Stable local dynamics: expansion, quasi-conformality and ergodicity

Abstract

In this paper, we study the stable ergodicity of the action of groups of diffeomorphisms on smooth manifolds. The existence of such actions is known only on one-dimensional manifolds. The aim of this paper is to overcome this restriction and to give a method for constructing higher-dimensional examples.In particular, we show that every closed manifold admits stably ergodic finitely generated group actions by diffeomorphisms of class C 1+α . 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 a new local and stable mechanism/phenomenon which we call quasi-conformal blender. This tool stably provides quasi-conformal orbits and yields stable local ergodicity. The quasi-conformal blender is developed in the context of pseudo-semigroup actions of locally defined smooth diffeomorphisms which allows for applications in several different settings.