Patterson-Sullivan construction of equilibrium states and weighted counting in nonpositive curvature
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866918142399217664 |
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| author | Wu, Weisheng |
| author_facet | Wu, Weisheng |
| contents | Consider the geodesic flow on a closed rank one manifold of nonpositive curvature. For certain Hölder continuous potential, there exists a unique equilibrium state by \cite{BCFT}. In this paper, we introduce the notions of core limit set, regular radial limit set and uniformly recurrent and regular vectors, and then construct a family of Patterson-Sullivan measures on the boundary at infinity in two separate settings. Then we give an explicit construction of the above unique equilibrium state using Patterson-Sullivan measures. This enables us to prove the Bernoulli property of the equilibrium states. Using the Patterson-Sullivan construction and mixing properties of equilibrium states, we count the number of free homotopy classes with weights in nonpositive curvature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09924 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Patterson-Sullivan construction of equilibrium states and weighted counting in nonpositive curvature Wu, Weisheng Dynamical Systems Consider the geodesic flow on a closed rank one manifold of nonpositive curvature. For certain Hölder continuous potential, there exists a unique equilibrium state by \cite{BCFT}. In this paper, we introduce the notions of core limit set, regular radial limit set and uniformly recurrent and regular vectors, and then construct a family of Patterson-Sullivan measures on the boundary at infinity in two separate settings. Then we give an explicit construction of the above unique equilibrium state using Patterson-Sullivan measures. This enables us to prove the Bernoulli property of the equilibrium states. Using the Patterson-Sullivan construction and mixing properties of equilibrium states, we count the number of free homotopy classes with weights in nonpositive curvature. |
| title | Patterson-Sullivan construction of equilibrium states and weighted counting in nonpositive curvature |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2509.09924 |