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arXiv · 2305.09023

Continuum-wise hyperbolic homeomorphisms on surfaces

Rodrigo Arruda·Bernardo Carvalho·Alberto Sarmiento
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Abstract

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, Pac\'ifico 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.

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Rodrigo Arruda, Bernardo Carvalho, Alberto Sarmiento. 2023-05-15. Continuum-wise hyperbolic homeomorphisms on surfaces. https://doi.org/10.3934/dcds.2023125

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