Random actions of homeomorphisms of Cantor sets embedded in a line and Tits alternative

2023/04/17 by Malicet, Dominique, Militon, Emmanuel
#Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2304.08070

Abstract

In 2000, Margulis proved that any group of homeomorphisms of the circle either preserves a probabilty measure on the circle or contains a free subgroup in two generators, which is reminiscent of the Tits alternatve for linear groups. In this article, we prove an analogous statement for groups of locally monotonic homeomorphisms of a compact subset of R.

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• Short proofs of theorems of Malyutin and Margulis
• Probabilistic Tits alternative for circle diffeomorphisms
• From invariant measures to orbit equivalence, via locally finite groups
• Dynamical simplices and minimal homeomorphisms
• Random walks on Homeo(S¹)