Probabilistic Tits alternative for circle diffeomorphisms

TL;DR

The paper proves a probabilistic Tits alternative for circle dynamics by showing that two independent random walks on proximal subgroups of $Homeo}_+(S^1)$ or $Diff^1}_+(S^1)$ almost surely generate free groups at large times. It develops exponential contraction-in-mean and Hölder regularity results for random walks on $Diff^1}_+(S^1)$, enabling an Aoun-style ping-pong construction in the diffeomorphism setting. For homeomorphisms, a density-1 ping-pong phenomenon is obtained under no invariant measure assumption, while for diffeomorphisms with moment conditions a stronger exponential convergence framework is established. Overall, the work extends probabilistic Tits-type results from linear groups to circle dynamics, connecting contracting random dynamics, repulsors, and ping-pong arguments to ensure freeness of random subgroup generation in a probabilistic sense.

Abstract

Let $μ_1, μ_2$ be probability measures on $\mathrm{Diff}^1_+(S^1)$ satisfying a suitable moment condition and such that their supports genererate discrete groups acting proximally on $S^1$. Let $(f^n_ω)_{n \in \mathbb{N}}, (f^n_{ω'})_{n \in \mathbb{N}}$ be two independent realizations of the random walk driven by $μ_1, μ_2$ respectively. We show that almost surely there is an $N \in \mathbb{N}$ such that for all $n \geq N$ the elements $f^n_ω, f^n_{ω'}$ generate a nonabelian free group. The proof is inspired by the strategy by R. Aoun for linear groups and uses work of A. Gorodetski, V. Kleptsyn and G. Monakov, and of P. Barrientos and D. Malicet. A weaker (and easier) statement holds for measures supported on $\mathrm{Homeo}_+(S^1)$ with no moment conditions.

Figures (1)

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Theorems & Definitions (34)

• Theorem 1.1: G. Margulis margulisCircle, see also V. Antonov antonov
• Theorem 1.2: R. Aoun aounaoun1
• Theorem 1
• Corollary 2
• Theorem 3
• Proposition 2.1: malicet
• Theorem 2.2: see ghysCircle
• Theorem 2.3: deroinKleptsynNavas
• Theorem 2.4
• Proposition 2.5