Open AccessEntropy-Expansiveness and Domination for Surface Diffeomorphisms
Author(s) -
María José Pacífico,
José Vieitez
Publication year - 2008
Publication title -
revista matemática complutense
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.093
H-Index - 24
eISSN - 1696-8220
pISSN - 1139-1138
DOI - 10.5209/rev_rema.2008.v21.n2.16370
Subject(s) - diffeomorphism , homoclinic orbit , expansive , mathematics , pure mathematics , manifold (fluid mechanics) , invariant (physics) , topological entropy , entropy (arrow of time) , mathematical analysis , combinatorics , physics , mathematical physics , thermodynamics , mechanical engineering , compressive strength , quantum mechanics , engineering , bifurcation , nonlinear system
Let f : M → M be a Cr-diffeomorphism, r ≥ 1, deffined 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).Let f : M → M be a Cr-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 TK M = 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 contin¬uation H(pg) of H(p) is entropy-expansive then there is a dominated splitting for f/H(p)