Exponential return times in a zero-entropy process

TL;DR

This paper constructs a zero-entropy process that exhibits exponential return time laws, challenging previous results limited to positive-entropy, strongly mixing processes, thus expanding understanding of recurrence behaviors.

Contribution

It introduces a zero-entropy, weakly mixing process with exponential return time laws, a phenomenon previously known only in positive-entropy systems with strong mixing.

Findings

Constructed a zero-entropy process with exponential return times

Demonstrated exponential laws hold in almost every point

Extended the class of systems known to exhibit exponential return times

Abstract

We construct a zero-entropy weakly mixing finite-valued process with the exponential limit law for return resp. hitting times. This limit law is obtained in almost every point, taking the limit along the full sequence of cylinders around the point. Till now, the exponential limit law for return resp. hitting times, taking the limit along the full sequence of cylinders, have been obtained only in positive-entropy processes satisfying some strong mixing conditions of Rossenblatt type.