Stable local dynamics: expansion, quasi-conformality and ergodicity

TL;DR

This paper introduces a geometric method called the quasi-conformal blender to construct higher-dimensional stably ergodic group actions on manifolds, expanding the understanding of stable ergodicity beyond one-dimensional cases.

Contribution

It develops the quasi-conformal blender mechanism and demonstrates its application to create stable ergodic actions on all closed manifolds, including algebraic actions near the identity.

Findings

Every closed manifold admits stably ergodic finitely generated diffeomorphism actions.

The stable ergodicity of certain algebraic actions, such as generic matrix pairs near the identity, is established.

Introduction of the quasi-conformal blender as a new tool for proving stable local ergodicity.

Abstract

In this paper, we study stable ergodicity of the action of groups of diffeomorphisms on smooth manifolds. Such actions are known to exist only on one-dimensional manifolds. The aim of this paper is to introduce a geometric method to overcome this restriction and to construct higher dimensional examples. In particular, we show that every closed manifold admits stably ergodic finitely generated group actions by diffeomorphisms of class $C^{1+\alpha}$. We also prove the stable ergodicity of certain algebraic actions, including the natural action of a generic pair of matrices near the identity on a sphere of arbitrary dimension. These are consequences of the quasi-conformal blender, a local and stable mechanism/phenomenon introduced in this paper, which encapsulates our method for proving stable local ergodicity by providing quasi-conformal orbits with fine controlled geometry. The quasi-conformal blender is developed in the context of pseudo-semigroup actions of locally defined smooth diffeomorphisms, which allows for applications in diverse settings.