Topological structure of (partially) hyperbolic sets with positive volume

AbstractLet $X$ be a compact metric space, $f:X\to X$ a homeomorphism and $\phi \in C(X,\mathbb {R})$. We construct a fundamental domain for the set of points with finite peaks with respect to the induced cocycle $\{\phi _n\}$. As applications, we give sufficient conditions for the transitive set of a non-conservative partially hyperbolic diffeomorphism to have positive Lebesgue measure, i.e., for an accessible partially hyperbolic diffeomorphism, if the set of points with finite peaks for the Jacobian cocycle is not of full volume, then the set of transitive points is of positive volume.

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

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748018000579 ◽

2019 ◽

Vol 19 (5) ◽

pp. 1765-1792 ◽

Cited By ~ 1

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.

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Partially hyperbolic sets from a co-dimension one bifurcation

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.1995.1.253 ◽

1995 ◽

Vol 1 (2) ◽

pp. 253-275 ◽

Cited By ~ 3

Author(s):

Todd Young ◽

Keyword(s):

Partially Hyperbolic ◽

Dimension One ◽

Hyperbolic Sets

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CENTER MANIFOLDS FOR PARTIALLY HYPERBOLIC SETS WITHOUT STRONG UNSTABLE CONNECTIONS

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748015000055 ◽

2015 ◽

Vol 15 (4) ◽

pp. 785-828 ◽

Cited By ~ 6

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.

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Partially hyperbolic sets with a dynamically minimal lamination

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.2018114 ◽

2018 ◽

Vol 38 (6) ◽

pp. 2717-2729

Author(s):

Luiz Felipe Nobili França ◽

Keyword(s):

Partially Hyperbolic ◽

Hyperbolic Sets

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A link between topological entropy and Lyapunov exponents

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2017.39 ◽

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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