Ergodic theory of generic continuous maps

TL;DR

This paper investigates the ergodic properties of generic continuous maps and homeomorphisms on compact manifolds, revealing contrasting behaviors in terms of Birkhoff averages and physical measures, especially in higher dimensions.

Contribution

It proves that generic homeomorphisms have convergent Birkhoff averages almost everywhere but lack physical measures in higher dimensions, contrasting with differentiable dynamics.

Findings

Generic homeomorphisms have convergent Birkhoff averages at almost every point.

In higher dimensions, generic homeomorphisms lack physical measures.

For conjugates of expanding circle maps, Birkhoff averages diverge almost everywhere.

Abstract

We study the ergodic properties of generic continuous dynamical systems on compact manifolds. As a main result we prove that generic homeomorphisms have convergent Birkhoff averages under continuous observables at Lebesgue almost every point. In spite of this, when the underlying manifold has dimension greater than one, generic homeomorphisms have no physical measure --- a somewhat strange result which stands in sharp contrast to current trends in generic differentiable dynamics. Similar results hold for generic continuous maps. To further explore the mysterious behaviour of $C^0$ generic dynamics, we also study the ergodic properties of continuous maps which are conjugated to expanding circle maps. In this context, generic maps have divergent Birkhoff averages along orbits starting from Lebesgue almost every point.