Continuum-wise hyperbolicity
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2020
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| _version_ | 1866913554545770496 |
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| author | Artigue, Alfonso Carvalho, Bernardo Cordeiro, Welington Vieitez, José |
| author_facet | Artigue, Alfonso Carvalho, Bernardo Cordeiro, Welington Vieitez, José |
| contents | We introduce continuum-wise hyperbolicity, a generalization of hyperbolicity with respect to the continuum theory. We discuss similarities and differences between topological hyperbolicity and continuum-wise hyperbolicity. A shadowing lemma for cw-hyperbolic homeomorphisms is proved in the form of the L-shadowing property and a Spectral Decomposition is obtained in this scenario. In the proof we generalize the construction of Fathi \cite{Fat89} of a hyperbolic metric using only cw-expansivity, obtaining a hyperbolic cw-metric. We also introduce cwN-hyperbolicity, exhibit examples of these systems for arbitrarily large $N\in\mathbb{N}$ and obtain further dynamical properties of these systems such as finiteness of periodic points with the same period. We prove that homeomorphisms of $\mathbb{S}^2$ that are induced by topologically hyperbolic homeomorphisms of $\mathbb{T}^2$ are continuum-wise-hyperbolic and topologically conjugate to linear cw-Anosov diffeomorphisms of $\mathbb{S}^2$, being in particular cw2-hyperbolic. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2011_08147 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Continuum-wise hyperbolicity Artigue, Alfonso Carvalho, Bernardo Cordeiro, Welington Vieitez, José Dynamical Systems We introduce continuum-wise hyperbolicity, a generalization of hyperbolicity with respect to the continuum theory. We discuss similarities and differences between topological hyperbolicity and continuum-wise hyperbolicity. A shadowing lemma for cw-hyperbolic homeomorphisms is proved in the form of the L-shadowing property and a Spectral Decomposition is obtained in this scenario. In the proof we generalize the construction of Fathi \cite{Fat89} of a hyperbolic metric using only cw-expansivity, obtaining a hyperbolic cw-metric. We also introduce cwN-hyperbolicity, exhibit examples of these systems for arbitrarily large $N\in\mathbb{N}$ and obtain further dynamical properties of these systems such as finiteness of periodic points with the same period. We prove that homeomorphisms of $\mathbb{S}^2$ that are induced by topologically hyperbolic homeomorphisms of $\mathbb{T}^2$ are continuum-wise-hyperbolic and topologically conjugate to linear cw-Anosov diffeomorphisms of $\mathbb{S}^2$, being in particular cw2-hyperbolic. |
| title | Continuum-wise hyperbolicity |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2011.08147 |