Statistical properties for compositions of standard maps with increasing coefficent

TL;DR

This paper investigates the statistical properties of compositions of standard maps with increasing coefficients, establishing limit theorems and mixing estimates under certain growth conditions, and extending results to a broader class of hyperbolic maps.

Contribution

It provides rigorous statistical results for compositions of standard maps with increasing parameters, a problem of intermediate difficulty, and introduces methods applicable to similar hyperbolic systems.

Findings

Established a Strong Law for compositions with fast-growing coefficients.

Proved a Central Limit Theorem for Holder observables in these systems.

Derived quantitative mixing estimates for the compositions.

Abstract

The Chirikov standard map family is a one-parameter family of volume-preserving maps exhibiting hyperbolicity on a `large' but noninvariant subset of phase space. Based on this predominant hyperbolicity and numerical experiments, it is anticipated that the standard map has positive metric entropy for many parameter values. However, rigorous analysis is notoriously difficult, and it remains an open question whether the standard map has positive metric entropy for any parameter value. Here we study a problem of intermediate difficulty: compositions of standard maps with increasing parameter. When the coefficients increase to infinity at a sufficiently fast polynomial rate, we obtain a Strong Law, Central Limit Theorem, and quantitative mixing estimate for Holder observables. The methods used are not specific to the standard map and apply to a class of compositions of `prototypical' 2D maps with hyperbolicity on `most' of phase space.