Partial hyperbolicity and central shadowing

We study shadowing property for a partially hyperbolic diffeomorphism $f$. It is proved that if $f$ is dynamically coherent then any pseudotrajectory can be shadowed by a pseudotrajectory with"jumps"along the central foliation. The proof is based on the Tikhonov-Shauder fixed point theorem.


Introduction
The theory of shadowing of approximate trajectories (pseudotrajectories) of dynamical systems is now a well developed part of the global theory of dynamical systems (see, for example, monographs [12], [13]). This theory is of special importance for numerical simulations and the classical theory of structural stability.
It is well known that a diffeomorphism has the shadowing property in a neighborhood of a hyperbolic set [2], [4] and a structurally stable diffeomorphism has the shadowing property on the whole manifold [11], [17], [19].
There are a lot of examples of non-hyperbolic diffeomorphisms, which have shadowing property (see for instance [14], [21]) at the same time this phenomena is not frequent. More precisely the following statements are correct. Diffeomophisms with C 1 -robust shadowing property are structurally stable [18]. In [1] Abdenur and Diaz conjectured that C 1 -generically shadowing is equivalent to structural stability, and proved this statement for socalled tame diffeomorphisms. Lipschitz shadowing is equivalent to structural stability [15] (see [21] for some generalizations).
In present article we study shadowing property for partially hyperbolic diffeomorphisms. Note that due to [7] one cannot expect that in general shadowing holds for partially hyperbolic diffeomorphisms. We use notion of central pseudotrajectory and prove that any pseudotrajectory of a partially hyperbolic diffeomorphism can be shadowed by a central pseudotrajectory. This result might be considered as a generalization of a classical shadowing lemma for the case of partially hyperbolic diffeomorphisms.

Definitions and the main result
Let M be a compact n -dimensional C ∞ smooth manifold, with a Riemannian metric dist. Let | · | be the Euclidean norm at R n and the induced norm on the leaves of the tangent bundle T M. For any x ∈ M, ε > 0 we denote Below in the text we use the following definition of partial hyperbolicity (see for example [6]).
that the mapping f m satisfies the following property. There exists a continuous invariant bundle x ∈ M and continuous positive functions ν,ν, γ,γ : M → R such that ν,ν < 1, ν < γ <γ <ν −1 and for all x ∈ M, v ∈ R n , |v| = 1 Denote For further considerations we need the notion of dynamical coherence.
Definition 2. We say that a k -dimensional distribution E over T M is uniquely integrable if there exists a k -dimensional continuous foliation W of the manifold M, whose leaves are tangent to E at every point. Also, any C 1 -smooth path tangent to E is embedded to a unique leaf of W .

Definition 3.
A partially hyperbolic diffeomorphism f is dynamically coherent if both the distributions E cs and E cu are uniquely integrable.
If f is dynamically coherent then distribution E c is also uniquely integrable and corresponding foliation W c is a subfoliation of both W cs and W cu . For a discussion how often partially hyperbolic diffeomorphisms are dynamically coherent see [5], [9].
In the text below we always assume that f is dynamically coherent. For τ ∈ {s, c, u, cs, cu} and y ∈ W τ (x) let dist τ (x, y) be the inner distance on W τ (x) from x to y. Note that Let us recall the definition of the shadowing property.
Definition 5. Diffeomorphism f satisfies the shadowing property if for any ε > 0 there exists d > 0 such that for any d-pseudotrajectory (3) Definition 6. Diffeomorphism f satisfies the Lipschitz shadowing property if there exist L, d 0 > 0 such that for any d ∈ (0, d 0 ), and any d-pseudotrajectory {x k : k ∈ Z} there exists a trajectory {y k } of the diffeomorphism f , satisfying (3) with ε = Ld.
As was mentioned before in a neighborhood of a hyperbolic set diffeomorphism satisfies the Lipschitz shadowing property [2], [4], [13].
We suggest the following generalization of the shadowing property for partially hyperbolic dynamically coherent diffeomorphisms.
Definition 7 (see for example [10]). An ε-pseudotrajectory {y k } is called central if for any k ∈ Z the inclusion f (y k ) ∈ W c ε (y k+1 ) holds (see Fig. 1).  Note that the Lipschitz central shadowing property implies the central shadowing property.
We prove the following analogue of the shadowing lemma for partially hyperbolic diffeomorphisms. Theorem 1. Let diffeomorphism f ∈ C 1 be partially hyperbolic and dynamically coherent. Then f satisfies the Lipschitz central shadowing property.
Note that for Anosov diffeomorphisms any central pseudotrajectory is a true trajectory.
Let us also mention the following related notion [10].
Definition 10. Partially hyperbolic, dynamically coherent diffeomorphism f is called plaque expansive if there exists ε > 0 such that for any ε-central In the theory of partially hyperbolic diffeomorphisms the following conjecture plays important role [3], [10].
Let us note that if the diffeomorphism f in Theorem 1 is additionally plaque expansive then leaves W c (y k ) are uniquely defined (see Remark 1 below).
Among results related to Theorem 1 we would like to mention that partially hyperbolic dynamically coherent diffeomorphisms, satisfying plaque expansivity property are leaf stable (see [10,Chapter 7], [16] for details).

Proof of Theorem 1
In what follows below we will use the following statement, which is consequence of transversality and continuity of foliations W s , W cu . Statement 1. There exists δ 0 > 0, L 0 > 1 such that for any δ ∈ (0, δ 0 ] such that for any x, y ∈ M satisfying dist(x, y) < δ there exists unique point Note that for a fixed diffeomorphism f , satisfying the assumptions of the theorem, it suffices to prove that its fixed power f m satisfies the Lipschitz central shadowing property. Since foliations W τ , τ ∈ {s, u, c, cs, cu} of f m coincide with the corresponding foliations of the initial diffeomorphism f we can assume without loss of generality that conditions (1) hold for m = 1. Note that a similar claim can be done using adapted metric, see [8].
Let us choose l so big that λ l > 2L 0 .
Arguing similarly to previous paragraph it is sufficient to prove that f l has the Lipschitz central shadowing property and hence, we can assume without loss of generality that l = 1. Decreasing δ 0 if necessarily we conclude from inequalities (1) that and dist u (f (x), f (y)) ≥ λ dist u (x, y), y ∈ W u δ 0 (x).
A3 For y ∈ W τ ε (x), τ ∈ {s, c, u, cs, cu} the following holds Consider small enough µ ∈ (0, 1) satisfying the following inequality Choose corresponding ε > 0 from Statement 2. Let δ = min(δ 0 , ε/L 0 ). For a pseudotrajectory {x k } consider maps h s k : U k ⊂ E s (x k ) → E s (x k+1 ) defined as the following: and U k is the set of points for which map h s k is well-defined (see Fig. 2). Note that maps h s k (z) are continuous. The following lemma plays a central role in the proof of Theorem 1.

Lemma 1.
There exists d 0 > 0, L > 1 such that for any d < d 0 and d-pseudotrajectory {x k } maps h s k are well-defined for z ∈ I s Ld (x k ) and the following inequalities hold Proof. Inequality (5) implies that there exists L > 0 such that Let us choose d 0 < δ 0 /2L. Fix d < d 0 , d-pseudotrajectory {x k }, k ∈ Z and z ∈ I s Ld (x k ). Condition A2 of Statement 2 implies that dist s (x k , exp s x k (z)) ≤ Ld(1 + µ). Inequality (4) implies the following Inequalities (2) and dist(f (x k ), x k+1 ) < d imply (see Fig. 3 for illustration)  (6) is well-defined and inequality (8) implies the following This inequality and Statement 2 imply |h s k (z)| < Ld, which completes the proof.
Let d 0 , L > 0 are constants provided by Lemma 1. Let d < d 0 and {x k } is a d-pseudotrajectory. Denote This set endowed with the Tikhonov product topology is compact and convex. Let us consider map H : X s → X s defined as following By Lemma 1 this map is well-defined. Since z ′ k+1 depends only on z k map H is continuous. Due to the Tikhonov-Schauder theorem [20], the mapping H has a (maybe non-unique) fixed point {z * k }. Denote y s k = exp s x k (z * k ). Since z * k+1 = h s k (z * k ), inequality (9) implies that y s k+1 ∈ W cu Ld (f (y s k )), k ∈ Z.
Since |z * k | < Ld we conclude dist(x k , y s k ) ≤ dist s (x k , y s k ) < (1 + µ)Ld < 2Ld, k ∈ Z. Similarly (decreasing d 0 and increasing L if necessarily) one may show that there exists a sequence {y u k ∈ W u 2Ld (x k )} such that y u k+1 ∈ W cs Ld (f (y u k )), k ∈ Z.
Hence dist(y s k , y u k ) < dist(y s k , x k ) + dist(x k , y u k ) < 4Ld. Decreasing d 0 if necessarily we can assume that 4L 0 Ld < δ 0 . Then there exists an unique point y k = W cu 4L 0 Ld (y s k ) ∩ W s 4L 0 Ld (y u k ) and inclusion (10) implies that for all k ∈ Z the following holds where R = sup x∈M | D f (x)| and L cu > 1 do not depends on d. Similarly for some constant L cs > 1 the following inequalities hold dist cs (y k+1 , f (y k )) < L cs d, k ∈ Z.
To complete the proof let us note that dist(x k , y k ) < dist(x k , y s k ) + dist(y s k , y k ) < 2Ld + 4L 0 Ld, k ∈ Z. Taking L = max(L 1 , 2L + 4L 0 ) we conclude that {y k } is an Ld-central pseudotrajectory which Ld shadows {x k }.
Remark 1. Note that we do not claim uniqueness of such sequences {y s k } and {y u k }. In fact it is easy to show (we leave details to the reader) that uniqueness of those sequences is equivalent to the plaque expansivity conjecture.

Acknowledgement
Sergey Kryzhevich was supported by the UK Royal Society (joint project with Aberdeen University), by the Russian Federal Program "Scientific and pedagogical cadres", grant no. 2010-1.1-111-128-033. Sergey Tikhomirov was supported by the Humboldt postdoctoral fellowship for postdoctoral researchers (Germany). Both the coauthors are grateful to the Chebyshev Laboratory (Department of Mathematics and Mechanics, Saint-Petersburg State University) for the support under the grant 11.G34.31.0026 of the Government of the Russian Federation.