Ergodic cocycles in hyperbolic and Hadamard spaces
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908524950323200 |
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| author | Bars, Corentin Le |
| author_facet | Bars, Corentin Le |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06797 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ergodic cocycles in hyperbolic and Hadamard spaces Bars, Corentin Le Dynamical Systems Group Theory 37D40 (primary), 37H05, 20F65, 60F15, 22D40 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. |
| title | Ergodic cocycles in hyperbolic and Hadamard spaces |
| topic | Dynamical Systems Group Theory 37D40 (primary), 37H05, 20F65, 60F15, 22D40 |
| url | https://arxiv.org/abs/2509.06797 |