Skew products, quantitative recurrence, shrinking targets and decay of correlations

TL;DR

This paper investigates how arithmetical properties influence recurrence, hitting times, and decay of correlations in toral extensions of hyperbolic systems, revealing polynomial decay rates and examples lacking shrinking target properties.

Contribution

It establishes the dependence of recurrence and correlation decay on arithmetical properties and provides estimations for decay exponents in these dynamical systems.

Findings

Decay of correlations is polynomial with respect to $C^{r}$ observables.

The decay exponent depends on $r$ and arithmetical properties.

Some systems lack the shrinking target property and have trivial return time distributions.

Abstract

We consider toral extensions of hyperbolic dynamical systems. We prove that its quantitative recurrence (also with respect to given observables) and hitting time scale behavior depend on the arithmetical properties of the extension. By this we show that those systems have a polynomial decay of correlations with respect to $C^{r}$ observables, and give estimations for its exponent, which depend on $r$ and on the arithmetical properties of the system. We also show examples of systems of this kind having not the shrinking target property, and having a trivial limit distribution of return time statistics.