Convergence of pushforward measures for certain countably piecewise linear Markov maps
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866916916299300864 |
|---|---|
| author | Kalocsai, Zoltán |
| author_facet | Kalocsai, Zoltán |
| contents | We study piecewise linear Markov maps, with countable Markov partitions, inspired by a problem of the Miklós Schweitzer competition in 2022. We introduce $\ell$-Markov partitions and apply ideas of symbolic dynamics to our systems, relating them to Markov shifts. We survey how the Frobenius--Perron operators of these systems can be represented by matrices, and adapt results to countable alphabets. We apply these statements to prove a convergence theorem on the pushforwards of absolutely continuous measures. This enables us to prove a variety of useful ergodic properties of our maps and study even non-$σ$-finite absolutely continuous invariant measures. We explain how our results are not implied by previous ones and apply the convergence theorem to solve the original problem in the competition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18172 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence of pushforward measures for certain countably piecewise linear Markov maps Kalocsai, Zoltán Dynamical Systems Primary: 37A50 Secondary: 37A05, 37A25, 37A30, 37A40, 37B10 We study piecewise linear Markov maps, with countable Markov partitions, inspired by a problem of the Miklós Schweitzer competition in 2022. We introduce $\ell$-Markov partitions and apply ideas of symbolic dynamics to our systems, relating them to Markov shifts. We survey how the Frobenius--Perron operators of these systems can be represented by matrices, and adapt results to countable alphabets. We apply these statements to prove a convergence theorem on the pushforwards of absolutely continuous measures. This enables us to prove a variety of useful ergodic properties of our maps and study even non-$σ$-finite absolutely continuous invariant measures. We explain how our results are not implied by previous ones and apply the convergence theorem to solve the original problem in the competition. |
| title | Convergence of pushforward measures for certain countably piecewise linear Markov maps |
| topic | Dynamical Systems Primary: 37A50 Secondary: 37A05, 37A25, 37A30, 37A40, 37B10 |
| url | https://arxiv.org/abs/2508.18172 |