Slow decay of correlations for generic mixing automorphisms

TL;DR

This paper demonstrates that for typical mixing automorphisms, correlations decay slowly, and mildly mixing automorphisms lack certain convergence properties of non-zero averages, highlighting nuanced behaviors in dynamical systems.

Contribution

It establishes that for generic mixing automorphisms, correlations decay slowly, and mildly mixing automorphisms do not exhibit specific convergence of averages, revealing new insights into their long-term behavior.

Findings

Correlations decay slowly for typical mixing automorphisms.

Mildly mixing automorphisms lack convergence of non-zero averages at certain rates.

Set of times with correlations above a threshold is infinite for generic mixing automorphisms.

Abstract

Let $\psi(n)\to +0$ and a square-integrable function $f$ be non-zero, then for the typical mixing automorphism $T$ the set $\{n:\, |(T^nf,f) |>\psi( n)\}$ is infinite. The mildly mixing automorphisms $T$ do not have convergences of non-zero averages $\frac 1 {k_n} \sum_{i=1}^{k_n}T^if (x)$ with the rate of $o\left(\frac 1 {k_n}\right)$.