Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Existence and uniqueness of maximizing measures for robust classes of local diffeomorphisms

Pages: 225 - 236, Volume 15, Issue 1, May 2006      doi:10.3934/dcds.2006.15.225

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Krerley Oliveira - Departamento de Matemática - UFAL, Campus A.C. Simões, s/n 57072-090 Maceió, Alagoas, Brazil (email)
Marcelo Viana - IMPA, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, RJ, Brazil (email)

Abstract: We prove existence of maximal entropy measures for an open set of non-uniformly expanding local diffeomorphisms on a compact Riemannian manifold. In this context the topological entropy coincides with the logarithm of the degree, and these maximizing measures are eigenmeasures of the transfer operator. When the map is topologically mixing, the maximizing measure is unique and positive on every open set.

Keywords:  Entropy, maximizing measure, non-uniform hyperbolicity.
Mathematics Subject Classification:  Primary: 37D25, 37D35, 37A35; Secondary: 37A60.

Received: September 2004;      Revised: February 2005;      Available Online: February 2006.