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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2510.22246 |
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| _version_ | 1866917042395807744 |
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| author | Lee, Keonhee Lee, Seunghee Morales, C. A. |
| author_facet | Lee, Keonhee Lee, Seunghee Morales, C. A. |
| contents | We introduce the {\em $μ$-topological stability}. This is a type of stability depending on the measure $μ$ different from the set-valued approach \cite{lm}. We prove that the map $f$ is $m_p$-topologically stable if and only if $p$ is a topologically stable point ($m_p$ is the Dirac measure supported on $p$). On closed manifolds of dimension $\geq2$ we prove that every $μ$-topologically stable map has the $μ$-shadowing property for finitely supported measures $μ$. Moreover the $μ$-topological stability is invariant under topological conjugacy or restriction to compact invariant sets of full measure. We also prove for expansive maps that the set of measures $μ$ for which the map is $μ$-topologically stable is convex. We analyze the relationship between $μ$-topological stability for absolutely continuous measures. In the nonatomic case we show that the $μ$-topological stability implies the set-valued stability approach in \cite{lm}. Finally, we show that every expansive map with the weak $μ$-shadowing property (c.f. \cite{lr}) is $μ$-topologically stable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_22246 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Topological stability from a measurable viewpoint Lee, Keonhee Lee, Seunghee Morales, C. A. Dynamical Systems Primary 37B25, Secondary 37C50 We introduce the {\em $μ$-topological stability}. This is a type of stability depending on the measure $μ$ different from the set-valued approach \cite{lm}. We prove that the map $f$ is $m_p$-topologically stable if and only if $p$ is a topologically stable point ($m_p$ is the Dirac measure supported on $p$). On closed manifolds of dimension $\geq2$ we prove that every $μ$-topologically stable map has the $μ$-shadowing property for finitely supported measures $μ$. Moreover the $μ$-topological stability is invariant under topological conjugacy or restriction to compact invariant sets of full measure. We also prove for expansive maps that the set of measures $μ$ for which the map is $μ$-topologically stable is convex. We analyze the relationship between $μ$-topological stability for absolutely continuous measures. In the nonatomic case we show that the $μ$-topological stability implies the set-valued stability approach in \cite{lm}. Finally, we show that every expansive map with the weak $μ$-shadowing property (c.f. \cite{lr}) is $μ$-topologically stable. |
| title | Topological stability from a measurable viewpoint |
| topic | Dynamical Systems Primary 37B25, Secondary 37C50 |
| url | https://arxiv.org/abs/2510.22246 |