Entrance Time and R\'enyi Entropy

TL;DR

This paper extends the understanding of recurrence and entrance times in ergodic systems, showing that entrance times grow exponentially at a rate equal to the entropy under certain conditions, and explores connections to Re9nyi entropy.

Contribution

It establishes the exponential growth rate of entrance times equals the entropy in ergodic systems with natural convergence conditions, extending classical results.

Findings

Entrance times grow exponentially at the entropy rate in ergodic systems.

Convergence of entrance times is in probability under natural conditions.

Links between Re9nyi entropy and sums over entrance times are demonstrated.

Abstract

For ergodic systems with generating partitions, the well known result of Ornstein and Weiss shows that the exponential growth rate of the recurrence time is almost surely equal to the metric entropy. Here we look at the exponential growth rate of entrance times, and show that it equals the entropy, where the convergence is in probability in the product measure. This is however under the assumptions that the limiting entrance times distribution exists almost surely. This condition looks natural in the light of an example by Shields in which the limsup in the exponential growth rate is infinite almost everywhere but where the limiting entrance times do not exist. We then also consider $\phi$-mixing systems and prove a result connecting the R\'enyi entropy to sums over the entrance times orbit segments.