 0 such that if ∥X_{1} − X_{0}∥ ≤ δ′_{1}, then ∠(E_{h_{1}}^{ω}(z), E_{h_{0}}^{ω}(z)) ≤ θ′_{1}. Finally set θ_{1} = min{θ′_{1}, θ′′_{1}} and δ_{1} = min{δ′_{1}, δ′′_{1}, θ_{1}}, where δ′′_{1} and θ′′_{1} are given by Lemma 5.3 below. We can show

(1) d(Ψ_{h_{1}}(z), Ψ_{h_{0}}(z)) ≤ 1/2^{8} for all z ∈ N × Z_{1}^{(1)};

(2) d(Θ_{ϑ,2,l,h_{1}}^{i}(z), z) ≤ ϵ_{2}/4 for all z ∈ W_{h_{1}}^{c}(z_{0}(2, l), K, Z_{l}^{(2)}), i = a, b, τ, ϑ ∈ [0, 1] and l = 1, …, k_{2};

(3) A_{h_{♮}}(z_{0}(1, 1)) ⊃ W_{h_{♮}}^{c}(z_{0}(1, 1), K̄, Z̄_{1}^{(1)}) for any gentle perturbation h_{♮}^{t} of f^{t}, close to h_{1}^{t}, with ∠(E_{h_{♮}}^{ω}(z), E_{h_{1}}^{ω}(z)) ≤ θ′_{1}, ω = u, s, c

and z ∈ N × Z_{1}^{(1)}. Moreover, the above statements imply that A_{h_{♮}}(z_{0}(1, 1)) ⊃ N × Z̃_{1}^{(1)}, in particular, h_{1}^{t} has the accessibility property on N × Z̃_{1}^{(1)}.

Proceeding inductively, we can choose δ_{n} such that Statements (5) and (6) of Proposition 5.1 hold. Furthermore, we have for i = a, b, τ, ϑ ∈ [0, 1] and l = 1, …, k_{n+1},

(1) d(Ψ_{h_{n}}(z), Ψ_{h_{n−1}}(z)) ≤ 1/2^{n+7} for all z ∈ N × Z_{l}^{(n)};

(2) d(Θ_{ϑ,n+1,l,h_{n}}^{i}(z), z) ≤ ϵ_{n+1}/4 for all z ∈ W_{h_{n}}^{c}(z_{0}(n+1, l), K, Z_{l}^{(n+1)});

(3) A_{h_{♮}}(z_{0}(n, l)) ⊃ W_{h_{♮}}^{c}(z_{0}(n, l), K̄, Z̄_{l}^{(n)}) for any gentle perturbation h_{♮}^{t} of f^{t}, close to h_{n}^{t}, with ∠(E_{h_{♮}}^{ω}(z), E_{h_{n}}^{ω}(z)) ≤ δ_{n}, ω =

u, s, c and z ∈ N × Z_{l}^{(n)}.

Therefore, A_{h_{♮}}(z_{0}(n, l) ⊃ N × Z̃_{l}^{(n)} for all l = 1, …, k_{n+1}. In other words, h_{♮}^{t} has the accessibility property on N × Z̃_{l}^{(n)}.

Note that

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

and the intersection of any two sets among Ũ_{n−1} and N × Z̃_{l}^{(n)}, l = 1, …, k_{n} contains a nonempty open set whenever they intersect. Since Ũ_{n} is connected, we obtain accessibility of h_{♮}^{t} on Ũ_{n}. In particular, h_{n}^{t} has the accessibility property on Ũ_{n} when we apply that h_{♮}^{t} = h_{n}^{t}.

5.4. A technical lemma. The proof of Proposition 5.1 heavily relies on the following technical statements.

Lemma 5.3. Suppose for some n > 0, d(Θ_{ϑ,n,l,h_{n−1}}^{i}(z), z) ≤ ϵ_{n}/4 for all i = a, b, τ, ϑ ∈ [0, 1], z ∈ W_{h_{n−1}}^{c}(z_{0}(n, l), K, Z_{l}^{(n)}), l = 1, …, k_{n}. Then there are δ′′_{n}, θ′′_{n} > 0 such that

(1) if ∥X_{n} − X_{n−1}∥_{C^{n}} ≤ δ′′_{n}, then we have

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

for all i = a, b, τ, ϑ ∈ [0, 1], and l = 1, …, k_{n+1};

(2) for any gentle perturbation h_{♮}^{t} of f^{t}, close to h_{n}^{t} and with

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

we have

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

In particular, (5.8) holds with h_{♮}^{t} = h_{n}^{t}.

This lemma is an adaptation of Lemma 5.2 in [10] to the flow case. It can be proved in a similar fashion subject to the following sublemma.

Sublemma 5.4. For each n > 0, there exists δ′′_{n} > 0 such that if ∥X_{n} − X_{n−1}∥_{C^{n}} = β_{n}∥X̂_{n}∥_{C^{n}} ≤ δ′′_{n}, then for all l = 1, …, k_{n}, we have

(1) Θ^{a}((q, 1/2, a, 0)) = (q, 1/2, a′, 0) with a′ < a for any a ∈ I_{n};

(2) Θ^{b}((q, 1/2, a, b)) = (q, 1/2, a, b′) with b′ < b for any a ∈ I_{n}, b ∈ J_{n};

(3) Θ^{τ}((q, τ, a, b)) = (q, τ′, a, b) with τ′ < τ for any a ∈ I_{n}, b ∈ J_{n} and τ ∈ K,

where q = q(n, l) is given by (5.2), and Θ^{i} = Θ_{n,l,h_{n}}^{i} for i = a, b, τ.

Proof. The proof is similar to that in [6] (see Lemma B.4; see also [10], Sublemma 5.3) but is adapted to the language of vector fields.

We will prove the first statement. Consider the coordinate system (u, s, τ, a, b) in Ω_{n,l}^{a} with the origin at (p^{a}(n, l), 1/2, y_{0}(n, l)). Write q = q(n, l), p^{a} = p^{a}(n, l) and y_{0} = y_{0}(n, l). We may assume that the square Z_{l}^{(n)} is parameterized as (a, b) ∈ I_{n} × J_{n}, that the center y_{0}(n, l) of Z_{l}^{(n)} is (0, 0) and that the local coordinates of the points q, [q, p^{a}], [p^{a}, q] and p^{a} are (u_{0}, s_{0}), (0, s_{0}), (u_{0}, 0) and (0, 0) respectively, where u_{0} = α_{u}^{a}(n, l) and s_{0} = α_{s}^{a}(n, l) given by (5.3).

Consider the case n > 1 first. Note that the vector field X_{n} inside Ω^{a}(n, l) is exactly X_{f} + β_{n}X_{n,l}^{a}. For any a_{1} = a ∈ I_{n}, b ∈ J_{n}, and τ ∈ (1/2 − ϵ_{τ}, 1/2 + ϵ_{τ}), choose the point \underline{z}_{1} = (q, τ, a_{1}, b) = (u_{0}, s_{0}, τ, a_{1}, b). Note that under the original flow f^{t}, we have a closed quadrilateral (u, s)_{f^{t}}-path \underline{γ} = \{\underline{z}_{1}, \underline{z}_{2}, \underline{z}_{3}, \underline{z}_{4}, \underline{z}_{5}\}, where

\begin{aligned}\underline{z}_{2} & = ([q, p^{a}], τ, \underline{a}_{2}, b) = (0, s_{0}, τ, \underline{a}_{2}, b), \\ \underline{z}_{3} & = (p^{a}, τ, \underline{a}_{3}, b) = (0, 0, τ, \underline{a}_{3}, b), \\ \underline{z}_{4} & = ([p^{a}, q], τ, \underline{a}_{4}, b) = (u_{0}, 0, τ, \underline{a}_{4}, b), \\ \underline{z}_{5} & = ([q, p^{a}], τ, \underline{a}_{5}, b) = (u_{0}, s_{0}, τ, \underline{a}_{5}, b) = \underline{z}_{1}, \end{aligned}

and \underline{a}_{k} = a_{1} = a for k = 1, 2, 3, 4, 5.

Let us compare the vector field X_{n} = X_{f} + β_{n}X_{n,l}^{a} on each leg L_{k} = [\underline{z}_{k}, \underline{z}_{k+1}] for k = 1, 2, 3, 4. In fact, X_{n,l}^{a} ≡ 0 on legs L_{1} and L_{4}. Since the u-component of every point on the leg L_{2} is 0, the u-component of the vector field X_{n,l}^{a} is 0, and the a-component does not depend on the u-coordinate. On the leg L_{3} = [\underline{z}_{3}, \underline{z}_{4}], the u-component of X_{n,l}^{a} is negative at the interior points and it is zero at two endpoints \underline{z}_{3} and \underline{z}_{4}, while the a-component is positive, with the value smoothly changing from a constant to zero.

Now choose the point z_{1} = \underline{z}_{1}, and we have the quadrilateral (u, s)_{h_{n}^{t}}- path γ = {z_{1}, z_{2}, z_{3}, z_{4}, z_{5}}. By the above comparison, the τ- and bcoordinate are the same for each z_{k}, k = 1, 2, 3, 4, 5. By the construction of the vector fields X_{n}, the image of the leg [z_{3}, z_{4}] under the flow h_{n}^{t} is contained in Ω^{a}. Now let a_{k} be the a-coordinate of z_{k}, k = 1, 2, 3, 4, 5, since the a-component of X_{n} are the same for all points on L_{1}, L_{2} and L_{4}, while it changes from a constant to zero along the unstable leg L_{3}, then we have a_{1} = a_{2} = a_{3} > a_{4} = a_{5}. This shows Statement (1) for the case n > 1.

In the case n = 1 similar arguments can be used with the following modification, and we will obtain a_{1} = a_{2} ≥ a_{3} > a_{4} = a_{5}. This completes the proof of Statement (1). Statements (2) and (3) can be proved in a similar way. □

6. Proof of Main Theorem.

Since each X_{n} is divergence free, so is X_{h}, and hence h^{t} is volume preserving. The first statement of the Main Theorem follows.

Note that h^{t} = f^{t} on U^{c} and is of the form

h^{t}((x, τ), y) = ((x + tα_{0}, τ), y)

for each z = ((x, τ), y) ∈ U^{c} = N × C where α_{0} is a diophantine vector (see Section 3). Hence, h^{t} preserves each 3-dimensional submanifold N × {y}, y ∈ C, and h^{t}|N × {y} is a non-identity linear flow since α(y) ≠ 0. Moreover, the frequency vector α(y) is Diophantine if y ∈ C. Thus Statements (1) and (3) of the Main Theorem follows.

It remains to prove the second statement. By Proposition 4.1, each diffeomorphism h_{n}^{t} is pointwise partially hyperbolic on U and uniformly partially hyperbolic on \overline{U}_{n}. By Theorem B.1 in the Appendix B, if the sequence δ_{n} decreases sufficiently fast, the limit flow h^{t} is pointwise partially hyperbolic on U.

We now claim that the one-dimensional strongly stable E_{h}^{s} and unstable E_{h}^{u} subbundles are integrable to invariant strongly stable W_{h}^{s} and unstable W_{h}^{u} foliations with smooth leaves, which are transversal and absolutely continuous. Recall that the “start-up” flow f^{t} has strongly stable and unstable local manifolds V_{f}^{s}(z) and V_{f}^{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 5.1(3), h_{n}^{t}|U_{n}^{c} = f^{t}|U_{n}^{c}, and thus V_{h_{n}}^{ω}(z) = V_{f}^{ω}(z) for all z ∈ U\U_{n}, ω = s, u. On the other hand, each h_{n}^{t} is a perturbation of h_{n−1}^{t} on the compact set U_{n}, on which both h_{n}^{t} and h_{n−1}^{t} are uniformly partially hyperbolic if δ_{n} is sufficiently small. Furthermore, let r_{n} be the size of V_{h_{n}}^{ω}(z) for z ∈ U_{n}, one can have r_{n}/r_{n−1} ≥ 2^{−1/2^{n}}, and thus by induction we have that the size of local manifolds of h_{n}^{t}|U_{n} is bigger than r. Therefore, given z ∈ U, we obtain that the size of V_{h_{n}}^{ω}(z) has a lower bound r > 0, which is independent of z and n.

Write each V_{h_{n}}^{s}(z) in the coordinate chart as follows

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

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

(1) ψ_{h_{n}}^{s}(0) = 0 and dψ_{h_{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_{h_{n}}^{s}∥_{C^{1}} ≤ Δ for all n ≥ 0.

This implies that z ∈ V_{h_{n}}^{s}(z) and T_{z}V_{h_{n}}^{s}(z) = E_{h_{n}}^{s}(z). Furthermore,

(1) h_{n}^{t}(V_{h_{n}}^{s}(z)) ⊂ V_{h_{n}}^{s}(h_{n}^{t}(z));

(2) d(h_{n}^{t}(z), h_{n}^{t}(y)) ≤ λ̃(z)d(z, y) for each y ∈ V_{h_{n}}^{s}(z) and some continuous function λ̃(z) on U 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_{h_{n}}^{s}(v), ∥v∥ ≤ r, is compact in the C^{1} topology and hence, there is a subsequence ψ_{h_{n_{k}}}^{s} that converges to a C^{1} function ψ satisfying ψ(0) = 0, dψ(0) = 0 and ∥ψ∥_{C^{1}} ≤ Δ. Setting

(6.1)  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_{h}^{s}(z);

(2) h^{t}(V (z)) ⊂ V (h^{t}(z));

(3) d(h^{t}(z), h^{t}(y)) ≤ λ̃(z)d(z, y) for each y ∈ V (z).

This implies that if m_{k} is any subsequence for which ψ_{h_{m_{k}}}^{s} converges in the C^{1} topology to a function ψ̃, then ψ̃ = ψ. Thus the formula (6.1) determines uniquely a local strongly stable manifold through z and the formula W (z) = ∪_{n≥0} h^{−t}(V (h^{t}(z)) defines the global strongly stable manifold through z. These manifolds form a continuous strongly stable foliation with smooth leaves for h^{t}. In a similar fashion we can obtain strongly unstable local manifolds and construct a strongly unstable foliation with smooth leaves for h^{t}. These two foliations are transverse at every point z ∈ U.

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

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

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

= ∫_{Z} log λ_{h}(z) dm(z) < 0

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

Since h^{t} is nonuniformly partially hyperbolic on U, by Theorem 8.6.1 in [2], we obtain that its strongly stable and unstable foliations are absolutely continuous.

Next we will show that the flow h^{t} has the accessibility property on U via its invariant foliations W_{h}^{s} and W_{h}^{u}. Indeed, by Proposition 5.1 (5), for any n > k, ω = s, u, c and any z ∈ U_{k} ⊂ U,

∠(E_{h_{n}}^{ω}(z), E_{h_{k}}^{ω}(z)) ≤ δ_{k}(1 − 1/2^{n−k}) < δ_{k}.

Taking the limit as n → ∞, we obtain that ∠(E_{h}^{ω}(z), E_{h_{k}}^{ω}(z)) ≤ δ_{k} on U_{k}. Hence, by Proposition 5.1(6), the flow h^{t} has the accessibility property on each Ũ_{k}. Since k is arbitrary, we obtain that the flow h^{t} has the accessibility property on U.

To show that the flow h^{t} has positive central Lyapunov exponent, we first recall that the average Lyapunov exponents of the flow g^{t} are arranged as in (4.2). Set c = L_{3}(g^{t}) − L_{2}(g^{t}) > 0. By the upper semicontinuity of L_{i}(·), we choose the number δ_{h} > 0 in Proposition 5.1 so small that L_{2}(h^{t}) < L_{2}(g^{t}) + c/2. On the other hand, it follows from Proposition 5.1(4) that

L_{4}(h_{n}^{t}) = ∫_{M} det(dh_{n}^{t}|E_{h_{n}}^{uabτ}(z))dm = ∫_{M} det(dg^{t}|E_{g}^{uabτ}(z))dm = L_{4}(g^{t}).

Taking the limit as n → ∞ we obtain L_{4}(h^{t}) = L_{4}(g^{t}) = L_{3}(g^{t}). Therefore,

L_{4}(h^{t}) − L_{2}(h^{t}) = ∫_{M}(λ_{3}(z, h^{t}) + λ_{4}(z, h^{t}))dm(z) ≥ c/2 > 0,

then there is a subset A ⊂ U such that λ_{3}(z, h^{t}) + λ_{4}(z, h^{t}) > 0 for all z ∈ A, and thus λ_{2}(z, h^{t}) ≥ λ_{3}(z, h^{t}) ≥ \frac{1}{2}[λ_{3}(z, h^{t}) + λ_{4}(z, h^{t})] > 0 for all z ∈ A. Since the center subspace E_{h}^{c}(z) is 3-dimensional and the flow direction Span{X_{h}} corresponds to zero exponent, we conclude that λ_{2}(z, h^{t}) and λ_{3}(z, h^{t}) correspond to vectors in E_{h}^{c}(z)\Span{X_{h}}. Thus the flow h^{t} has positive central Lyapunov exponents.

By Theorem 2.4, we obtain that h^{t} has positive central exponents at almost every point in U, and h^{t}|U is an ergodic flow.

Appendix A. The differential and metric structures of the suspension manifold

We specify the differential and metric structure of the suspension manifold N and the 5-dimensional manifold M in Section 3. Associated to the Anosov automorphism A of X = 𝕋^{2}, one can find smooth local charts (U_{x}, ϕ_{x}) around each x ∈ X such that

ϕ_{x} : U_{x} → (−u_{0}(x), u_{0}(x)) × (−s_{0}(x), s_{0}(x))

satisfies ϕ_{x}(x) = (0, 0) and

ϕ_{Ax} ∘ A ∘ ϕ_{x}^{−1}(u, s) = (ηu, η^{−1}s),

where u_{0}(x), s_{0}(x) > 0 are sizes of charts depending on x, and η > 1 is the expanding rate along the unstable direction. In fact, \frac{∂}{∂u} and \frac{∂}{∂s} are the unstable and stable directions of A respectively.

Recall that the suspension manifold N is the quotient space X × ℝ/ ∼ with the equivalence relation (x, τ + 1) ∼ (Ax, τ). Let π : X ×ℝ → N be the natural projection. Following [12] there is a natural differential structure on N with atlas (U_{(x,τ)}^{1}, ϕ_{(x,τ)}^{1}) for τ ∈ (−1/4, 3/4) and (U_{(x,τ)}^{2}, ϕ_{(x,τ)}^{2}) for τ ∈ (1/4, 5/4), where

U_{(x,τ)}^{1} = π(U_{x} × (−1/4, 3/4)), ϕ_{(x,τ)}^{1}(π(ϕ_{x}^{−1}(u, s), τ)) = (u, s, τ);

U_{(x,τ)}^{2} = π(U_{x} × ( 1/4, 5/4)), ϕ_{(x,τ)}^{2}(π(ϕ_{x}^{−1}(u, s), τ)) = (u, s, τ). It is easy to verify that

\begin{aligned}ϕ_{(x′,τ′)}^{i} ∘ ϕ_{(x,τ)}^{i}(π(ϕ_{x}^{−1}(u, s), τ) & = (ϕ_{x′} ∘ ϕ_{x}^{−1}(u, s), τ), i = 1, 2; \\ ϕ_{(x′,τ′)}^{1} ∘ ϕ_{(x,τ)}^{2}(π(ϕ_{x}^{−1}(u, s), τ) & = (ϕ_{Ax′} ∘ A ∘ ϕ_{x}^{−1}(u, s), τ − 1). \end{aligned}

In particular,

(A.1)  ϕ_{(x,τ′)}^{1} ∘ ϕ_{(x,τ)}^{2}(π(ϕ_{x}^{−1}(u, s), τ) = (ηu, η^{−1}s, τ − 1).

There are three subbundles E^{u}, E^{s} and E^{τ} on N generated by independent vector fields dπ(\frac{∂}{∂u}), dπ(\frac{∂}{∂s}) and dπ(\frac{∂}{∂τ}) respectively. By [3], we can choose the Riemannian metric on N which has the local representation η^{2τ}du^{2} + η^{−2τ}ds^{2} + dτ^{2}. Under this metric, the suspension flow S^{t} : N → N satisfies

∥dS^{t}v∥ = η^{t}∥v∥,  v ∈ E^{u},
∥dS^{t}v∥ = η^{−t}∥v∥,  v ∈ E^{s},
∥dS^{t}v∥ = ∥v∥,  v ∈ E^{τ}.

For the 2-torus Y = 𝕋^{2}, we choose the local coordinate (a, b) centered at each y ∈ Y. Given z = (x, τ, y) ∈ M = N × Y, we can hence choose a local coordinate system (u, s, τ, a, b), endowed with the product Riemannian metric η^{2τ}du^{2} + η^{−2τ}ds^{2} + dτ^{2} + da^{2} + db^{2}. In particular, the metric on the cross-section X × {0} × Y is given by the flat metric

(A.2)  du^{2} + ds^{2} + da^{2} + db^{2}.

Appendix B. Partial Hyperbolicity of the Limit Flow

Let M be a compact smooth Riemannian manifold and S ⊂ M an open subset. Let also H^{t} be the flow on M 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^{t}-invariant;

(3) H^{t}|\overline{U_{n}} is uniformly partially hyperbolic. The goal of this appendix is to prove the following statement.

Theorem B.1. There exists a sequence of positive numbers ε_{n} such that if smooth vector fields X_{n} on M satisfy

X_{0} = X_{H}, X_{n} = X_{n−1} on M\\overline{U_{n}}

and

∥X_{n} − X_{n−1}∥_{C^{1}} ≤ ε_{n},

then for every n ≥ 1 the corresponding flow h_{n}^{t} is uniformly partially hyperbolic on the invariant set \overline{U_{n}} and hence pointwise partially hyperbolic on S. Moreover, the limit vector field X = lim_{n→∞} X_{n} is of class C^{1} and generates a pointwise partially hyperbolic flow h^{t} on S.

We need the following technical statements.

Lemma B.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}, ∏_{n=1}^{∞}(1 − a_{n}) ≥ 1 − 2 ∑_{n=1}^{∞} a_{n}.

Lemma B.3 (Gronwall’s inequality). Let η(t) be a nonnegative C^{1} function on [0, T] satisfying

η′(t) ≤ ϕ(t)η(t) + ψ(t),

where ϕ(t) and ψ(t) are nonnegative integrable functions, then for all 0 ≤ t ≤ T,

η(t) ≤ e^{∫_{0}^{t} ϕ(s)ds}[η(0) + ∫_{0}^{t} ψ(s)ds].

Lemma B.4. Set K := 2∥X_{H}∥_{C^{1}}. If ε_{n} < \frac{K}{2^{n+1}} , then ∥X_{n}∥_{C^{1}} ≤ K for all n ≥ 0. Moreover, given a flow F^{t} with ∥X_{F} ∥_{C^{1}} ≤ K, we have for any x ∈ M and t ∈ ℝ^{+} that,

e^{−tK} ≤ m(d_{x}F^{t}) ≤ ∥d_{x}F^{t}∥ ≤ e^{tK}.

In particular,

e^{−K} ≤ min_{0≤t≤1} m(d_{x}F^{t}) ≤ max_{0≤t≤1} ∥d_{x}F^{t}∥ ≤ e^{K}.

Proof of the lemma. Since ∥X_{H}∥_{C^{1}} = K/2, we find that

∥X_{n} − X_{H}∥_{C^{1}} ≤ ∑_{k=1}^{n} ∥X_{k} − X_{k−1}∥_{C^{1}} ≤ ∑_{k=1}^{n} ε_{k} < K/2.

This implies that ∥X_{n}∥_{C^{1}} ≤ K.

Let F^{t} be a flow and X_{F} the corresponding vector field. Consider the variational differential equation

\frac{d}{dt}d_{x}F^{t} = DX_{F} (F^{t}(x))d_{x}F^{t},

for any x ∈ M and t ∈ ℝ^{+}. Then

\frac{d}{dt}∥d_{x}F^{t}∥ ≤ ∥\frac{d}{dt}d_{x}F^{t}∥ ≤ ∥DX_{F} ∥∥d_{x}F^{t}∥ ≤ K∥d_{x}F^{t}∥.

Since ∥d_{x}F^{0}∥ = 1, by Lemma B.3, we obtain that ∥d_{x}F^{t}∥ ≤ e^{tK}. Noting that m(d_{x}F^{t}) = ∥d_{x}F^{−t}∥^{−1} and the flow F^{−t} corresponds to the vector field −X_{F}, we get that m(d_{x}F^{t}) ≥ e^{−tK}. □

Let F^{t} and G^{t} be flows on M and X_{F} and X_{G} the corresponding vector fields. Assume that ∥X_{F} ∥_{C^{1}}, ∥X_{G}∥_{C^{1}} ≤ K, where K is given in Lemma B.4.

Lemma B.5. Set

M := e^{3K}∥X_{F} ∥_{C^{2}} + e^{2K}, ε_{F,G} := ∥X_{G} − X_{F} ∥_{C^{1}}.

Then for t ∈ [0, 1],

(B.1)  ρ_{C^{1}}(G^{t}, F^{t}) ≤ 2tMε_{F,G},

where ρ_{C^{1}}(G^{t}, F^{t}) = max_{x∈M}(dist(G^{t}(x), F^{t}(x)) + ∥d_{x}F^{t} − d_{x}G^{t}∥) is the distance between the flows in the C^{1} topology.

Proof of the lemma. Consider the family of flows F^{t}(τ) generated by the family of vector fields (1 − τ)X_{F} + τX_{G} with τ ∈ [0, 1]. Given x ∈ M and t ∈ [0, 1], the curve c_{t} = c_{t}^{x} : τ ↦ F^{t}(τ)(x) is of length

L(c_{t}) = ∫_{0}^{1} ∥\frac{∂c_{t}}{∂τ} ∥ dτ = ∫_{0}^{1} ∥\frac{∂}{∂τ} F^{t}(τ)(x)∥ dτ,

and hence,

\frac{d}{dt}L(c_{t}) ≤ ∫_{0}^{1} ∥\frac{∂}{∂τ} \frac{d}{dt}F^{t}(τ)(x)∥ dτ
≤ ∥X_{G} − X_{F} ∥ + [(1 − τ)∥DX_{F} ∥ + τ∥DX_{G}∥] ∫_{0}^{1} ∥\frac{∂c_{t}}{∂τ} ∥ dτ
≤ ε_{F,G} + KL(c_{t}).

Recall that L(c_{0}) = 0, c_{t}(0) = F^{t}x and c_{t}(1) = G^{t}x. By Lemma B.3, we obtain

(B.2)  dist(F^{t}x, G^{t}x) ≤ L(c_{t}) ≤ te^{tK}ε_{F,G} ≤ tMε_{F,G}.

On the other hand,

\frac{d}{dt}∥d_{x}F^{t} − d_{x}G^{t}∥ ≤ ∥DX_{F} (F^{t}x)d_{x}F^{t} − DX_{G}(G^{t}x)d_{x}G^{t}∥
≤ ∥DX_{F} (F^{t}x)d_{x}F^{t} − DX_{F} (F^{t}x)d_{x}G^{t}∥
+ ∥DX_{F} (F^{t}x)d_{x}G^{t} − DX_{F} (G^{t}x)d_{x}G^{t}∥
+ ∥DX_{F} (G^{t}x)d_{x}G^{t} − DX_{G}(G^{t}x)d_{x}G^{t}∥
≤ ∥DX_{F} ∥∥d_{x}F^{t} − d_{x}G^{t}∥
+ ∥D^{2}X_{F} ∥dist(F^{t}x, G^{t}x)∥d_{x}G^{t}∥
+ ∥DX_{F} − DX_{G}∥∥d_{x}G^{t}∥
≤ K∥d_{x}F^{t} − d_{x}G^{t}∥ + Me^{−K}ε_{F,G},

where in the last inequality we use Lemma B.4 and the inequalities (B.2). By Lemma B.3, we obtain

(B.3)  ∥d_{x}F^{t} − d_{x}G^{t}∥ ≤ tMe^{(t−1)K}ε_{F,G} ≤ tMε_{F,G}.

Now (B.1) follows by combining (B.2) and (B.3). □

Given flows F^{t} and G^{t} and invariant distributions E_{F} and E_{G} on S respectively, let

\begin{matrix}Δ_{F^{t},G^{t},E_{F} ,E_{G}}(x) = & max \{|\frac{∥d_{x}G^{t}|E_{G}(x)∥}{∥d_{x}F^{t}|E_{F} (x)∥} − 1| , |\frac{m(d_{x}G^{t}|E_{G}(x))}{m(d_{x}F^{t}|E_{F} (x))} − 1|\} , \\ δ_{F^{t},G^{t}} = & ∥G^{t} − F^{t}∥_{C^{1}}, θ_{E_{F} ,E_{G}}(x) = ∠(E_{F} (x), E_{G}(x)). \end{matrix} (B.4)

Lemma B.6. Assume that ∥X_{F} ∥_{C^{1}} ≤ K, then

Δ_{F^{t},G^{t},E_{F} ,E_{G}}(x) ≤ e^{K}[δ_{F^{t},G^{t}} + Ce^{K}θ_{E_{F} ,E_{G}}(x)]

for any x ∈ S and t ∈ [0, 1], where C > 0 is a constant which depends only on the Riemannian metric of M.

Proof of the lemma. Similarly to the proof of Lemma B.4, we can show that if ∥X_{F} ∥_{C^{1}} ≤ K then for any x ∈ S and t ∈ [0, 1],

e^{−K} ≤ m(d_{x}F^{t}) ≤ ∥d_{x}F^{t}∥ ≤ e^{K}.

We have that

|∥d_{x}G^{t}|E_{G}(x)∥ − ∥d_{x}F^{t}|E_{F} (x)∥| ≤ |∥d_{x}G^{t}|E_{G}(x)∥ − ∥d_{x}F^{t}|E_{G}(x)∥|
+ |∥d_{x}F^{t}|E_{G}(x)∥ − ∥d_{x}F^{t}|E_{F} (x)∥|
≤ ∥d_{x}G^{t} − d_{x}F^{t}∥ + ∥d_{x}F^{t}∥dist(E_{G}(x), E_{F} (x))
≤ ∥d_{x}G^{t} − d_{x}F^{t}∥ + C∥d_{x}F^{t}∥∠(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^{t}|E_{F} (x)∥ and noting that ∥d_{x}F^{t}|E_{F} (x)∥ ≥ m(d_{x}F^{t}), we obtain that

|\frac{∥d_{x}G^{t}|E_{G}(x)∥}{∥d_{x}F^{t}|E_{F} (x)∥} − 1| ≤ \frac{1}{m(d_{x}F^{t})}[∥d_{x}G^{t} − d_{x}F^{t}∥
+ C∥d_{x}F^{t}∥∠(E_{G}(x), E_{F} (x))]
≤ e^{K}[δ_{F^{t},G^{t}} + Ce^{K}θ_{E_{F} ,E_{G}}(x)].

Similarly, one can show that |\frac{m(d_{x}G^{t}|E_{G}(x))}{m(d_{x}F^{t}|E_{F} (x))} − 1| admits the same

upper bound. □

Lemma B.7. A flow F^{t} is uniformly partially hyperbolic on a compact invariant subset Λ ⊂ S if and only if the time-1 map F^{1}|Λ is uniformly partially hyperbolic.

Proof of the lemma. See [9]. □

Lemma B.8. Suppose that F^{t} is uniformly partially hyperbolic on a compact invariant subset Λ ⊂ S. Pick numbers 0 < λ < λ̃ ≤ 1 ≤ μ̃ < μ such that

λ ≥ λ(F^{1}, Λ) = sup_{x∈Λ} ∥d_{x}^{s}F^{1}∥, λ̃ ≤ λ̃(F^{1}, Λ) = inf_{x∈Λ} m(d_{x}^{c}F^{1}),
μ̃ ≥ μ̃(F^{1}, Λ) = sup_{x∈Λ} ∥d_{x}^{c}F^{1}∥, μ ≤ μ(F^{1}, Λ) = inf_{x∈Λ} m(d_{x}^{u}F^{1}),

where d_{x}^{ω}F^{t} = d_{x}F^{t}|E_{F}^{ω}(x), ω = s, c, u. Given Δ > 0, there is ε = ε(Δ, λ, λ̃, μ̃, μ) such that if ∥X_{G} − X_{F} ∥_{C^{1}} < ε and X_{G} = X_{F} on S\Λ, then G^{t}|Λ is also a uniformly partially hyperbolic flow and (B.5) Δ_{F^{t},G^{t}}^{ω}(x) := Δ_{F^{t},G^{t},E_{F}^{ω},E_{G}^{ω}}(x) ≤ Δt, ω = s, c, u, x ∈ Λ, t ∈ [0.5, 1].

In particular,

(B.6) 1 − Δ ≤ \frac{λ(G^{1}, Λ)}{λ(F^{1}, Λ)}, \frac{λ̃(G^{1}, Λ)}{λ̃(F^{1}, Λ)}, \frac{μ̃(G^{1}, Λ)}{μ̃(F^{1}, Λ)}, \frac{μ(G^{1}, Λ)}{μ(F^{1}, Λ)} ≤ 1 + Δ.

Proof of the lemma. Consider the time-1 map F^{1}. By [13], there is ε < Δe^{−K}/4M depending on Δ, λ, λ̃, μ̃, μ such that if ∥X_{G} −X_{F} ∥_{C^{1}} < ε and X_{G} = X_{F} on S\Λ, then G^{1}|Λ is uniformly partially hyperbolic on Λ with

(B.7)  sup_{x∈Λ} ∠(E_{G^{1}}^{ω}(x), E_{F^{1}}^{ω}(x)) < \frac{Δ}{4Ce^{2K}} .

By Lemma B.7, the flow G^{t} is uniformly partially hyperbolic on Λ with the same invariant distributions as its time-1 map G^{1}. Moreover, it follows from Lemma B.5 and B.6 that

Δ_{F^{t},G^{t},E_{F}^{ω},E_{G}^{ω}}(x) ≤ \frac{Δt}{2} + \frac{Δ}{4} ≤ Δt, ω = s, c, u, x ∈ Λ, t ∈ [0.5, 1].

In particular,

∥d_{x}^{s}G^{1}∥ ≤ ∥d_{x}^{s}F^{1}∥(1 + Δ) ≤ λ(1 + Δ),

and hence \frac{λ(G^{1}, Λ)}{λ(F^{1}, Λ)} ≤ 1 + Δ. The other inequalities in (B.6) can be

shown in a similar fashion. □

We will 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^{1}, \overline{U_{n}}), λ̃_{n} ≤ λ̃(H^{1}, \overline{U_{n}}), μ̃_{n} ≥ μ̃(H^{1}, \overline{U_{n}}), μ_{n} ≤ μ(H^{1}, \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}^{c}H^{1})\}}{∥d_{x}^{s}H^{1}∥}, \frac{m(d_{x}^{u}H^{1})}{max\{1, ∥d_{x}^{c}H^{1}∥\}}\} ,

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

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

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

(B.9) 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}} \};

(B.10)  Δ_{n} < \frac{1}{2^{n+2}} , ∑_{k=n}^{∞} Δ_{k} < γ_{n}.

Finally, choose

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

where ε(Δ, λ, λ̃, μ̃, μ) is given by Lemma B.8.

Proof of Theorem B.1. First we will show that for every n > 0 the map h_{n}^{t} is uniformly partially hyperbolic on \overline{U_{n}}. This is clearly true for h_{0}^{t} and we will use induction assuming that h_{k}^{t}|\overline{U_{k}} for k = 1, …, n − 1 are uniformly partially hyperbolic. By Lemma B.6, we obtain that

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

Note that

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

The fact that λ(h_{0}^{1}, \overline{U_{1}}) ≤ λ_{1} and the choice of Δ_{n} in (B.9) guarantee that

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

Similarly, we have

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

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 ∥X_{n} − X_{n−1}∥_{C^{1}} ≤ ε_{n}, by Lemma B.8, we obtain that h_{n}^{t}|\overline{U_{n}} is uniformly partially hyperbolic.

Next we will show that X = lim_{n→∞} X_{n} exists and is smooth. In fact,

{X_{n}} is a Cauchy sequence in the C^{1} topology since for any n, m ∈ ℕ,

∥X_{n+m} − X_{n}∥_{C^{1}} ≤ ∑_{l=1}^{m} ∥X_{n+l} − X_{n+l−1}∥_{C^{1}} ≤ ∑_{l=1}^{m} ε_{n+l} ≤ \frac{K}{2^{n+1}} .

Hence X = lim_{n→∞} X_{n} exists and is C^{1}.

It remains to show that the flow h^{t} generated by X is pointwise partially hyperbolic on S. First we construct invariant distributions for h^{t}. Given x ∈ S, we have

∠(E_{h_{n}}^{ω}(x), E_{h_{n−1}}^{ω}(x)) ≤ \frac{Δ_{n}}{4Ce^{2K}} < \frac{1}{2^{n+4}Ce^{2K}} , ω = s, c, u.

Hence the sequence of subspaces E_{h_{n}}^{ω}(x) is Cauchy and it converges to

E_{h}^{ω}(x) = lim_{n→∞} E_{h_{n}}^{ω}(x),

which is clearly dh^{t}-invariant for all t ∈ ℝ^{+}.

Now we would like to estimate Δ_{h^{1},H^{1}}^{ω}(x). Fix x ∈ U_{n}\\overline{U_{n−1}}, we have

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

Note that

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

and ∑ Δ_{k} < 1/4, we obtain by Lemma B.2,

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

Letting l → ∞, we have

Δ_{h^{1},H^{1}}^{ω}(x) ≤ 2 ∑_{k=n}^{∞} Δ_{k}, ω = s, c, u, x ∈ U_{n}\\overline{U_{n−1}}.

Therefore,

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

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

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

Similarly, one can show that m(d_{x}^{u}h^{1}) > max{1, ∥d_{x}^{c}h^{1}∥}. It follows that h^{1} is pointwise partially hyperbolic on S, and so is the flow h^{t} by definition. □

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Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA

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

Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA