Quantitative exponential mixing for the randomized Chirikov standard map

TL;DR

This paper proves explicit exponential mixing for a randomized version of the Chirikov standard map on the torus, overcoming deterministic obstructions to ergodicity by establishing verifiable conditions.

Contribution

It introduces a criterion for exponential mixing in incompressible random dynamical systems and applies it to the randomized Chirikov map with explicit conditions.

Findings

Established almost-sure exponential mixing for large kicking strengths.

Reduced mixing proof to verifiable conditions for the random dynamical system.

Provided milder parameters for qualitative exponential mixing and dissipation.

Abstract

We investigate the mixing properties of a randomized Chirikov standard map on $\mathbb{T}^2$. While the deterministic dynamics exhibit obstructions to global ergodicity, we establish explicit almost-sure quantitative exponential mixing when kicking strengths are sufficiently large. To achieve this, we formulate a criterion for incompressible random dynamical systems, reducing quantitative exponential mixing to serval verifiable conditions. Additionally, we provide a milder parameter condition to derive qualitative exponential mixing and enhanced dissipation.