An almost sure ergodic theorem for quasistatic dynamical systems

TL;DR

This paper establishes almost sure ergodic theorems for quasistatic dynamical systems, introducing the concept of physical families of measures and verifying these for specific models like expanding systems and dispersing billiards.

Contribution

It extends ergodic theory to quasistatic systems lacking invariant measures by proving new almost sure ergodic theorems and defining physical families of measures.

Findings

Proved almost sure ergodic theorems for quasistatic systems.

Introduced and verified physical families of measures.

Applied results to expanding systems and dispersing billiards.

Abstract

We prove almost sure ergodic theorems for a class of systems called quasistatic dynamical systems. These results are needed, because the usual theorem due to Birkhoff does not apply in the absence of invariant measures. We also introduce the concept of a physical family of measures for a quasistatic dynamical system. These objects manifest themselves, for instance, in numerical experiments. We then verify the conditions of the theorems and identify physical families of measures for two concrete models, quasistatic expanding systems and quasistatic dispersing billiards.