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Fully essential dynamics for area-preserving surface homeomorphisms

Published online by Cambridge University Press:  14 February 2017

ANDRES KOROPECKI  and
FABIO ARMANDO TAL
ANDRES KOROPECKI
Affiliation:
Universidade Federal Fluminense, Instituto de Matemática e Estatística, Rua Mário Santos Braga S/N, 24020-140 Niteroi, RJ, Brasil email ak@id.uff.br
FABIO ARMANDO TAL
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brazil email fabiotal@ime.usp.br
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Abstract

We study the interplay between the dynamics of area-preserving surface homeomorphisms homotopic to the identity and the topology of the surface. We define fully essential dynamics and generalize the results previously obtained on strictly toral dynamics to surfaces of higher genus. Non-fully essential dynamics are, in a way, reducible to surfaces of lower genus, while in the fully essential case the dynamics is decomposed into a disjoint union of periodic bounded disks and a complementary invariant externally transitive continuum $C$. When the Misiurewicz–Ziemian rotation set has non-empty interior the dynamics is fully essential, and the set $C$ is (externally) sensitive on initial conditions and realizes all of the rotational dynamics. As a fundamental tool we introduce the notion of homotopically bounded sets and we prove a general boundedness result for invariant open sets when the fixed point set is inessential.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 5 , August 2018 , pp. 1791 - 1836
Copyright
© Cambridge University Press, 2017 

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