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This article proves a correlation between groups of homeomorphisms and measure preservation, suggesting new insights into dynamical systems.
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 ℝ . The proof relies on dynamical properties of random walks on the group, which may be of independent interest.
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Malicet et al. (2026) studied this question.
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