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

Przemysław Berk, Łukasz Kotlewski

TL;DR

This work studies skew products over symmetric interval exchange transformations with a non-integrable cocycle $f(x)=-\frac{1}{x^{a}}+\frac{1}{(1-x)^{a}}$ for $a>1$. The main result shows a coexistence phenomenon: for a typical symmetric IET base, the system is measure-theoretically dissipative (almost every orbit escapes a fixed vertical bound) yet is topologically recurrent (every open rectangle experiences infinite returns along a subsequence of iterates). The analysis combines Diophantine-type structure of IETs, lower bounds on Birkhoff sums of $f'$, a Borel–Cantelli escape argument, and a symmetry-based construction to produce infinite rectangle intersections. These findings reveal a novel contrast between measure-theoretic escape and topological recurrence in the non-integrable cocycle regime, with implications for infinite ergodic theory and Poincaré sections of flows on surfaces.

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 $T^{q_n}_f(A)\cap A\neq\emptyset$.

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

TL;DR

This work studies skew products over symmetric interval exchange transformations with a non-integrable cocycle for . The main result shows a coexistence phenomenon: for a typical symmetric IET base, the system is measure-theoretically dissipative (almost every orbit escapes a fixed vertical bound) yet is topologically recurrent (every open rectangle experiences infinite returns along a subsequence of iterates). The analysis combines Diophantine-type structure of IETs, lower bounds on Birkhoff sums of , a Borel–Cantelli escape argument, and a symmetry-based construction to produce infinite rectangle intersections. These findings reveal a novel contrast between measure-theoretic escape and topological recurrence in the non-integrable cocycle regime, with implications for infinite ergodic theory and Poincaré sections of flows on surfaces.

Abstract

We study the recurrence properties of certain skew products over symmetric interval exchange transformations, including rotations, with cocycles of the form , where . We prove that typically, such systems are dissipative. However, at the same time they are \emph{topologically recurrent}, i.e. for every open rectangle , there exists an infinite sequence such that .
Paper Structure (5 sections, 4 theorems, 25 equations)

This paper contains 5 sections, 4 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Interval exchange transformations and the necessary Diophantine conditions
  3. Lower bounds for the derivative.
  4. Escaping points.
  5. Intersecting rectangles

Key Result

Theorem 1.1

Let $f:[0,1)\to \mathbb{R}$ be given by $f(x)=-\frac{1}{x^{a}}+\frac{1}{(1-x)^{a}}$, with $a>1$. For almost every symmetric interval exchange transformation $T$ on $[0,1)$, the skew product $T_{f}$ on $[0,1)\times\mathbb{R}$ given by has the following properties: In particular, by Property A, $T_f$ is not ergodic with respect to $Leb_{[0,1)}\otimes Leb_{\mathbb{R}}$.

Theorems & Definitions (8)

  • Theorem 1.1
  • Proposition 2.1
  • Lemma 3.1
  • proof
  • proof : Proof of Property A in Theorem \ref{['thm: main']}.
  • Lemma 5.1
  • proof
  • proof : Proof of Property B in Theorem \ref{['thm: main']}.