scholarly journals Sharp polynomial bounds on decay of correlations for multidimensional nonuniformly hyperbolic systems and billiards

2021 ◽  
Vol 4 ◽  
pp. 407-451
Author(s):  
Henk Bruin ◽  
Ian Melbourne ◽  
Dalia Terhesiu
Keyword(s):  
Hyperbolic Systems ◽  
Decay Of Correlations ◽  
Polynomial Bounds ◽  
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.

scholarly journals POLYNOMIAL DECAY OF CORRELATIONS IN THE GENERALIZED BAKER'S TRANSFORMATION

2013 ◽  
Vol 23 (08) ◽  
pp. 1350130 ◽  
Author(s):  
CHRISTOPHER BOSE ◽  
RUA MURRAY
Keyword(s):  
Hyperbolic Systems ◽  
Decay Of Correlations ◽  
Polynomial Decay ◽  
Mixing Rate ◽  
New Techniques ◽  
Correlation Decay ◽  
Simple Geometry ◽  
Uniformly Hyperbolic Systems ◽  
Mixing Rates ◽  
Area Preserving

We introduce a family of area preserving generalized baker's transformations acting on the unit square and having sharp polynomial rates of mixing for Hölder data. The construction is geometric, relying on the graph of a single variable "cut function". Each baker's map B is nonuniformly hyperbolic and while the exact mixing rate depends on B, all polynomial rates can be attained. The analysis of mixing rates depends on building a suitable Young tower for an expanding factor. The mechanisms leading to a slow rate of correlation decay are especially transparent in our examples due to the simple geometry in the construction. For this reason, we propose this class of maps as an excellent testing ground for new techniques for the analysis of decay of correlations in non-uniformly hyperbolic systems. Finally, some of our examples can be seen to be extensions of certain 1D non-uniformly expanding maps that have appeared in the literature over the last twenty years, thereby providing a unified treatment of these interesting and well-studied examples.

Decay of correlations in certain hyperbolic systems

1982 ◽  
Vol 26 (1) ◽  
pp. 717-719 ◽  
Author(s):  
Giulio Casati ◽  
Giorgio Comparin ◽  
Italo Guarneri
Keyword(s):  
Hyperbolic Systems ◽  
Decay Of Correlations

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].

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