Quantitative recurrence of some dynamical systems with an infinite measure in dimension one

TL;DR

This paper investigates the asymptotic behavior of first return times in one-dimensional dynamical systems with infinite invariant measures, focusing on $ Z$-extensions of subshifts of finite type and using a probabilistic model for analysis.

Contribution

It provides new insights into the recurrence properties of infinite measure-preserving systems, particularly for $ Z$-extensions of subshifts of finite type, with a probabilistic approach.

Findings

Asymptotic behavior characterized for return times in infinite measure systems

Extension of recurrence results to $ Z$-extensions of subshifts of finite type

Probabilistic model clarifies the proof strategy and dynamics

Abstract

We are interested in the asymptotic behaviour of the first return time of the orbits of a dynamical system into a small neighbourhood of their starting points. We study this quantity in the context of dynamical systems preserving an infinite measure. More precisely, we consider the case of $\mathbb{Z}$-extensions of subshifts of finite type. We also consider a toy probabilistic model to enlight the strategy of our proofs.