G\'en\'ericit\'e au sens probabiliste dans les diff\'eomorphismes du cercle

TL;DR

This paper explores the probabilistic dynamics of circle diffeomorphisms using Malliavin-Shavgulidze measures, revealing that hyperbolic behavior is typical in a probabilistic setting and bridging stochastic processes with one-dimensional dynamics.

Contribution

It introduces and analyzes Malliavin-Shavgulidze measures on circle diffeomorphisms, connecting stochastic measures with dynamical properties in a novel way.

Findings

Hyperbolic dynamics are prevalent in the probabilistic framework.

Malliavin-Shavgulidze measures are quasi-invariant on diffeomorphism groups.

Probabilistic and topological dynamics show significant similarities.

Abstract

What kind of dynamics do we observe in general on the circle? It depends somehow on the interpretation of "in general". Everything is quite well understood in the topological (Baire) setting, but what about the probabilistic sense? The main problem is that on an infinite dimensional group there is no analogue of the Lebesgue measure, in a strict sense. There are however some analogues, quite natural and easy to define: the Malliavin-Shavgulidze measures provide an example and constitute the main character of this text. The first results show that there is no actual disagreement of general features of the dynamics in the topological and probabilistic frames: it is the realm of hyperbolicity! The most interesting questions remain however unanswered... This work, coming out from the author's Ph.D. thesis, constitutes an opportunity to review interesting results in mathematical topics that could interact more often: stochastic processes and one-dimensional dynamics. After an introductory overview, the following three chapters are a pedagogical summary of classical results about measure theory on topological groups, Brownian Motion, theory of circle diffeomorphisms. Then we present the construction of the Malliavin-Shavgulidze measures on the space of interval and circle $C^1$ diffeomorphisms, and discuss their key property of quasi-invariance. The last chapter is devoted to the study of dynamical features of a random Malliavin-Shavgulidze diffeomorphism.