Long hitting time, slow decay of correlations and arithmetical properties

TL;DR

This paper constructs specific torus translations with Liouville properties that defy typical logarithm laws for hitting times and demonstrates that such systems cannot have polynomial decay of correlations, revealing new dynamical behaviors.

Contribution

It introduces a class of torus translations with arithmetical properties that violate standard logarithm laws and shows these systems cannot be polynomially mixing.

Findings

Constructed translations with infinite lower hitting time indicator.

Established a mixing system on three torus with these properties.

Proved systems with polynomial decay of correlations cannot have these arithmetical properties.

Abstract

Let $\tau_r(x,x_0)$ be the time needed for a point $x$ to enter for the first time in a ball $B_r(x_0)$ centered in $x_0$, with small radius $r$. We construct a class of translations on the two torus having particular arithmetic properties (Liouville components with intertwined denominators of convergents) not satisfying a logarithm law, i.e. such that for generic $x,x_0$ \liminf_{r\to 0} \frac{\log \tau_r(x,x_0)}{-\log r} = \infty. By considering a suitable reparametrization of the flow generated by a suspension of this translation, using a previous construction by Fayad, we show the existence of a mixing system on three torus having the same properties. The speed of mixing of this example must be subpolynomial, because we also show that: in a system having polynomial decay of correlations the above ratio of logarithms (which is also called the lower hitting time indicator) is bounded (it is a function of the local dimension and the speed of correlation decay). More generally, this shows that reparametrizations of torus translations having a Liouville component cannot be polynomially mixing.