Continuum-wise hyperbolic homeomorphisms on surfaces
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917809254039552 |
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| author | Arruda, Rodrigo Carvalho, Bernardo Sarmiento, Alberto |
| author_facet | Arruda, Rodrigo Carvalho, Bernardo Sarmiento, Alberto |
| contents | This paper discusses the dynamics of continuum-wise hyperbolic surface homeomorphisms. We prove that $cw_F$-hyperbolic surface homeomorphisms containing only a finite set of spines are $cw_2$-hyperbolic. In the case of $cw_3$-hyperbolic homeomorphisms we prove the finiteness of spines and, hence, that $cw_3$-hyperbolicity implies $cw_2$-hyperbolicity. In the proof, we adapt techniques of Hiraide [11] and Lewowicz [15] in the case of expansive surface homeomorphisms to prove that local stable/unstable continua of $cw_F$-hyperbolic homeomorphisms are continuous arcs. We also adapt techniques of Artigue, Pacífico and Vieitez [6] in the case of N-expansive surface homeomorphisms to prove that the existence of spines is strongly related to the existence of bi-asymptotic sectors and conclude that spines are necessarily isolated from other spines. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_09023 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Continuum-wise hyperbolic homeomorphisms on surfaces Arruda, Rodrigo Carvalho, Bernardo Sarmiento, Alberto Dynamical Systems Primary 37B45, Secondary 37D10 This paper discusses the dynamics of continuum-wise hyperbolic surface homeomorphisms. We prove that $cw_F$-hyperbolic surface homeomorphisms containing only a finite set of spines are $cw_2$-hyperbolic. In the case of $cw_3$-hyperbolic homeomorphisms we prove the finiteness of spines and, hence, that $cw_3$-hyperbolicity implies $cw_2$-hyperbolicity. In the proof, we adapt techniques of Hiraide [11] and Lewowicz [15] in the case of expansive surface homeomorphisms to prove that local stable/unstable continua of $cw_F$-hyperbolic homeomorphisms are continuous arcs. We also adapt techniques of Artigue, Pacífico and Vieitez [6] in the case of N-expansive surface homeomorphisms to prove that the existence of spines is strongly related to the existence of bi-asymptotic sectors and conclude that spines are necessarily isolated from other spines. |
| title | Continuum-wise hyperbolic homeomorphisms on surfaces |
| topic | Dynamical Systems Primary 37B45, Secondary 37D10 |
| url | https://arxiv.org/abs/2305.09023 |