Can points of bounded orbits surround points of unbounded orbits ?
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929323491983360 |
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| author | Mai, Jiehua Shi, Enhui Yan, Kesong Zeng, Fanping |
| author_facet | Mai, Jiehua Shi, Enhui Yan, Kesong Zeng, Fanping |
| contents | We show a somewhat surprising result: if $E$ is a disk in the plane $\mathbb R^2$, then there is a homeomorphism $h:\mathbb R^2\rightarrow\mathbb R^2$ such that, for every $x\in\partial E$, the orbit $O(x, h)$ is bounded, but for every $y\in {\rm Int}(E)$, the orbit $O(y, h)$ is doubly divergent. To prove this, we define a class of homeomorphisms on the square $[-1, 1]^2$, called normally rising homeomorphisms, and show that a normally rising homeomorphism can have very complex $ω$-limit sets and $α$-limt sets, though the homeomorphism itself looks very simple. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13642 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Can points of bounded orbits surround points of unbounded orbits ? Mai, Jiehua Shi, Enhui Yan, Kesong Zeng, Fanping Dynamical Systems 37E30 We show a somewhat surprising result: if $E$ is a disk in the plane $\mathbb R^2$, then there is a homeomorphism $h:\mathbb R^2\rightarrow\mathbb R^2$ such that, for every $x\in\partial E$, the orbit $O(x, h)$ is bounded, but for every $y\in {\rm Int}(E)$, the orbit $O(y, h)$ is doubly divergent. To prove this, we define a class of homeomorphisms on the square $[-1, 1]^2$, called normally rising homeomorphisms, and show that a normally rising homeomorphism can have very complex $ω$-limit sets and $α$-limt sets, though the homeomorphism itself looks very simple. |
| title | Can points of bounded orbits surround points of unbounded orbits ? |
| topic | Dynamical Systems 37E30 |
| url | https://arxiv.org/abs/2404.13642 |