Statistical stability for systems semi-conjugate to pre-piecewise \textit{convex or expanding} maps with countably many branches
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909047987372032 |
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| author | Lucena, Rafael |
| author_facet | Lucena, Rafael |
| contents | We investigate the statistical stability of a class of dynamical systems semi-conjugate to pre-piecewise \textit{convex or expanding} maps with countably many branches. These systems naturally arise in the study of transformations with unbounded derivatives, discontinuities, or infinite Markov partitions; features that pose significant challenges for stability analysis. Specifically, we consider one-parameter families of transformations $\{F_δ\}_{δ\in [0,1)}$ and their corresponding invariant measures $\{μ_δ\}$. We provide general conditions ensuring that the unperturbed measure $μ_0$ is statistically stable, meaning the map $δ\mapsto μ_δ$ is continuous at $δ= 0$ in the appropriate topology. Furthermore, we establish explicit quantitative estimates for the modulus of continuity of $μ_δ$ in terms of the perturbation parameter $δ$. Our results apply to a broad class of maps, including those semi-conjugate to classical examples such as the Gauss and Lüroth maps. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_11878 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Statistical stability for systems semi-conjugate to pre-piecewise \textit{convex or expanding} maps with countably many branches Lucena, Rafael Dynamical Systems Functional Analysis 37A25, 37A10, 37C30, 37D50 We investigate the statistical stability of a class of dynamical systems semi-conjugate to pre-piecewise \textit{convex or expanding} maps with countably many branches. These systems naturally arise in the study of transformations with unbounded derivatives, discontinuities, or infinite Markov partitions; features that pose significant challenges for stability analysis. Specifically, we consider one-parameter families of transformations $\{F_δ\}_{δ\in [0,1)}$ and their corresponding invariant measures $\{μ_δ\}$. We provide general conditions ensuring that the unperturbed measure $μ_0$ is statistically stable, meaning the map $δ\mapsto μ_δ$ is continuous at $δ= 0$ in the appropriate topology. Furthermore, we establish explicit quantitative estimates for the modulus of continuity of $μ_δ$ in terms of the perturbation parameter $δ$. Our results apply to a broad class of maps, including those semi-conjugate to classical examples such as the Gauss and Lüroth maps. |
| title | Statistical stability for systems semi-conjugate to pre-piecewise \textit{convex or expanding} maps with countably many branches |
| topic | Dynamical Systems Functional Analysis 37A25, 37A10, 37C30, 37D50 |
| url | https://arxiv.org/abs/2508.11878 |