Quasi-compactness of Frobenius-Perron Operator for Piecewise Convex Maps with Countable Branches
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908481558151168 |
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| author | Gora, Pawel Rajput, Aparna |
| author_facet | Gora, Pawel Rajput, Aparna |
| contents | In this paper, we prove the quasi-compactness of the Frobenius-Perron operator for a piecewise convex map $τ$ with a countably infinite number of branches on the interval $I=[0,1]$. We establish that for high enough $n$ iterates of $τ$, $τ^n$ are piecewise expanding. Using the Lasota-Yorke Inequality derived from references \cite{hofbauer1982} and \cite{keller1985}, adapted to meet the assumptions of the Ionescu-Tulcea and Marinescu ergodic theorem, we demonstrate the existence of absolutely continuous invariant measure (ACIM) $μ$ for $τ$, the exactness of the dynamical system $(I, τ,μ)$ and the quasi-compactness of Frobenius-Perron operator $P_τ$ induced by $τ$. The last fact implies a multitude of strong ergodic properties of $τ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19929 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quasi-compactness of Frobenius-Perron Operator for Piecewise Convex Maps with Countable Branches Gora, Pawel Rajput, Aparna Dynamical Systems 37A05, 37E05 In this paper, we prove the quasi-compactness of the Frobenius-Perron operator for a piecewise convex map $τ$ with a countably infinite number of branches on the interval $I=[0,1]$. We establish that for high enough $n$ iterates of $τ$, $τ^n$ are piecewise expanding. Using the Lasota-Yorke Inequality derived from references \cite{hofbauer1982} and \cite{keller1985}, adapted to meet the assumptions of the Ionescu-Tulcea and Marinescu ergodic theorem, we demonstrate the existence of absolutely continuous invariant measure (ACIM) $μ$ for $τ$, the exactness of the dynamical system $(I, τ,μ)$ and the quasi-compactness of Frobenius-Perron operator $P_τ$ induced by $τ$. The last fact implies a multitude of strong ergodic properties of $τ$. |
| title | Quasi-compactness of Frobenius-Perron Operator for Piecewise Convex Maps with Countable Branches |
| topic | Dynamical Systems 37A05, 37E05 |
| url | https://arxiv.org/abs/2406.19929 |