Abstract:Let $\Gamma$ be a non-elementary Gromov-hyperbolic group, and $\partial \Gamma$ denote its Gromov boundary. We consider $\Gamma$-invariant proper $\delta$-hyperbolic, quasi-convex metric $d$ on $\Gamma$, and the associated Patterson-Sullivan measure class $[\nu]$ on $\partial^{(2)}\Gamma$, and its square $[\nu\times\nu]$ on $\partial^{(2)}\Gamma$ -- the space of distinct pairs of points on the boundary. We construct an analogue of a geodesic flow to study ergodicity properties of the $\Gamma$-actions on $(\partial\Gamma,\nu)$ and on $(\partial^{(2)}\Gamma,[\nu\times\nu])$. We also prove some ergodic theorems for $\Gamma$-actions guided by the geometry of $(\Gamma,d)$.
| Comments: | This expended version includes some new results, and more details of the constructions and the proofs |
| Subjects: | Dynamical Systems (math.DS) |
| MSC classes: | 37Axx, 37Dxx |
| Cite as: | arXiv:1707.02020 [math.DS] |
| (or arXiv:1707.02020v3 [math.DS] for this version) | |
| https://doi.org/10.48550/arXiv.1707.02020 arXiv-issued DOI via DataCite |
From: Alex Furman [view email]
[v1]
Fri, 7 Jul 2017 02:20:27 UTC (26 KB)
[v2]
Tue, 8 Aug 2017 16:15:46 UTC (26 KB)
[v3]
Sat, 23 May 2026 22:52:14 UTC (38 KB)
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