Coexistence of two contrasting recurrence properties of certain non-integrable cocycles

TL;DR

This paper investigates the recurrence behavior of certain non-integrable cocycles over interval exchange transformations, revealing that such systems are generally dissipative yet topologically recurrent, highlighting contrasting dynamical properties.

Contribution

It demonstrates the coexistence of dissipative and topologically recurrent properties in non-integrable cocycles over interval exchange transformations.

Findings

Systems are typically dissipative.

Systems are topologically recurrent.

Recurrence occurs for every open rectangle.

Abstract

We study the recurrence properties of certain skew products over symmetric interval exchange transformations, including rotations, with cocycles of the form $f(x)=-\frac{1}{x^a}+\frac{1}{(1-x)^a}$, where $a>1$. We prove that typically, such systems are dissipative. However, at the same time they are \emph{topologically recurrent}, i.e. for every open rectangle $A\subset[0,1)\times \R$, there exists an infinite sequence $(q_n)_{n=1}^{\infty}$ such that $T^{q_n}_f(A)\cap A\neq\emptyset$.