Expanding actions: minimality and ergodicity

TL;DR

This paper proves that expanding minimal semigroup actions of certain diffeomorphisms on compact manifolds are robustly minimal or ergodic, with blending regions playing a key role in these properties.

Contribution

It establishes the robustness of minimality and ergodicity for expanding semigroup actions, linking local blending regions to global dynamical properties.

Findings

Robust minimality for expanding $C^1$ diffeomorphism actions.

Ergodicity with respect to Lebesgue measure for conformal actions.

Blending regions ensure robustness of minimality and ergodicity.

Abstract

We prove that every expanding minimal semigroup action of $C^1$ diffeomorphisms of a compact manifold (resp. $C^{1+\alpha}$ conformal) is robustly minimal (resp. ergodic with respect to Lebesgue measure). We also show how, locally, a blending region yields the robustness of the minimality and implies ergodicity.