Many works in the theory of dynamical systems deal with the question of whether a given system is indecomposable or can be decomposed into say `smaller' systems. The concept of indecomposability can be studied from different aspects. In this talk, our focus is on the stable ergodicity of smooth (semi)group actions on manifolds (w.r.t natural volume) which concerns the indecomposability from a measure theoretical point of view. Examples of stably ergodic actions were known in dimension one where the conformality of smooth one-dimensional maps plays a crucial role. Generalizations of one-dimensional arguments work for ergodicity of conformal actions in higher dimensions. However, such generalizations do not provide stably ergodicity, since conformality is not stable in higher dimensions. I will start the talk with a brief introduction of the subject and discuss the ideas of proofs in one-dimensional cases and obstructions for generalization to higher dimensions. Then, I will introduce a local mechanism based on a covering property that guarantees stable `quasi-conformality" for certain higher-dimensional actions and can be used to derive stable ergodicity of such actions. If time permits I will discuss an application of these tools to provide stable ergodic actions on spheres induced by the matrices.

Reference:
A. Fakhari, M. Nassiri and H. Rajabzadeh, Stable local dynamics: expansion, quasi-conformality, and ergodicity, arXiv: 2102.09259 (2021).

[IPM Youth Seminars on Topology and Dynamics]
Venue: meet.google.com/qix-iqiy-txt