Ergodic cocycles in hyperbolic and Hadamard spaces

2025/09/08 by Corentin Le Bars, Bars, Corentin Le
Mathematics · #20F65 #22D40 #37D40 (primary) #37H05 #60F15 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Mathematical Dynamics and Fractals #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2509.06797

openalex publication_date 2025/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider discrete random dynamical systems induced by a non-elementary group action on a non-proper hyperbolic space. We prove that if the system is ergodic and satisfies the ``asymptotic past and future independence condition'' as defined by Bader and Furman, the associated ergodic cocycle converges to the Gromov boundary almost surely. If the cocycle has finite first moment, we show that its drift is positive. Using hyperbolic models introduced by Petyt-Spriano-Zalloum, we prove analogous statements for groups acting on Hadamard spaces with a pair of contracting elements.

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