Potential well in Poincar\'e recurrence

TL;DR

This paper reviews the concept of potential well in dynamical systems and introduces new exponential approximation results for recurrence times in mixing processes, along with an analysis of the potential well's positivity.

Contribution

It provides new exponential approximation formulas for recurrence times in $\

Findings

Explicit and sharp error bounds for hitting and return time approximations.

New results on the uniform positivity of the potential well.

Enhanced understanding of recurrence time scaling in mixing processes.

Abstract

From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence time distributions in mixing stochastic processes and dynamical systems. Besides providing a review of the different scaling factors used in the literature in recurrence times, the present work contributes with two new results: (1) for $\phi$-mixing and $\psi$-mixing processes, we give a new exponential approximation for hitting and return times using the potential well as scaling parameter. The error terms are explicit and sharp. (2) We analyse the uniform positivity of the potential well.