Volume 9 (2026)
Malicet, Dominique;  Militon, Emmanuel
Random actions of homeomorphisms of Cantor sets embedded in a line and Tits alternative
Annales Henri Lebesgue, Volume 9 (2026), pp. 33-64

Metadata

DOI10.5802/ahl.259
Keywords Random walks ,  Homeomorphisms ,  Cantor sets ,  Tits alternative

Abstract

In 2000, Margulis proved that any group of homeomorphisms of the circle either preserves a probability measure on the circle or contains a free subgroup on two generators, which is reminiscent of the Tits alternative for linear groups. In this article, we prove an analogous statement for groups of locally monotonic homeomorphisms of a compact subset of $\mathbb{R}$. The proof relies on dynamical properties of random walks on the group, which may be of independent interest.


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