Convergence of Dynamics and the Perron-Frobenius Operator

TL;DR

This paper establishes a fundamental link between the asymptotic behavior of dynamical systems and the properties of their Perron-Frobenius operators, showing that strong operator convergence corresponds to setwise convergence of the system.

Contribution

It provides a new characterization of the convergence of Perron-Frobenius operators in terms of setwise convergence and mixing properties of the underlying dynamical system.

Findings

Strong convergence of Perron-Frobenius operators is equivalent to setwise convergence of the dynamics.

The convergence is characterized by a uniform mixing-like property.

The results unify operator-theoretic and measure-theoretic perspectives on dynamical systems.

Abstract

We complete the picture how the asymptotic behavior of a dynamical system is reflected by properties of the associated Perron-Frobenius operator. Our main result states that strong convergence of the powers of the Perron-Frobenius operator is equivalent to setwise convergence of the underlying dynamic in the measure algebra. This situation is furthermore characterized by a uniform mixing-like property of the system.