Dynamics on Hyperspace of Pointwise Periodic Homeomorphisms
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910062326317056 |
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| author | Naghmouchi, Issam |
| author_facet | Naghmouchi, Issam |
| contents | In this paper, we first prove that the topological entropy of induced map of any distal homeomorphism of a compact metric space is null. Then we consider induced map $2^f$ of an arbitrary pointwise periodic homeomorphism $f:X\to X$ of a compact metric space $X$, we show that the set of almost periodic points coincides with the set of uniformly recurrent points, i.e. $AP(2^f)=UR(2^f)$. Furthermore, we prove that inside any infinite $ω$-limit set $ω_{2^f}(A)$ there is a unique minimal set and this minimal set is an adding machine. As a consequence, $(2^X,2^f)$ has no Devaney chaotic subsystems. In contrast to these rigidity properties, we obtain some results with chaotic flavor. In fact, we prove the following dichotomy, the hyperspace system $(2^X,2^f)$ is either equicontinuous or choatic with respect to Li-Yorke chaos and $ω$-chaos. It is shown that the later case occurs if and only if $R(2^f)\setminus AP(2^f)\neq\emptyset$. This enables us to provide simple examples of pointwise periodic homeomorphisms with chaotic induced systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_08836 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dynamics on Hyperspace of Pointwise Periodic Homeomorphisms Naghmouchi, Issam Dynamical Systems In this paper, we first prove that the topological entropy of induced map of any distal homeomorphism of a compact metric space is null. Then we consider induced map $2^f$ of an arbitrary pointwise periodic homeomorphism $f:X\to X$ of a compact metric space $X$, we show that the set of almost periodic points coincides with the set of uniformly recurrent points, i.e. $AP(2^f)=UR(2^f)$. Furthermore, we prove that inside any infinite $ω$-limit set $ω_{2^f}(A)$ there is a unique minimal set and this minimal set is an adding machine. As a consequence, $(2^X,2^f)$ has no Devaney chaotic subsystems. In contrast to these rigidity properties, we obtain some results with chaotic flavor. In fact, we prove the following dichotomy, the hyperspace system $(2^X,2^f)$ is either equicontinuous or choatic with respect to Li-Yorke chaos and $ω$-chaos. It is shown that the later case occurs if and only if $R(2^f)\setminus AP(2^f)\neq\emptyset$. This enables us to provide simple examples of pointwise periodic homeomorphisms with chaotic induced systems. |
| title | Dynamics on Hyperspace of Pointwise Periodic Homeomorphisms |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2512.08836 |