Continuum-wise hyperbolic homeomorphisms on surfaces

Rodrigo Arruda; Bernardo Carvalho; Alberto Sarmiento

doi:10.3934/dcds.2023125

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2024, Volume 44, Issue 3: 768-790. Doi: 10.3934/dcds.2023125

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Continuum-wise hyperbolic homeomorphisms on surfaces

1.
Universidade Federal de Minas Gerais, Brazil
2.
Università degli Studi di Roma Tor Vergata, Italy

Received: May 2023

Revised: September 2023

Early access: October 25, 2023

Published online: October 25, 2023

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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, Pacífico and Vieitez [7] 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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Mathematics Subject Classification: Primary: 37D10; Secondary: 37B99.

Citation:

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Figure 2. Lemma 2.10

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Figure 8. Two arcs topologically transverse at the point $ x $

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Figure 10.

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Figure 11. Non-comparable stable arcs

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Figure 19. Bi-asymptotic sectors close to $ C^s_ \varepsilon(x) $

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Figure 20. Examples of bi-asymptotic sectors and their spines

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References

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