Quasi-Stability of Partially Hyperbolic Diffeomorphisms

Huyi Hu^{1} and Yujun Zhu^{2}

1. Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA 2. College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, 050024, P.R.China

Abstract A partially hyperbolic diffeomorphism f is structurally quasi-stable if for any diffeomorphism g C^{1}-close to f, there is a homeomorphism π of M such that π ∘g and f ∘π differ only by a motion τ along center directions. f is topologically quasi-stable if for any homeomorphism g C^{0}-close to f, the above holds for a continuous map π instead of a homeomorphism. We show that any partially hyperbolic diffeomorphism f is topologically quasi-stable, and if f has C^{1} center foliation W_{f}^{c}, then f is structurally quasi-stable. As applications we obtain continuity of topological entropy for certain partially hyperbolic diffeomorphisms with one or two dimensional center foliation.

0 Introduction

The motivation of this paper is to study topological properties of partially hyperbolic systems which are similar to those of uniformly hyperbolic systems.

Partial hyperbolicity theory was first studied in the work of Brin and Pesin ([5]) which emerged in attempts to extend the notion of complete hyperbolicity. A closely related notion of normal hyperbolicity was introduced earlier by Hirsh, Pugh and Shub [6]. For general theory of partial hyperbolicity and normal hyperbolicity, we refer to [12], [7], [2] and [3].

It is well known that Anosov diffeomorphisms are structurally stable([1]), that is, if f is an Anosov diffeomorphism on a compact manifold M then any diffeomorphism g C^{1}-close to f is topologically conjugate to f, i.e., there exists a homeomorphism π on M such that

π ∘ g = f ∘ π.  (0.1)

Moreover, f is also topologically stable([15]), that is, for any homeomorphism g C^{0}-close to f, there exists a continuous map π from M onto M such that equation (0.1) holds. For partially hyperbolic diffeomorphisms, we can not expect such stabilities because of the existence of the center direction. However, since the systems have both stable and unstable directions, we should be able to obtain some similar properties if we look at the behavior of the hyperbolic part, and “ignore” the motions along the center direction.

2000 Mathematics Subject Classification:37C20,37C50,37D30 Keywords and phrases: partially hyperbolicity; quasi-stability; entropy.

The second author is supported by NSFC(No:11071054), the Key Project of Chinese Ministry of Education(No:211020), NCET and the SRF for ROCS, SEM.

In this paper, we shall investigate the “stability” property of partially hyperbolic systems under C^{0} and C^{1} perturbations. Let f be a partially hyperbolic diffeomorphism. We show in Theorem A that for any homeomorphism g C^{0}-close to f, there exist a continuous map π from M to itself and a family of locally defined continuous maps {τ_{x} : x ∈ M}, which send points along the center direction, such that

π ∘ g(x) = τ_{f(x)} ∘ f ∘ π(x) for all x ∈ M.  (0.2)

In this case we say that f is topologically quasi-stable. Moreover, if center foliation W_{f}^{c} of f exists and is of C^{1}, then we can choose a new family {τ_{x} : x ∈ M}, which map points along the center leaves such that an equation similar to (0.2) holds. The results are given in Theorem B. Theorem B′ deal with a particular case, i.e., one dimensional center foliation, in which the map τ can be determined by a flow along the foliation. In Theorem C we obtain structural quasi-stability property of f under C^{1} perturbation. That is, if the center foliation W_{f}^{c} is C^{1}, then for any diffeomorphism g C^{1}-close to f, π is a leaf conjugacy between f and g.

As applications of the results, we obtain that if f is the time 1 map of an Anosov flow generated by a C^{1} vector field, then any diffeomorphism g C^{1}-close to f is a time 1+τ ∘f map of a flow, and the topological entropy of f and g are close (Theorem D). Also, if f has almost parallel center foliation (see next section for the precise definition), then so does any diffeomorphism g C^{1}-close to f, and the topological entropy function is locally constant at f in Diff^{1}(M) in the case of one dimensional center foliation, and continuous at f in Diff^{∞}(M) in the case of two dimensional center foliation (Theorem E).

Our results concerning topological quasi-stability and structural quasi-stability can be regarded as generalizations of topological stability and structural stability for hyperbolic systems ([1] and [15]) to partially hyperbolic systems. They can also be regarded as generalizations of leaf conjugacy for the case that f has C^{1} center foliation ([7] and [12]). However, for topologically quasi-stability, we do not require any additional assumption on f except for partial hyperbolicity. The methods we use for topological and structural quasi-stability are basically the same. We construct an operator in the Banach space consisting all continuous sections of the tangent bundle T M which is contracting in a neighborhood of the zero section such that under the inverse of the exponential map exp^{−1}, π and τ are given by the fixed point of the operator. The methods are adopted from [15, 11] and are different from that used in [7]. We notice that there is another strategy to investigate the topological quasi-stability of partially hyperbolic diffeomorphism using the similar idea in [16], in which Walters obtained topological stability for expansive homeomorphisms with shadowing property. Actually, adapting the unified approach in this paper we will show in a forthcoming paper [8] that any partially hyperbolic diffeomorphism has the so-called “quasi-shadowing” property. Hence, we can obtain the similar results in Theorem A, Theorem B and Theorem B′.

With some additional condition Y. Hua showed that the topological entropy is continuous near the time one map f of an Anosov flow ([9]). Now the fact becomes a direct consequence of our result and a result in [14]. Our results for topological entropy of diffeomorphisms with almost parallel center foliation are similar to that in [10], which is under the assumption that the strongly unstable and stable foliations stably carry some unique nontrivial homologies.

This paper is organized as the following. The statements of results are given in Section 1. We also define some words and notations in the section. In Section 2 we deal with topological quasi-stability, including the proof of Theorem A, Theorem B and Theorem B′. The case of structural quasi-stability is discussed in Section 3, where we use the facts obtained in Section 2 to prove that the map π is a homeomorphism to obtain Theorem C. Section 4 is concerning the applications to topological entropy, where Theorem D and Theorem E are proved.

1 Definition, statement of results and notations

Let M be an m-dimensional C^{∞} compact Riemannian manifold. We denote by ∥·∥ and d(·, ·) the norm on T M and the metric on M induced by the Riemannian metric respectively. Denote by Diff^{r}(M) the set of C^{r} diffeomorphisms of M, 1 ≤ r ≤ ∞.

A diffeomorphism f ∈ Diff^{r}(M) is said to be (uniformly) partially hyperbolic if there exist numbers λ, λ′, μ and μ′ with 0 < λ < 1 < μ and λ < λ′ ≤ μ′ < μ, and an invariant decomposition T_{x}M = E_{x}^{s} ⊕ E_{x}^{c} ⊕ E_{x}^{u} ∀x ∈ M, such that for any n ≥ 0,

∥d_{x}f^{n}v∥ ≤ Cλ^{n}∥v∥  as v ∈ E^{s}(x),
C^{−1}(λ′)^{n}∥v∥ ≤ ∥d_{x}f^{n}v∥ ≤ C(μ′)^{n}∥v∥ as v ∈ E^{c}(x),
C^{−1}μ^{n} ∥v∥ ≤ ∥d_{x}f^{n}v∥  as v ∈ E^{u}(x)

hold for some number C > 0. E_{x}^{s}, E_{x}^{c} and E_{x}^{u} are called stable, center and unstable subspace, respectively. Via a change of Riemannian metric we always assume that C = 1. Moreover, for simplicity of notation, we assume that λ = \frac{1}{μ}.

Since M is compact, we can take constant ρ_{0} > 0 such that for any x ∈ M, the standard exponential mapping exp_{x} : {v ∈ T_{x}M : ∥v∥ < ρ_{0}} → M is a C^{∞} diffeomorphism to the image. Clearly, we have d(x, exp_{x} v) = ∥v∥ for v ∈ T_{x}M with ∥v∥ < ρ_{0}. For any diffeomorphism f : M → M, we take ρ = ρ_{f} ∈ (0, ρ_{0}/2) such that for any x, y ∈ M with d(f^{−1}(x), y) ≤ ρ, v ∈ T_{y}M with ∥v∥ ≤ ρ,

d(x, f ∘ exp_{y} v) ≤ ρ_{0}/2.

Reduce ρ if necessary such that both sides in equation (2.3) and (2.20), in the proof of Theorem A and Theorem B respectively, are contained in the set {v ∈ T_{x}M : ∥v∥ < ρ_{0}}.

For any given continuous center section u = {u(x) ∈ E_{x}^{c} : x ∈ M} with sup_{x∈M} ∥u(x)∥ < ρ, we define a smooth map τ_{x}^{(1)} = τ_{x}^{(1)}(·, u) on B(x, ρ) for any x ∈ M by

τ_{x}^{(1)}(y) = exp_{x}(u(x) + exp_{x}^{−1} y).

Theorem A. Let f ∈ Diff^{r}(M) be a partially hyperbolic diffeomorphism. Then there exists ε_{0} ∈ (0, ρ) satisfying the following conditions: For any ε ∈ (0, ε_{0}) there exists δ > 0 such that for any homeomorphism g of M with d(f, g) < δ there exist a continuous center section u and a surjective continuous map π : M → M such that

π ∘ g(x) = τ_{f(x)}^{(1)} ∘ f ∘ π(x), x ∈ M.  (1.1)

Moreover, u and π can be chosen uniquely so as to satisfy the following conditions:

d(π, id_{M}) < ε,
exp_{x}^{−1}(π(x)) ∈ E_{x}^{s} ⊕ E_{x}^{u} for x ∈ M.

(1.2)

We mention again here that the theorem does not require any additional condition, provided f is a partially hyperbolic diffeomorphism, and g is a homeomorphisms C^{0} close to f.

If f has C^{1} center foliation W_{f}^{c}, then we can require τ to move along the center foliation. In this case, we denote for any ε > 0, Σ_{ε}(x) = exp_{x}(H_{x}(ε)), where H_{x}(ε) is the ε−ball in E_{x}^{s} ⊕ E_{x}^{u}. Obviously, Σ_{ε}(x) is a smooth disk transversal to E_{x}^{c} at x. Since the center foliation W_{f}^{c} is C^{1}, we can conclude that if y is close enough to x, then there exist a locally defined smooth map τ_{x}^{(2)} on some neighborhood U(x) of x and a constant K_{1} > 1 independent of x such that for any y ∈ U(x), we have

τ_{x}^{(2)}(y) ∈ Σ_{ε}(x) ∩ W_{f}^{c}(y)  (1.3)

and

d(τ_{x}^{(2)}(y), x) < K_{1}d(y, x).  (1.4)

Theorem B. Assume that f ∈ Diff^{r}(M) is a partially hyperbolic diffeomorphism with C^{1} center foliation W_{f}^{c}. Then there exists ε_{0} ∈ (0, ρ) satisfying the following conditions: For any ε ∈ (0, ε_{0}) there exists δ > 0 such that for any homeomorphism g of M with d(f, g) < δ there exists a surjective continuous map π : M → M such that

π ∘ g(x) = τ_{g(x)}^{(2)} ∘ f ∘ π(x), x ∈ M.  (1.5)

Moreover, π can be chosen uniquely so as to satisfy the conditions in (1.2).

As a special case, if the center foliation W_{f}^{c} is C^{1} and of dimension one, then we can define τ more directly. Let u ∈ 𝔛^{c} with ∥u(x)∥ = 1 for any x ∈ M, and φ^{t} be the flow generated by u. For any continuous function τ̃ : M → ℝ, define a smooth map τ_{x}^{(3)} = τ_{x}^{(3)}(·, τ̃) of B(x, ρ) for any x ∈ M by

τ_{x}^{(3)}(y) = φ^{τ̃(x)}(y).

Theorem B′. Assume that f ∈ Diff^{r}(M) is a partially hyperbolic diffeomorphism with one dimensional C^{1} center foliation W_{f}^{c}. Then there exists ε_{0} ∈ (0, ρ) satisfying the following conditions: For any ε ∈ (0, ε_{0}) there exists δ > 0 such that for any homeomorphism g of M with d(f, g) < δ there exists a continuous function τ̃: M → ℝ and a surjective continuous map π : M → M such that

π ∘ g(x) = τ_{f(x)}^{(3)} ∘ f ∘ π(x), x ∈ M.  (1.6)

Moreover, τ̃ and π can be chosen uniquely so as to satisfy the conditions in (1.2).

Now we consider the structural quasi-stability as g is C^{1}-close to f.

A diffeomorphism f is called dynamically coherent if E^{cu} := E^{c} ⊕ E^{u}, E^{c}, and E^{cs} := E^{c} ⊕ E^{s} are integrable, and everywhere tangent to W_{f}^{cu}, W_{f}^{c} and W_{f}^{cs}, the center-unstable, center and center-stable foliations, respectively; and W_{f}^{c} and W_{f}^{u} are subfoliations of W_{f}^{cu}, while W_{f}^{c} and W_{f}^{s} are subfoliations of W_{f}^{cs}. By Theorem 2.3 of [13], if f is as in Theorem B then it is dynamically coherent and this property is permanent under C^{1} perturbation.

Theorem C. Under the assumption of Theorem B (resp. Theorem B′), if g is a diffeomorphism C^{1}-close to f, then π can be chosen to be a homeomorphism and hence there exists a homeomorphism τ^{(2)} in Theorem B (resp. τ^{(3)} in Theorem B′) such that π ∘ g = τ^{(2)} ∘ f ∘ π (resp. π ∘ g = τ^{(3)} ∘ f ∘ π). Also, π and τ^{(2)} (resp. τ^{(3)}) can be chosen uniquely so as to satisfy the conditions in (1.2) if we replace E_{x}^{s} and E_{x}^{u} in (1.2) by their smooth approximation Ẽ_{x}^{s} and Ẽ_{x}^{u} respectively.

Moreover, π sends W_{g}^{cu}, W_{g}^{c} and W_{g}^{cs} to W_{f}^{cu}, W_{f}^{c} and W_{f}^{cs} respectively. In particular, π is a leaf conjugacy from (g, W_{g}^{c}) to (f, W_{f}^{c}).

Remark 1.1. In fact, if π is one to one, then for any y ∈ M, x = π^{−1}(f^{−1}(y)) is uniquely determined. Hence we can define τ^{(2)} : M → M by τ^{(2)}(y) = τ_{g(π^{−1}∘f^{−1}(y))}^{(2)}(y) and obtain

π ∘ g(x) = τ^{(2)} ∘ f ∘ π(x), x ∈ M.

τ^{(3)} can be defined in a similar way.

The main result of Theorem C, in particular, leaf conjugacy, is well known ([7], [12]). Our proof provides a different approach.

Example. Let N be a smooth compact Riemannian manifold and h : N ⟶ N be an Anosov diffeomorphism. Then the diffeomorphism

f = h × id_{S^{1}} : N × S^{1} ⟶ N × S^{1}

is quasi-stable. In particular, if R is a rotation on S^{1} close to the identity, and

g = h × R : N × S^{1} ⟶ N × S^{1},

then we can take π = id_{N×S^{1}} and τ^{(2)} = id_{N} × R in Theorem C.

Moreover, Theorem E below gives that the topological entropy is constant in a neighborhood of f in C^{1} topology.

As applications of Theorem C, we have the following results about the continuity of the entropy. We say that a diffeomorphism g is a time 1 + τ map of a flow ψ for some real function τ on M if

g(x) = ψ^{1+τ(x)}(x) for any x ∈ M.

Theorem D. Let f be the time one map of an Anosov flow φ. Then for any diffeomorphism g C^{1}- close to f, there is a flow ψ and a continuous real function τ on M such that g is the time 1 + τ ∘ f map of the flow ψ. Hence the topological entropy function is continuous at f in Diff^{1}(M) with C^{1} topology.

Y. Hua proved the second part of the theorem under the condition that f is topologically transitive ([9]).

Let f be a partially hyperbolic diffeomorphism with center foliation W_{f}^{c}. For a smooth surface Σ, we denote Σ ⊥ W_{f}^{c} if for any x ∈ Σ ∩ W_{f}^{c}, T_{x}Σ ⊥ E_{f}^{c}(x). For any two surfaces Σ_{1} and Σ_{2} that are smooth or are images of homeomorphisms of some smooth surfaces, the holonomy map θ^{c} : Σ_{1} → Σ_{2} is a continuous map defined by sliding along the W_{f}^{c}-leaves, i.e. for x ∈ Σ_{1}, θ^{c}(x) ∈ Σ_{2} ∩ W_{f}^{c}(x).

Definition 1.2. A partially hyperbolic diffeomorphism f with integrable center foliation W_{f}^{c} is said to have almost parallel center foliation if for any α > 0, there exists constant β > 0, such that for any smooth surfaces Σ_{1}, Σ_{2} with Σ_{1} ⊥ W_{f}^{c} and Σ_{2} ⊥ W_{f}^{c}, for any x, y ∈ Σ_{1} with d(x, y) ≤ β, we have d(θ^{c}(x), θ^{c}(y)) ≤ α whenever they are defined.

Remark 1.3. In the definition we require Σ_{1} ⊥ W_{f}^{c} and Σ_{2} ⊥ W_{f}^{c} only for convenience. It is clear that we can change the definition by requiring the angles between Σ_{1}, Σ_{2} and W_{f}^{c} uniformly bounded from below.

Remark 1.4. The requirements of the definition mean that the holonomy maps along the center foliation are equicontinuous for all possible holonomy maps whenever they are defined.

It is clear that each of the maps f and g given by the above example has almost parallel center foliation, while Anosov flows do not.

Theorem E. Let f be a partially hyperbolic diffeomorphism as in Theorem B. If the center foliation of f is almost parallel, then any diffeomorphism g that is C^{1}-close to f also has almost parallel center foliation.

Moreover, if the center foliation of f is one dimensional, then the topological entropy function is locally constant in Diff^{1}(M); if f ∈ Diff^{∞}(M) and the center foliation of f is two dimensional, then the topological entropy function is continuous near f in Diff^{∞}(M) with C^{1} topology.

For the case that the dimension of center subbundle of f is one or two, the same conclusions are obtained in [10] under the assumption that the strongly stable and unstable foliations of f stably carry some unique nontrivial homologies. Our proof uses some idea in the paper.

Denote by 𝔛 the Banach space of continuous vector fields on M with the norm

∥ω∥ = sup_{x∈M} ∥ω(x)∥, w ∈ 𝔛.

In other words, each element of 𝔛 is a continuous section of the tangent bundle T M. Similarly, we denote by 𝔛^{s}, 𝔛^{c} and 𝔛^{u} the space of continuous sections of the stable, center and unstable bundles E^{s}, E^{c} and E^{u} respectively. Also, we denote 𝔛^{us} = 𝔛^{u} ⊕ 𝔛^{s}. Let Π_{x}^{s} : T_{x}M → E_{x}^{s} be the projection onto E_{x}^{s} along E_{x}^{c} ⊕ E_{x}^{u}. Π_{x}^{c} and Π_{x}^{u} are defined in a similar way.

Recall that ∥ · ∥ is the norm on T M. We define the norm ∥ · ∥_{1} on T M by ∥w∥_{1} = ∥u∥ + ∥v∥ if w = u + v ∈ T_{x}M with u ∈ E_{x}^{c} and v ∈ E_{x}^{u} ⊕ E_{x}^{s}. Similarly, if w = u + v ∈ 𝔛 with u ∈ 𝔛^{c} and v ∈ 𝔛^{us}, we also define ∥w∥_{1} = ∥u∥ + ∥v∥. By triangle inequality and the fact that the angles between E^{c} and E^{u} ⊕ E^{s} are uniformly bounded away from zero, we know that there exists a constant L such that

∥w∥ ≤ ∥w∥_{1} ≤ L∥w∥.  (1.7)

For any ε > 0, we denote

𝔅(ε) = {w ∈ 𝔛 : ∥w∥ ≤ ε}, 𝔅^{us}(ε) = {w ∈ 𝔛^{us} : ∥w∥ ≤ ε},
𝔅_{1}(ε) = {w ∈ 𝔛 : ∥w∥_{1} ≤ ε}.

2 Topological quasi-stability

2.1 The general case

Proof of Theorem A. We choose

ε_{0} ∈ (0, ρ)  (2.1)

small enough such that any map π with d(π, id_{M}) < ε_{0} must be surjective (see e.g. Lemma 3 of [15] for existence of such ε_{0}).

To find a continuous center section u ∈ 𝔛^{c} and a surjective continuous map π : M ⟶ M satisfying (1.1) and the conditions in (1.2) of this theorem, we shall first try to solve the equation

π ∘ g(x) = τ_{f(x)}^{(1)} ∘ f ∘ π(x)  (2.2)

for unknown u and π. Putting h = g ∘ f^{−1} and π(x) = exp_{x}(v(x)) for v ∈ 𝔅^{us}(ρ), and replacing x by f^{−1}(x), we see that (1.1) is equivalent to

exp_{x}^{−1} ∘ exp_{h(x)}(v(h(x))) = exp_{x}^{−1} ∘τ_{x}^{(1)} ∘ f ∘ exp_{f^{−1}(x)}(v(f^{−1}(x))).  (2.3)

By the definition of τ_{x}^{(1)}, we have

exp_{x}^{−1} ∘τ_{x}^{(1)} ∘ f ∘ exp_{f^{−1}(x)}(v(f^{−1}(x))) = u(x) + exp_{x}^{−1} ∘f ∘ exp_{f^{−1}(x)}((v(f^{−1}(x))).

Define an operator β : 𝔅(ρ) → 𝔛 and a linear operator F : 𝔛 → 𝔛 by

β(w)(x) = exp_{x}^{−1} ∘f ∘ exp_{f^{−1}(x)}((w(f^{−1}(x))),  (2.4)
(F w)(x) = d_{f^{−1}(x)}fw(f^{−1}(x)).  (2.5)

Clearly, F w = d_{0}βw. Let

η(w)(x) = β(w)(x) − (d_{0}βw)(x).  (2.6)

Then we can write

exp_{x}^{−1} ∘τ_{x}^{(1)} ∘ f ∘ exp_{f^{−1}(x)}(v(f^{−1}(x))) = (F v)(x) + u(x) + η(v)(x).  (2.7)

Define a linear operator J_{h} : 𝔅(ρ) → 𝔛 by

(J_{h}w)(x) = ∑_{i=s,c,u} Π_{x}^{i} ∘ d_{0}(exp_{x}^{−1} ∘ exp_{h(x)}) ∘ Π_{h(x)}^{i}w(h(x))  (2.8)

for any w ∈ 𝔅(ρ). Set

θ_{h}(w)(x) = exp_{x}^{−1} ∘ exp_{h(x)}(w(h(x))) − d_{0}(exp_{x}^{−1} ∘ exp_{h(x)})w(h(x))
+ ∑_{i,j=s,c,u,i≠j} Π_{x}^{i} ∘ d_{0}(exp_{x}^{−1} ∘ exp_{h(x)}) ∘ Π_{h(x)}^{j}w(h(x)).

Then we have

exp_{x}^{−1} ∘ exp_{h(x)}(v(h(x))) = (J_{h}v)(x) + θ_{h}(v)(x).  (2.9)

Also we mention that by definition,

θ_{h}(0)(x) = exp_{x}^{−1} ∘ exp_{h(x)} 0 = exp_{x}^{−1} h(x).  (2.10)

Therefore, by (2.9) and (2.7), (2.3) is equivalent to

J_{h}v + θ_{h}(v) = F v + u + η(v),

further, is equivalent to

−J_{h}^{−1}u + (id_{𝔛} − J_{h}^{−1}F)v = J_{h}^{−1}(η(v) − θ_{h}(v)).

Define a linear operator P_{h} from a neighborhood of 0 ∈ 𝔛 to 𝔛 by

P_{h}ω = −J_{h}^{−1}u + (id_{𝔛} − J_{h}^{−1}F)v  (2.11)

for ω = u + v ∈ 𝔛, where u ∈ 𝔛^{c} and v ∈ 𝔛^{us}.

Define an operator Φ_{h} from a neighborhood of 0 ∈ 𝔛 to 𝔛 by

Φ_{h}(u + v) = P_{h}^{−1}J_{h}^{−1}(η(v) − θ_{h}(v)).

Hence, equation (2.2) is equivalent to

Φ_{h}(u + v) = u + v,  (2.12)

namely, u + v is a fixed point of Φ_{h}.

By Lemma 2.1 below, we know that for any ε ∈ (0, ε_{0}) there exists δ = δ(ε) such that for any homeomorphism h with d(h, id_{M}) ≤ δ, Φ_{h} : 𝔅_{1}(ε) → 𝔅_{1}(ε) is a contracting map, and therefore has a fixed point in 𝔅_{1}(ε). Hence, (2.2) has a unique solution.

Lemma 2.1. We can reduce ε_{0} in (2.1) if necessary such that for any ε ∈ (0, ε_{0}) there exists δ = δ(ε) > 0 such that for any homeomorphism h of M with d(h, id_{M}) ≤ δ, Φ_{h}(𝔅_{1}(ε)) ⊂ 𝔅_{1}(ε) and for any ω, ω′ ∈ 𝔅_{1}(ε),

∥Φ_{h}(ω) − Φ_{h}(ω′)∥_{1} ≤ \frac{1}{2}∥ω − ω′∥_{1}.

Proof. Recall that the constant L is given in (1.7).

Reduce ε_{0} if necessary such that for any ε ∈ (0, ε_{0}),

\frac{4L}{1 − λ}C(ε) < \frac{1}{4},  (2.13)

where C(ε) is the Lipschitz constant of η(v) given in Sublemma 2.3. This is possible since by the sublemma, C(ε) → 0 as ε → 0. Note that ε only depends on f.

Then we take δ = δ(ε) such that

\frac{4L}{1 − λ}δ < \frac{1}{4}ε;  (2.14)

and such that for any homeomorphism h with d(h, id_{M}) < δ,

max\{∥J_{h}∥, ∥J_{h}^{−1}∥\} ≤ min \{2, \frac{1 + λ^{−1}}{2} \} ,  (2.15)

where J_{h} is defined in (2.8), and

\frac{4L}{1 − λ}K(h) < \frac{1}{4},  (2.16)

where K(h) is the Lipschitz constant of θ_{h}(·) given in Sublemma 2.4. This is possible since by the sublemma, K(h) → 0 as ε → 0.

By (2.15), Sublemma 2.2 below can be applied and therefore we get

∥P_{h}^{−1}∥_{1} ≤ \frac{2}{1 − λ}.  (2.17)

Note that J_{h}(𝔛^{i}) = 𝔛^{i} for i = s, c, u. Then it is easy to check that ∥J_{h}^{−1}∥_{1} ≤ ∥J_{h}^{−1}∥. Hence by (2.15), we have ∥J_{h}^{−1}∥_{1} ≤ 2. Also, by (2.4), β(0) = 0 and therefore by (2.6), η(0) = 0; and by (2.10), ∥θ_{h}(0)∥ ≤ δ.

Take ω = u + v ∈ 𝔅_{1}(ε) with u ∈ 𝔛^{c} and v ∈ 𝔛^{us}. By using the above estimates, Sublemma 2.3 and Sublemma 2.4, and then (2.13), (2.14) and (2.16), we can get

∥Φ_{h}(ω)∥_{1} ≤ ∥P_{h}^{−1}∥_{1} · ∥J_{h}^{−1}∥_{1} · ∥η(v) − θ_{h}(v)∥_{1}

≤ \frac{2}{1 − λ} · 2 · L∥η(v) − θ_{h}(v)∥

≤ \frac{4L}{1 − λ}(∥η(v) − η(0)∥ + ∥θ_{h}(v) − θ_{h}(0)∥ + ∥θ_{h}(0)∥)

≤ \frac{4L}{1 − λ}(C(ε)∥ω∥_{1} + K(h)∥ω∥_{1} + δ)

≤ \frac{1}{4}∥ω∥_{1} + \frac{1}{4}∥ω∥_{1} + \frac{1}{4}ε ≤ \frac{3ε}{4},

which implies that Φ_{h}(𝔅_{1}(ε)) ⊂ 𝔅_{1}(ε).

Similarly, for two elements ω = u + v, ω′ = u′ + v′ ∈ 𝔅_{1}(ε) with u, u′ ∈ 𝔛^{c} and v, v′ ∈ 𝔛^{us}, we have

∥Φ_{h}(ω) − Φ_{h}(ω′)∥_{1} ≤ \frac{4}{1 − λ}(∥η(v) − η(v′)∥_{1} + ∥θ_{h}(v) − θ_{h}(v′)∥_{1})

≤ \frac{4L}{1 − λ}(∥η(v) − η(v′)∥ + ∥θ_{h}(v) − θ_{h}(v′)∥)

≤ \frac{4L}{1 − λ}(C(ε_{0})∥ω − ω′∥_{1} + K(h)∥ω − ω′∥_{1})

≤ \frac{1}{2}∥ω − ω′∥_{1}.

This proves that Φ_{h} : 𝔅_{1}(ε) → 𝔅_{1}(ε) is a contraction.

Sublemma 2.2. For any homeomorphism h of M such that J_{h} satisfies (2.15), P_{h} is invertible and

∥P_{h}^{−1}∥_{1} ≤ \frac{2}{1 − λ}.  (2.18)

Proof. By the definitions of F and J_{h}, we have F (𝔛^{i}) = 𝔛^{i} and J_{h}(𝔛^{i}) = 𝔛^{i} for i = u, s, c. Let F^{i} = F |_{𝔛^{i}}, J_{h}^{i} = J_{h}|_{𝔛^{i}} for i = u, s, c. By the definition of P_{h} we have P_{h}|_{𝔛^{i}} = id_{𝔛^{i}} − (J_{h}^{i})^{−1} ∘ F^{i}, i = s, u, and P_{h}|_{𝔛^{c}} = −(J_{h}^{c})^{−1}. So we also have P_{h}(𝔛^{i}) = 𝔛^{i} for i = s, c, u.

Since ∥F^{s}∥, ∥(F^{u})^{−1}∥ ≤ λ, by (2.15) we know that

∥(J_{h}^{s})^{−1} ∘ F^{s}∥, ∥((F^{(u)})^{−1} ∘ J_{h}^{(u)})∥ ≤ λ · (1 + λ^{−1})/2 = (1 + λ)/2 < 1.

Hence, both P_{h}|_{𝔛^{s}} and P_{h}|_{𝔛^{u}} are invertible and

\begin{aligned}(P_{h}|_{𝔛^{s}})^{−1} & = (id_{𝔛^{s}} − (J_{h}^{s})^{−1} ∘ F^{s})^{−1} = ∑_{k=0}^{∞}((J_{h}^{s})^{−1} ∘ F^{s})^{k}, \\ (P_{h}|_{𝔛^{u}})^{−1} & = (id_{𝔛^{u}} − (J_{h}^{u})^{−1} ∘ F^{u})^{−1} = − ∑_{k=1}^{∞}((F^{u})^{−1} ∘ J_{h}^{u})^{k}. \end{aligned}

It follows that

∥(P_{h}|_{𝔛^{us}})^{−1}∥ ≤ max \{∥(P_{h}|_{𝔛^{s}})^{−1}∥, ∥(P_{h}|_{𝔛^{u}})^{−1}∥\} ≤ \frac{1}{1 − (1 + λ)/2} = \frac{2}{1 − λ}.

By (2.15) we also have

∥(P_{h}|_{𝔛^{c}})^{−1}∥ ≤ ∥J_{h}∥ ≤ 2.

So we obtain

∥P_{h}^{−1}∥_{1} ≤ max \{∥(P_{h}|_{𝔛^{us}})^{−1}∥, ∥(P_{h}|_{𝔛^{c}})^{−1}∥\} ≤ \frac{2}{1 − λ}.

This is what we need.

Sublemma 2.3. For any 0 < ε ≤ ρ, there exists constant C(ε) > 0 such that for any w, w′ ∈ 𝔅(ε),

∥η(w′) − η(w)∥ ≤ C(ε)(∥w′ − w∥).

Moreover, C(ε) can be chosen in such a way that C(ε) → 0 as ε → 0.

Proof. Let β_{x} : T_{f^{−1}(x)}M ⟶ T_{x}M be the map defined by β_{x}(ξ) = exp_{x}^{−1} ∘f ∘ exp_{f^{−1}(x)} ξ. Therefore β(w)(f^{−1}(x)) = β_{x}(w(f^{−1}(x))) for any w ∈ 𝔛. Then by (2.6), for any w, w′ ∈ 𝔛 with ∥w∥, ∥w′∥ < ε,

∥η(w′)(x) − η(w)(x)∥
=∥β_{x}(w′(f^{−1}(x))) − β_{x}(w(f^{−1}(x))) − d_{0}β_{x}(w′(f^{−1}(x)) − w(f^{−1}(x)))∥
=∥∫_{0}^{1} d_{w(f^{−1}(x))+t(w′(f^{−1}(x))−w(f^{−1}(x)))}β_{x}(w′(f^{−1}(x)) − w(f^{−1}(x)))dt
− d_{0}β_{x}(w′(f^{−1}(x)) − w(f^{−1}(x)))∥
≤ sup_{w^{∗}∈I_{x}} ∥d_{w^{∗}}β_{x} − d_{0}β_{x}∥ · ∥w′(f^{−1}(x)) − w(f^{−1}(x))∥,

where I_{x} = {w(f^{−1}(x)) + t(w′(f^{−1}(x)) − w(f^{−1}(x))) : t ∈ [0, 1]}. Since d_{w^{∗}}β_{x} is continuous with w^{∗} and the continuity is uniform with respect to x, we can take

C(ε) = sup{∥d_{w(f^{−1}(x))}β_{x} − d_{0}β_{x}∥ : w ∈ 𝔅_{1}(ε), x ∈ M}.

Now the results of the lemma are clear.

Sublemma 2.4. For any h with d(h, id_{M}) ≤ ρ, there exists a constant K = K(h) > 0 such that for any w, w′ ∈ 𝔅(ε),

∥θ_{h}(w′) − θ_{h}(w)∥ ≤ K(h)∥w′ − w∥.

Moreover, K(h) can be chosen in such a way that K(h) → 0 as d(h, id_{M}) → 0.

Proof. Since the map θ is C^{∞} with respect to w and h, we can use the same method in the proof of the previous lemma to get the inequality.

Note that if h = id_{M}, then the partial derivative of θ with respect to w is zero. So we get K(h) → 0 as d(h, id_{M}) → 0.

2.2 The center foliation W_{f}^{c} is C^{1}

2.2.1 The general case

Proof of Theorem B. The proof is similar to that of Theorem A.

To find π satisfying (1.5) and the conditions in (1.2) of this theorem, we shall try to solve the equation

π ∘ g(x) = τ_{g(x)}^{(2)} ∘ f ∘ π(x)  (2.19)

for unknown π. Putting π(x) = exp_{x}(v(x)) with v ∈ 𝔅^{us}(ρ), we see that (2.19) is equivalent to

v(x) = exp_{x}^{−1} ∘τ_{x}^{(2)} ∘ f ∘ π(v(g^{−1}(x))).  (2.20)

Define an operator β : 𝔅^{us}(ρ) → 𝔛^{us} and a linear operator A : 𝔅^{us}(ρ) → 𝔛^{us} by

(β(v))(x) = exp_{x}^{−1} ∘τ_{x}^{(2)} ∘ f ∘ π(v(g^{−1}(x))),  (2.21)
(Av)(x) = (A_{g^{−1}(x)}^{s} + A_{g^{−1}(x)}^{u})v(g^{−1}(x)),  (2.22)

where

A_{g^{−1}(x)}^{s} = Π_{x}^{s} ∘ d_{0}(exp_{x}^{−1} ∘τ_{x}^{(2)} ∘ f ∘ exp_{g^{−1}(x)}) ∘ Π_{g^{−1}(x)}^{s}

and

A_{g^{−1}(x)}^{u} = Π_{x}^{u} ∘ d_{0}(exp_{x}^{−1} ∘τ_{x}^{(2)} ∘ f ∘ exp_{g^{−1}(x)}) ∘ Π_{g^{−1}(x)}^{u}.

Let η = β − A. By (2.21) and (2.22), (2.20) is equivalent to

v = Av + η(v),

further, is equivalent to

v − Av = η(v).

Define a linear operator P from a neighborhood of 0 ∈ 𝔛^{us} to 𝔛^{us} by

P v = (id_{𝔛^{us}} − A)v  (2.23)

for v ∈ 𝔛^{us}.

Define an operator Φ from a neighborhood of 0 ∈ 𝔛^{us} to 𝔛^{us} by

Φ(v) = P^{−1}η(v).

Hence, the equation (2.19) is equivalent to

Φ(v) = v,  (2.24)

namely, v is a fixed point of Φ.

The remaining work is to show that for any ε ∈ (0, ε_{0}) there exists δ = δ(ε) such that for any homeomorphism g with d(g, f) ≤ δ, Φ : 𝔅(ε) → 𝔅(ε) is a contracting map, and therefore has a fixed point in 𝔅(ε). Hence, (2.19) has a unique solution. To this end we only need to modify the proof of Lemma 2.1 to a easer version since in this case we need not to concern with the center direction. We leave the details to the reader.

2.2.2 W_{f}^{c} is of one dimensional

Proof of Theorem B′. The proof is also similar to that of Theorem A.

To find π and τ̃ satisfying (1.6) and the conditions in (1.2) of this theorem, we shall try to solve the equation

π ∘ g(x) = τ_{f(x)}^{(3)} ∘ f ∘ π(x)  (2.25)

for unknown τ̃ and π. Putting h = g ∘ f^{−1} and π(x) = exp_{x}(v(x)) with v ∈ 𝔛^{us}, we see that (2.25) is equivalent to

exp_{x}^{−1} ∘ exp_{h(x)}(v(h(x))) = exp_{x}^{−1} ∘φ^{τ̃(x)} ∘ f ∘ exp_{f^{−1}(x)}(v(f^{−1}(x))).  (2.26)

Define β : 𝔅(ρ) × ℭ(ρ) → 𝔛, where ℭ(ρ) = {τ̃ ∈ C^{0}(M) : ∥τ̃∥ ≤ ρ}, by

β(ω, τ̃)(x) = exp_{x}^{−1} ∘φ^{τ̃(x)} ∘ f ∘ exp_{f^{−1}(x)}(ω(f^{−1}(x))).  (2.27)

It is easy to see that

(d_{(0,0)}β(ω, τ̃))(x) = (F ω)(x) + τ̃(x) · u(x),

where F : 𝔛 → 𝔛 is defined in (2.5) (also recall that in this case u is a unit center vector field). Let

η(ω, τ̃)(x) = β(ω, τ̃)(x) − (d_{(0,0)}β(ω, τ̃))(x).  (2.28)

Then we can write

exp_{x}^{−1} ∘φ^{τ̃(x)} ∘ f ∘ exp_{f^{−1}(x)}(v(f^{−1}(x))) = (F v)(x) + τ̃(x) · u(x) + η(v, τ̃)(x).  (2.29)

By (2.9) and (2.29), (2.26) is equivalent to

J_{h}v + θ_{h}(v) = F v + τ̃ · u + η(v, τ̃).

where J_{h} is a linear operator defined in (2.8). Further, the equation is equivalent to

−J_{h}^{−1}(τ̃ · u) + (id_{𝔛} − J_{h}^{−1}F)v = J_{h}^{−1}(η(v, τ̃) − θ_{h}(v)).

Similarly we define a linear map P_{h} by

P_{h}ω = −J_{h}^{−1}(τ̃ · u) + (id_{𝔛} − J_{h}^{−1}F)v  (2.30)

for ω = τ̃ · u + v ∈ 𝔛 with v ∈ 𝔅^{us}(ρ) and τ̃ ∈ ℭ(ρ). Hence, the above equation becomes

P_{h}(τ̃ · u + v) = J_{h}^{−1}(η(v, τ̃) − θ_{h}(v)).

Define a map Φ_{h} from a neighborhood of 0 ∈ 𝔛 to 𝔛 by

Φ_{h}(τ̃ · u + v) = P_{h}^{−1}J_{h}^{−1}(η(v, τ̃) − θ_{h}(v)).

Hence, the equation (2.25) is equivalent to

Φ_{h}(τ̃ · u + v) = τ̃ · u + v,  (2.31)

namely, τ̃ · u + v is a fixed point of Φ_{h}.

Also similar to what we have done in the proof of Theorem A, there exists ε_{0} ∈ (0, ρ) such that for any ε ∈ (0, ε_{0}) there exists δ = δ(ε) such that for any homeomorphism h with d(h, id_{M}) ≤ δ, Φ_{h} : 𝔅_{1}(ε) → 𝔅_{1}(ε) is a contracting map, and therefore has a fixed point in 𝔅_{1}(ε). Hence, (2.25) has a unique solution. We leave the details to the reader.

3 Structural quasi-stability

Proof of Theorem C. We only prove this theorem under the assumption of Theorem B. We shall find π and u using the similar strategy in the proof of Theorem B. Furthermore, in order to obtain a leaf conjugacy π we shall give some necessary modification in techniques.

Since the center foliation of f is C^{1}, hence from Theorem 5.10 of [12] and Section 6 of [6], we know that if a diffeomorphism g is sufficiently close to f in C^{1} topology, then it is also partially hyperbolic, the corresponding splitting E_{g}^{s} ⊕E_{g}^{c} ⊕E_{g}^{u} is near that of f and the center distribution E_{g}^{c} is integrable. Now choose C^{1} bundle Ẽ^{s} ⊕ Ẽ^{u} sufficiently close to E_{g}^{s} ⊕ E_{g}^{u}, and hence close to E_{f}^{s} ⊕ E_{f}^{u}. We want to find a continuous center section u ∈ 𝔛^{c} and a homeomorphism π : M ⟶ M ε close to id_{M} such that (1.5) holds and

exp_{x}^{−1}(π(x)) ∈ Ẽ_{x}^{s} ⊕ Ẽ_{x}^{u}  (3.1)

for x ∈ M. Put h = g ∘ f^{−1} and π(x) = exp_{x}(v(x)) for v ∈ 𝔛̃^{s} ⊕ 𝔛̃^{u}, where 𝔛̃^{s} and 𝔛̃^{u} denote the spaces of continuous sections of Ẽ^{s} and Ẽ^{u} respectively. We see that (1.5) is equivalent to

exp_{x}^{−1} ∘ exp_{h(x)}(v(h(x))) = exp_{x}^{−1} ∘τ_{x}^{(2)} ∘ f ∘ exp_{f^{−1}(x)}(v(f^{−1}(x))).  (3.2)

Then we can write

exp_{x}^{−1} ∘τ_{x}^{(2)} ∘ f ∘ exp_{f^{−1}(x)}(v(f^{−1}(x))) = (F v)(x) + u(x) + η(w)(x),  (3.3)

where ω = u + v ∈ 𝔛 with u ∈ 𝔛^{c} and v ∈ 𝔛̃^{us},

(F v)(x) = ∑_{i=s,u} Π̃_{x}^{i} ∘ d_{f^{−1}(x)}f ∘ Π̃_{f^{−1}(x)}^{i}v(f^{−1}(x))

and

η(w)(x) = exp_{x}^{−1} ∘τ_{x}^{(2)} ∘ f ∘ exp_{f^{−1}(x)}(v(f^{−1}(x)))
−d_{0}(exp_{x}^{−1} ∘τ_{x}^{(2)} ∘ f ∘ exp_{f^{−1}(x)})v(f^{−1}(x))
+ ∑_{i=s,c,u,j=s,u,i≠j} Π̃_{x}^{i} ∘ d_{f^{−1}(x)}f ∘ Π̃_{f^{−1}(x)}^{j}v(f^{−1}(x)),  (3.4)

in which Π̃_{x}^{s} is the projection from T_{x}M onto Ẽ_{x}^{s} along Ẽ_{x}^{c} ⊕ Ẽ_{x}^{u}, where Π̃_{x}^{c} = Π_{x}^{c}. Π̃_{x}^{c} and Π̃_{x}^{u} are defined in the similar manner. It is clear that F (𝔛̃^{s}) = 𝔛̃^{s}, F (𝔛̃^{u}) = 𝔛̃^{u}. Moreover, for any λ < λ̃ < 1, we can choose g sufficiently close to f, Ẽ^{s} ⊕ Ẽ^{u} sufficiently close to E_{f}^{s} ⊕ E_{f}^{u} such that

∥F |_{𝔛^{s}}∥, ∥F^{−1}|_{𝔛^{u}}∥^{−1} ≤ λ̃.

Similar to (2.9), we have

exp_{x}^{−1} ∘ exp_{h(x)}(v(h(x))) = (J_{h}v + θ_{h}(v))(x),  (3.5)

where J_{h} and θ_{h} is redefined with respect to 𝔛̃ = 𝔛̃^{s} ⊕ 𝔛̃^{c} ⊕ 𝔛̃^{u}. By (3.3) and (3.5), (3.2) is equivalent to

J_{h}v + θ_{h}(v) = F v + u + η(w).

From now, we can find π and u in a similar way as we have done in the proof of Theorem B. We omit the details.

In the following, we prove that π obtained above is a leaf conjugacy from (g, W_{g}^{c}) to (f, W_{f}^{c}). Since the bundle Ẽ^{s} ⊕ Ẽ^{u} is C^{1}, (3.1) implies that the restriction of π to each center leaf of g is one-to-one. If we can get that π sends center leaves of g to that of f then by the same arguments of Pesin in Lemma 5.11 of [12] we can conclude that π is a leaf conjugacy. Therefore, the remaining work is to prove that π sends center leaves of g to that of f.

Now we show that for any x ∈ M, π(W_{g}^{c}(x)) ⊂ W_{f}^{c}(πx). It is enough to show that the set π(W_{g}^{c}(x)) is tangent to E_{f}^{c}(πx) for any x ∈ M. Suppose not, then there exist a small number c_{1} > 0 and a sequence of points {y_{k}} ⊂ W_{g}^{c}(x) with y_{k} → x as k → ∞ such that

d(πz′_{k}, πz_{k}) + d(πz_{k}, πy_{k}) ≥ c_{1}d(πx, πy_{k}),  (3.6)

where z′_{k} and z_{k} are the unique points such that πz_{k} ∈ W_{f}^{u}(πy_{k}) and πz′_{k} ∈ W_{f}^{s}(πz_{k}) ∩ W_{f}^{c}(πx). By taking a subsequence we may assume that d(πz_{k}, πy_{k}) ≥ d(πz′_{k}, πz_{k}) for all k > 0, and the other case can be discussed similarly by using f^{−1}.

For each k > 0, there exists n = n(k) > 0 such that

d(g^{i}(x), g^{i}(y_{k})) ≤ (μ′_{g})^{i}d(x, y_{k}) ≤ ε ∀ 0 ≤ i ≤ n,

where μ′_{g} is the upper bounds of ∥Dg|_{E_{g}^{c}}∥ given in the definition of partially hyperbolic diffeomorphism. We can see that n(k) ∼ − log d(x, y_{k})/ log μ′_{g} if μ′_{g} > 1, and we regard n(k) = ∞ if otherwise. Since d(π, id_{M}) < ε, we have

d(π(g^{i}x), π(g^{i}y_{k})) ≤ 3ε  ∀ 0 ≤ i ≤ n.  (3.7)

Since the foliation W_{f}^{c} is smooth, for any x ∈ M there is a coordinate chart U_{x} at x of size r > 0 such that the local leaves of the center foliation can be viewed as parallel disks. For any z ∈ U_{x}, denote such a local center disk passing through z by B_{f}^{c}(z). We consider the coordinate charts U_{π(g^{i}x)} at π(g^{i}x) of size r, i = 0, 1, ⋯, n. We assume that r > 0 is small and the coordinates are taken in a way such that the metrics on the charts are close to the metric on the manifold. Also, we assume that ε and δ are small such that as d(f, g) < δ, all the points π(g^{i}w) and f(π(g^{i−1}w)) are in the chart U_{π(g^{i}x)}, where w = x, y_{k}, z_{k}, z′_{k}. Since τ is a motion along leaves, we have B_{f}^{c}(π(gx)) = B_{f}^{c}(f(πx)). So by the fact that d(f(πz_{k}), f(πy_{k})) ≥ μ_{f}d(πz_{k}, πy_{k}), we get

d(B_{f}^{c}(π(gz_{k})), B_{f}^{c}(π(gy_{k}))) ≥ μ̃d(B_{f}^{c}(πz_{k}), B_{f}^{c}(πy_{k}))

for some max{1, μ′_{g}} < μ̃ < μ_{f}. Inductively, we have

d(B_{f}^{c}(π(g^{n}z_{k})), B_{f}^{c}(π(g^{n}y_{k})) ≥ μ̃^{n}d(B_{f}^{c}(πz_{k}), B_{f}^{c}(πy_{k})).

Since πz′_{k} ∈ W_{f}^{s}(πz_{k}) and B_{f}^{c}(π(g^{n}z′_{k})) = B_{f}^{c}(π(g^{n}x), we have

d(B_{f}^{c}(π(g^{i}z_{k})), B_{f}^{c}(π(g^{i}x))) → 0 as i → ∞.

Hence

d(B_{f}^{c}(π(g^{n}x)), B_{f}^{c}(π(g^{n}y_{k}))) ≥ μ̃^{n}d(B_{f}^{c}(πz_{k}), B_{f}^{c}(πy_{k})).

Since πz_{k} ∈ W_{f}^{u}(πy_{k}), it is easy to see that d(B_{f}^{c}(πz_{k}), B_{f}^{c}(πy_{k})) ≥ c_{2}d(πz_{k}, πy_{k}) for some constant c_{2} > 0 only depends on the system. Also by (3.6) and the fact that d(πz_{k}, πy_{k}) ≥ d(πz′_{k}, πz_{k}), we have d(πz_{k}, πy_{k}) ≥ 0.5c_{1}d(πx, πy_{k}). Since y_{k} ∈ W_{g}^{c}(x) and the map π(x) is along Ẽ_{x}^{u} ⊕ Ẽ_{x}^{s}, which is a smooth tangent subbundle, we have d(πx, πy_{k}) ≥ c_{3}d(x, y_{k}) for some c_{3} > 0 independent of x and y_{k}. Therefore, we have

d(B_{f}^{c}(π(g^{n}x)), B_{f}^{c}(π(g^{n}y_{k}))) ≥ Cμ̃^{n}d(x, y_{k}),

where C is a constant independent of x, y_{k} and n. Since n = n(k) increases like − log d(x, y_{k})/ log μ′_{g} and min{1, μ′_{g}} < μ̃, we have μ̃^{n}d(x, y_{k}) → ∞ as d(x, y_{k}) → 0. This contrdicts to (3.7) which implies that d(B_{f}^{c}(π(g^{n}x)), B_{f}^{c}(π(g^{n}y_{k}))) is bounded.

Replacing the center leaves by the center-stable leaves or the center-unstable leaves and using similar arguments in the above paragraph, we can prove that π sends W_{g}^{cs} and W_{g}^{cu} to W_{f}^{cs} and W_{f}^{cu} respectively.

4 Applications in the entropy theory

In this section, we apply our results to continuity of entropy. It is well known that continuity properties of entropy are very delicate. Obviously, the topological entropy of Anosov diffeomorphisms is locally a constant since it is structurally stable. For partially hyperbolic systems, Hua, Saghin and Xia ([10]) proved that for the case that the unstable and stable foliations stably carry some unique nontrivial homologies, the topological entropy is locally constant if the center foliation is one dimensional, and continuous if the center foliation is two dimensional. Hua ([9]) showed that the topological entropy is continuous at the time one map of transitive Anosov flows. In this section, we will use our results on the structural quasi-stability to investigate continuity of entropy for some partially hyperbolic diffeomorphisms.

4.1 Time one map of Anosov flow

Proof of Theorem D. Let g be a diffeomorphism sufficiently close to f. By Theorem B′ and Theorem C, there exist a homeomorphism π : M → M with d(π, id_{M}) sufficiently small and a small τ̃ : M → ℝ such that

π ∘ g(x) = φ^{τ̃∘f(x)}(f ∘ π(x)) ∀x ∈ M.

Now we can define a flow ψ by ψ^{t}(x) = π^{−1}φ^{t}(π(x)) for x ∈ M and t ∈ ℝ. Obviously, φ and ψ are conjugate and

g(x) = ψ^{1+τ̃∘f(x)}(x)  (4.1)

for any x ∈ M. By Theorem B of [14], we have that

(1 + min_{x∈M} τ̃(x))h(φ) ≤ h(ψ) ≤ (1 + max_{x∈M} τ̃(x))h(φ),  (4.2)

where h(φ) and h(ψ) are the topological entropies of φ and ψ respectively. From Proposition 21 of [4], we have that for any t ∈ ℝ,

h(φ^{t}) = |t|h(φ^{1}) = |t|h(f) and h(ψ^{t}) = |t|h(ψ^{1}).  (4.3)

By (4.1),

h(ψ^{1+ min_{x∈M} τ̃(x)}) ≤ h(g) ≤ h(ψ^{1+max_{x∈M} τ̃(x)}).  (4.4)

Therefore, by (4.2), (4.3) and (4.4), we have

(1 + min_{x∈M} τ̃(x))^{2}h(f) ≤ h(g) ≤ (1 + max_{x∈M} τ̃(x))^{2}h(f).

Note that |τ̃| → 0 as g → f. Hence we conclude that the topological entropy function is continuous at f.

4.2 Systems with almost parallel center foliation

For a smooth surface Σ, y ∈ Σ and r > 0, we denote

Σ(y, α) = {z ∈ Σ : d(z, y) < r}.

The volume growth rate of the unstable foliation of f is defined by

χ^{u}(x, r) = lim sup_{n→∞} \frac{1}{n} log Vol(f^{n}W_{f}^{u}(x, r)),

and

χ^{u}(f) = sup_{x∈M} χ^{u}(x, r).

(See [10].) Clearly, χ^{u}(f) is independent of r. χ^{s}(f) is defined similarly by using stable manifolds W_{f}^{s}.

Proof of Theorem E. The first part of the theorem follows from Lemma 4.1 below. By Lemma 4.2 below, the volume growth satisfies χ^{u}(f) = χ^{u}(g). So, following the same arguments in the proof of Theorem 1.1 in [10], we can obtain the second part of the theorem.

Recall that θ^{c} is a holonomy map defined by sliding the center leaves. When we use the map, we will allow the domain to be a nonsmooth surface or even an arbitrary set. Also, we use θ_{f}^{c} and θ_{g}^{c} to denote the maps along the center leaves of f and g respectively.

Lemma 4.1. Let f be a partially hyperbolic diffeomorphism as in Theorem B and g be a diffeomorphism that is C^{1}-close to f. If f has almost parallel center foliation, then so does g.

Moreover, for any x, y ∈ M with y ∈ W_{f}^{c}(π(x)), and any smooth surfaces Σ_{f}(x) and Σ_{g}(y) with Σ_{f}(x) ⊥ W_{f}^{c} and Σ_{g}(y) ⊥ W_{g}^{c}, the map θ_{g}^{c} ∘ π : Σ_{f}(x) → Σ_{g}(y) is uniformly continuous with respect to x and y.

Proof. Take β > 0. Take x ∈ M and y ∈ W_{g}^{c}(x). Let Σ_{g}(x) and Σ_{g}(y) denote the smooth surfaces with Σ_{g}(x) ⊥ W_{g}^{c} and Σ_{g}(y) ⊥ W_{g}^{c}. Denote Σ_{g}(x, β) = {z ∈ Σ_{g}(x) : d(x, z) ≤ α}. We need to show that there exists α > 0 independent of x and y such that θ_{g}^{c}(Σ_{g}(x, α)) ⊂ Σ_{g}(y, β).

Denote R_{g}(y, β) = ∪_{z∈Σ_{g}(y,β)}W_{g}^{c}(z, β), where W_{g}^{c}(z, β) is the local center leaf of g at z of size β. Clearly, R_{g}(y, β) contains a ball about y of radius β. Since π is a homeomorphism, π^{−1} is uniformly continuous on M. So there is β′ > 0 independent of y such that π(R_{g}(y, β)) contains a ball of radius β′ about π(y). In particular, π(R_{g}(y, β)) ⊃ Σ_{f}(π(y), β′), where Σ_{f}(π(y), β′) is the part of a smooth surface Σ_{f}(π(y)) that is contained in a ball of radius β′, and Σ_{f}(π(y)) ⊥ W_{f}^{c}.

Note that by Theorem C, π(W_{g}^{c}(y)) = W_{f}^{c}(π(y)). Since f has almost parallel center foliation, there is α′ > 0 independent of π(y) such that θ_{f}^{c}(Σ_{f}(π(x), α′)) ⊂ Σ_{f}(π(y), β′).

Consider the set R_{f}(π(x), α′) = ∪_{z∈Σ_{f} (π(x),β′)}W_{f}^{c}(z, α′). It contains a ball of radius α′ about π(x). Since π is uniformly continuous, there exists α > 0 only depending on α′, such that π^{−1}(R_{f}(π(x), α′)) contains a ball of radius α about x. In particular, it contains Σ_{g}(x, α).

Now it is easy to check that θ_{g}^{c}(Σ_{g}(x, α)) ⊂ Σ_{g}(y, β).

The proof of the second part of the lemma can be obtained in a similar way.

Lemma 4.2. Let f and g be as in Lemma 4.1. Then we have

χ^{u}(g) = χ^{u}(f).

Proof. Take x ∈ M and r > 0.

By the last lemma we know that there exists r^{∗} ≥ r′ > 0 such that

W_{f}^{u}(π(x), r′) ⊂ (θ^{c} ∘ π)(W_{g}^{u}(x, r)) ⊂ W_{f}^{u}(π(x), r^{∗}),  (4.5)

where θ^{c} is the holonomy map into W_{f}^{u}(π(x), r^{∗}).

Define ψ_{n} = θ_{n}^{c} ∘ π, where θ_{n}^{c} is the holonomy map into f^{n}W_{f}^{u}(π(x), r^{∗}). It is easy to check that (4.5) implies

f^{n}W_{f}^{u}(π(x), r′) ⊂ ψ_{n}(g^{n}W_{g}^{u}(x, r)) ⊂ f^{n}W_{f}^{u}(π(x), r^{∗})

for any n > 0. Moreover, by the second part of Lemma 4.1, we know that for any α > 0, there exists β > 0 such that for any x ∈ M, n > 0, and y ∈ g^{n}W_{g}^{u}(x, r),

ψ_{n}W_{g}^{u}(y, β) ⊂ W_{f}^{u}(ψ_{n}(y), α).

The inclusions mean that Condition (b) of Sublemma 4.3 below is satisfied with ψ_{n} = ψ, W = g^{n}W_{g}^{u}(x, r) and W′ = f^{n}W_{f}^{u}(π(x), r′) for all n ≥ 0. Since W_{f}^{u} and W_{g}^{u} are smooth submanifolds with bounded curvature, Condition (a) of the sublemma holds.

Now we apply Sublemma 4.3 to get that there is C > 0 independent of n such that for all n ≥ 0,

Vol(f^{n}W_{f}^{u}(x, r′)) ≤ C Vol(g^{n}W_{g}^{u}(x, r)).

Since this is true for any x, we get χ^{u}(f) ≤ χ^{u}(g).

We can apply similar arguments, by using the inverse of π and the fact that g has almost parallel center foliation, to get χ^{u}(g) ≤ χ^{u}(f).

Sublemma 4.3. Let W, W′, W^{∗} ⊂ M be k dimensional manifolds with W′ ⊂ W^{∗} and ψ : W → W^{∗} be a one to one map such that W′ ⊂ ψ(W). Suppose that for all n ≥ 0,

(a) there are constants \overline{C} and \underline{C} such that for any small α > 0, y′ ∈ W′ and y ∈ W,

Vol W′(y′, α) ≤ \overline{C}α^{k}, \underline{C}α^{k} ≤ Vol W (y, α);

(b) for any α > 0, there is a constant β > 0 such that for any y ∈ W with ψ(y) ∈ W′,

ψ(W (y, β)) ⊂ W′(ψ(y), α).

Then there exists C > 0, only depending on \underline{C}, \overline{C}, α and β, such that

Vol(W′) ≤ C Vol(W).

Proof. Fix α > 0. Take a 2α separated set y′_{1}, ⋯, y′_{ℓ_{n}} ∈ W′, that is, d(y′_{i}, y′_{j}) ≥ 2α for any 1 ≤ i, j ≤ ℓ_{n}. We also require that the set has maximal cardinality. Hence, {B_{W′}(y′_{i}, 2α)} form a cover of W′. So by part (a) we have

Vol(W′) ≤ ℓ_{n} · \overline{C}(2α)^{k}.

Take β > 0 as in Condition (b) of the sublemma. Since the balls in {W′(y′_{i}, α)} are pairwise disjoint, and ψW (y_{i}, β) ⊂ B_{W′}(y′_{i}, α), where y_{i} = ψ^{−1}y′_{i} ∈ W, we see that {W (y_{i}, β)} are pairwise disjoint. Hence,

Vol(W) ≥ ℓ_{n} · \underline{C}β^{k}.

So we have

Vol(W′) ≤ C Vol(W),

where C = \overline{C}(2α)^{k}/\underline{C}β^{k}.

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