RMC - Vol. 21 Num. 2 - Pacifico/Vieitez

Full paper in PDF format:
$% M. J. Pacifico and J. L. Vieitez, Entropy-expansiveness and domination for surface diffeomorphisms, Rev. Mat. Complut. 21 (2008), no. 2, 293–317.%$

Entropy-Expansiveness and Domination
for Surface Diffeomorphisms

María José PACIFICO and José L. VIEITEZ

Instituto de Matematica

Universidade Federal do Rio de Janeiro

C. P. 68.530, CEP 21.945-970

Rio de Janeiro, R. J. — Brazil

pacifico@im.ufrj.br

Instituto de Matematica

Facultad de Ingenieria

Universidad de la Republica

CC30, CP 11300 Montevideo — Uruguay

jvieitez@fing.edu.uy

Received: July 3, 2006
Accepted: March 10, 2007

ABSTRACT

Let $f :M → M$ be a $r C$ -diffeomorphism, $r ≥ 1$ , defined on a closed manifold $M$ . We prove that if $M$ is a surface and $K ⊂ M$ is a compact invariant set such that $TKM = E ⊕F$ is a dominated splitting then $f∕K$ is entropy expansive. Moreover $C1$ generically in any dimension, isolated homoclinic classes $H (p)$ , $p$ hyperbolic, are entropy expansive.

Conversely, if there exists a $C1$ neighborhood $U$ of a surface diffeomorphism $f$ and a homoclinic class $H(p)$ , $p$ hyperbolic, such that for every $g ∈ U$ the continuation $H (pg)$ of $H (p)$ is entropy-expansive then there is a dominated splitting for $f∕H (p)$ .

Key words: entropy-expansiveness, homoclinic classes, dominated splitting, homoclinic tangency, symbolic extension.
2000 Mathematics Subject Classification: 37D30, 37C29, 37E30.