General harmonic measures for distance-expanding dynamical systems
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866914785325481984 |
|---|---|
| author | Li, Zhiqiang Qiu, Ruicen |
| author_facet | Li, Zhiqiang Qiu, Ruicen |
| contents | Partially motivated by the study of I. Binder, N. Makarov, and S. Smirnov [BMS03] on dimension spectra of polynomial Cantor sets, we initiate the investigation on some general harmonic measures, inspired by Sullivan's dictionary, for distance-expanding dynamical systems. Let $f\colon X\to X$ be an open distance-expanding map on a compact metric space $(X,ρ)$. A Gromov hyperbolic tile graph $Γ$ associated to the dynamical system $(X,f)$ is constructed following the ideas from M. Bonk, D. Meyer [BM17] and P. Haïssinsky, K. M. Pilgrim [HP09]. We consider a class of one-sided random walks associated with $(X,f)$ on $Γ$. They induce a Martin boundary of the tile graph, which may be different from the hyperbolic boundary. We show that the Martin boundary of such a random walk admits a surjection to $X$. We provide a class of examples to show that the surjection may not be a homeomorphism. Such random walks also induce measures on $X$ called harmonic measures. When $ρ$ is a visual metric, we establish an equality between the fractal dimension of the harmonic measure and the asymptotic quantities of the random walk. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02987 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | General harmonic measures for distance-expanding dynamical systems Li, Zhiqiang Qiu, Ruicen Dynamical Systems Metric Geometry Probability Primary: 37D20, Secondary: 31C35, 28A78, 05C81, 31E05 Partially motivated by the study of I. Binder, N. Makarov, and S. Smirnov [BMS03] on dimension spectra of polynomial Cantor sets, we initiate the investigation on some general harmonic measures, inspired by Sullivan's dictionary, for distance-expanding dynamical systems. Let $f\colon X\to X$ be an open distance-expanding map on a compact metric space $(X,ρ)$. A Gromov hyperbolic tile graph $Γ$ associated to the dynamical system $(X,f)$ is constructed following the ideas from M. Bonk, D. Meyer [BM17] and P. Haïssinsky, K. M. Pilgrim [HP09]. We consider a class of one-sided random walks associated with $(X,f)$ on $Γ$. They induce a Martin boundary of the tile graph, which may be different from the hyperbolic boundary. We show that the Martin boundary of such a random walk admits a surjection to $X$. We provide a class of examples to show that the surjection may not be a homeomorphism. Such random walks also induce measures on $X$ called harmonic measures. When $ρ$ is a visual metric, we establish an equality between the fractal dimension of the harmonic measure and the asymptotic quantities of the random walk. |
| title | General harmonic measures for distance-expanding dynamical systems |
| topic | Dynamical Systems Metric Geometry Probability Primary: 37D20, Secondary: 31C35, 28A78, 05C81, 31E05 |
| url | https://arxiv.org/abs/2405.02987 |