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

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.

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 Invariant measures for higher-rank hyperbolic abelian actions

1996 ◽  
Vol 16 (4) ◽  
pp. 751-778 ◽  
Author(s):  
A. Katok ◽  
R. J. Spatzier
Keyword(s):  
Invariant Measures ◽  
Metric Entropy ◽  
Ergodic Measures ◽  
Anosov Actions ◽  
Higher Rank ◽  
Partially Hyperbolic

AbstractWe investigate invariant ergodic measures for certain partially hyperbolic and Anosov actions of ℝk, ℤkandWe show that they are either Haar measures or that every element of the action has zero metric 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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