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Statistical properties for compositions of standard maps with increasing coefficient

Part of: Dynamical systems with hyperbolic behavior Smooth dynamical systems: general theory Limit theorems Ergodic theory

Published online by Cambridge University Press:  07 February 2020

ALEX BLUMENTHAL Open the ORCID record for ALEX BLUMENTHAL [Opens in a new window]
ALEX BLUMENTHAL*
Affiliation:
University of Maryland, College Park, USA email alexb123@math.umd.edu
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Abstract

The Chirikov standard map is a prototypical example of a one-parameter family of volume-preserving maps for which one anticipates chaotic behavior on a non-negligible (positive-volume) subset of phase space for a large set of parameters. Rigorous analysis is notoriously difficult and it remains an open question whether this chaotic region, the stochastic sea, has positive Lebesgue measure for any parameter value. Here we study a problem of intermediate difficulty: compositions of standard maps with increasing coefficient. When the coefficients increase to infinity at a sufficiently fast polynomial rate, we obtain a strong law, a central limit theorem, and quantitative mixing estimates for Holder observables. The methods used are not specific to the standard map and apply to a class of compositions of ‘prototypical’ two-dimensional maps with hyperbolicity on ‘most’ of phase space.

Keywords

non-autonomous dynamicsnon-uniform hyperbolicitystatistical properties of deterministic dynamics

MSC classification

Primary: 37C60: Nonautonomous smooth dynamical systems 37A25: Ergodicity, mixing, rates of mixing 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Secondary: 60F05: Central limit and other weak theorems
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 4 , April 2021 , pp. 981 - 1024
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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