Quasi-shadowing Property on Random Partially Hyperbolic Sets

2018 ◽  
Vol 34 (9) ◽  
pp. 1429-1444
Author(s):  
Lin Wang ◽  
Xin Sheng Wang ◽  
Yu Jun Zhu
Keyword(s):  
Shadowing Property ◽  
Partially Hyperbolic ◽  
Hyperbolic Sets

scholarly journals Quasi-shadowing for partially hyperbolic diffeomorphisms

2014 ◽  
Vol 35 (2) ◽  
pp. 412-430 ◽  
Author(s):  
HUYI HU ◽  
YUNHUA ZHOU ◽  
YUJUN ZHU
Keyword(s):  
Spectral Decomposition ◽  
Hyperbolic Systems ◽  
Decomposition Theorem ◽  
Closing Lemma ◽  
Shadowing Property ◽  
Hyperbolic Diffeomorphisms ◽  
Uniformly Hyperbolic Systems ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic Diffeomorphism

AbstractA partially hyperbolic diffeomorphism $f$ has the quasi-shadowing property if for any pseudo orbit $\{x_{k}\}_{k\in \mathbb{Z}}$, there is a sequence of points $\{y_{k}\}_{k\in \mathbb{Z}}$ tracing it in which $y_{k+1}$ is obtained from $f(y_{k})$ by a motion ${\it\tau}$ along the center direction. We show that any partially hyperbolic diffeomorphism has the quasi-shadowing property, and if $f$ has a $C^{1}$ center foliation then we can require ${\it\tau}$ to move the points along the center foliation. As applications, we show that any partially hyperbolic diffeomorphism is topologically quasi-stable under $C^{0}$-perturbation. When $f$ has a uniformly compact $C^{1}$ center foliation, we also give partially hyperbolic diffeomorphism versions of some theorems which hold for uniformly hyperbolic systems, such as the Anosov closing lemma, the cloud lemma and the spectral decomposition theorem.

scholarly journals The structure of the space of ergodic measures of transitive partially hyperbolic sets

2019 ◽  
Vol 190 (3) ◽  
pp. 441-479
Author(s):  
L. J. Díaz ◽  
K. Gelfert ◽  
T. Marcarini ◽  
M. Rams
Keyword(s):  
Ergodic Measures ◽  
Partially Hyperbolic ◽  
Hyperbolic Sets

Partially hyperbolic diffeomorphisms with a uniformly compact center foliation: the quotient dynamics

2015 ◽  
Vol 36 (4) ◽  
pp. 1067-1105 ◽  
Author(s):  
DORIS BOHNET ◽  
CHRISTIAN BONATTI
Keyword(s):  
Quotient Space ◽  
Shadowing Property ◽  
Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Chain Recurrent Set ◽  
Recurrent Set

We show that a partially hyperbolic $C^{1}$-diffeomorphism $f:M\rightarrow M$ with a uniformly compact $f$-invariant center foliation ${\mathcal{F}}^{c}$ is dynamically coherent. Further, the induced homeomorphism $F:M/{\mathcal{F}}^{c}\rightarrow M/{\mathcal{F}}^{c}$ on the quotient space of the center foliation has the shadowing property, i.e. for every ${\it\epsilon}>0$ there exists ${\it\delta}>0$ such that every ${\it\delta}$-pseudo-orbit of center leaves is ${\it\epsilon}$-shadowed by an orbit of center leaves. Although the shadowing orbit is not necessarily unique, we prove the density of periodic center leaves inside the chain recurrent set of the quotient dynamics. Other interesting properties of the quotient dynamics are also discussed.

scholarly journals NON-HYPERBOLIC ERGODIC MEASURES AND HORSESHOES IN PARTIALLY HYPERBOLIC HOMOCLINIC CLASSES

2019 ◽  
Vol 19 (5) ◽  
pp. 1765-1792 ◽  
Author(s):  
Dawei Yang ◽  
Jinhua Zhang
Keyword(s):  
Topological Entropy ◽  
Classical Result ◽  
Ergodic Measure ◽  
Rich Family ◽  
Ergodic Measures ◽  
Partially Hyperbolic ◽  
Hyperbolic Sets ◽  
Homoclinic Classes

We study a rich family of robustly non-hyperbolic transitive diffeomorphisms and we show that each ergodic measure is approached by hyperbolic sets in weak$\ast$-topology and in entropy. For hyperbolic ergodic measures, it is a classical result of A. Katok. The novelty here is to deal with non-hyperbolic ergodic measures. As a consequence, we obtain the continuity of topological entropy.

scholarly journals CENTER MANIFOLDS FOR PARTIALLY HYPERBOLIC SETS WITHOUT STRONG UNSTABLE CONNECTIONS

2015 ◽  
Vol 15 (4) ◽  
pp. 785-828 ◽  
Author(s):  
Christian Bonatti ◽  
Sylvain Crovisier
Keyword(s):  
Center Manifolds ◽  
Compact Sets ◽  
Partially Hyperbolic ◽  
Hyperbolic Sets

We consider compact sets which are invariant and partially hyperbolic under the dynamics of a diffeomorphism of a manifold. We prove that such a set $K$ is contained in a locally invariant center submanifold if and only if each strong stable and strong unstable leaf intersects $K$ at exactly one point.

scholarly journals Partially hyperbolic sets with a dynamically minimal lamination

2018 ◽  
Vol 38 (6) ◽  
pp. 2717-2729
Author(s):  
Luiz Felipe Nobili França ◽  
Keyword(s):  
Partially Hyperbolic ◽  
Hyperbolic Sets

scholarly journals A link between topological entropy and Lyapunov exponents

2017 ◽  
Vol 39 (3) ◽  
pp. 620-637
Author(s):  
THIAGO CATALAN
Keyword(s):  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Lower Semicontinuous ◽  
Periodic Points ◽  
Partial Hyperbolicity ◽  
Symplectic Diffeomorphisms ◽  
Symplectic Diffeomorphism ◽  
Partially Hyperbolic ◽  
Generic Set ◽  
Hyperbolic Sets

We show that a $C^{1}$-generic non-partially hyperbolic symplectic diffeomorphism $f$ has topological entropy equal to the supremum of the sum of the positive Lyapunov exponents of its hyperbolic periodic points. Moreover, we also prove that $f$ has topological entropy approximated by the topological entropy of $f$ restricted to basic hyperbolic sets. In particular, the topological entropy map is lower semicontinuous in a $C^{1}$-generic set of symplectic diffeomorphisms far from partial hyperbolicity.

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