scholarly journals Conditional variational principle for the irregular set in some nonuniformly hyperbolic systems

2016 ◽  
Vol 36 (11) ◽  
pp. 6581-6597
Author(s):  
Zheng Yin ◽  
Ercai Chen
Keyword(s):  
Variational Principle ◽  
Hyperbolic Systems ◽  
Nonuniformly Hyperbolic

scholarly journals A note on statistical properties for nonuniformly hyperbolic systems with slow contraction and expansion

2016 ◽  
Vol 16 (03) ◽  
pp. 1660012 ◽  
Author(s):  
Ian Melbourne ◽  
Paulo Varandas
Keyword(s):  
Invariance Principle ◽  
Hyperbolic Systems ◽  
Return Time ◽  
Limit Laws ◽  
Weak Invariance Principle ◽  
Weak Invariance ◽  
The Central Limit Theorem ◽  
Statistical Limit ◽  
Contraction And Expansion ◽  
Nonuniformly Hyperbolic

We provide a systematic approach for deducing statistical limit laws via martingale-coboundary decomposition, for nonuniformly hyperbolic systems with slowly contracting and expanding directions. In particular, if the associated return time function is square-integrable, then we obtain the central limit theorem, the weak invariance principle, and an iterated version of the weak invariance principle.

Upper semi-continuity of entropy map for nonuniformly hyperbolic systems

Nonlinearity ◽  
2015 ◽  
Vol 28 (8) ◽  
pp. 2977-2992 ◽  
Author(s):  
Gang Liao ◽  
Wenxiang Sun ◽  
Shirou Wang
Keyword(s):  
Hyperbolic Systems ◽  
Nonuniformly Hyperbolic

Livšic theorem for matrix cocycles over nonuniformly hyperbolic systems

2019 ◽  
Vol 19 (02) ◽  
pp. 1950010 ◽  
Author(s):  
Rui Zou ◽  
Yongluo Cao
Keyword(s):  
Periodic Point ◽  
Measurable Function ◽  
Hyperbolic Systems ◽  
Type Theorem ◽  
Hölder Continuous ◽  
Nonuniformly Hyperbolic

We prove a nonuniformly hyperbolic version of the Livšic-type theorem, with cocycles taking values in [Formula: see text]. To be more precise, let [Formula: see text] Diff[Formula: see text] preserving an ergodic hyperbolic measure [Formula: see text], and [Formula: see text] be Hölder continuous satisfying [Formula: see text] for each periodic point [Formula: see text], then there exists a measurable function [Formula: see text] satisfying [Formula: see text] for [Formula: see text]-almost every [Formula: see text].

scholarly journals Unstable entropies and dimension theory of partially hyperbolic systems

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 658-680
Author(s):  
Xueting Tian ◽  
Weisheng Wu
Keyword(s):  
Variational Principle ◽  
Topological Entropy ◽  
Multifractal Analysis ◽  
Hyperbolic Systems ◽  
General Setting ◽  
Dimension Theory ◽  
Partially Hyperbolic ◽  
Distribution Principle ◽  
Saturated Sets ◽  
Dimension Characteristic

Abstract In this paper we define unstable topological entropy for any subsets (not necessarily compact or invariant) in partially hyperbolic systems as a Carathéodory–Pesin dimension characteristic, motivated by the work of Bowen and Pesin etc. We then establish some basic results in dimension theory for Bowen unstable topological entropy, including an entropy distribution principle and a variational principle in general setting. As applications of this new concept, we study unstable topological entropy of saturated sets and extend some results in Bowen (1973 Trans. Am. Math. Soc. 184 125–36); Pfister and Sullivan (2007 Ergod. Theor. Dynam. Syst. 27 929–56). Our results give new insights to the multifractal analysis for partially hyperbolic systems.

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