Support of maximizing measures for typical \mathcal{C}^0 dynamics on compact manifolds

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September 2010, 26(3): 795-804. doi: 10.3934/dcds.2010.26.795

Support of maximizing measures for typical $\mathcal{C}^0$ dynamics on compact manifolds

Salvador Addas-Zanata ^1, and Fábio A. Tal ^1,

1.	Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brazil

Received December 2008 Revised August 2009 Published December 2009

Abstract
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Given a compact manifold $X,$ a continuous function $g:X\to \R{},$ and a map $T:X\to X,$ we study properties of the $T$-invariant Borel probability measures that maximize the integral of $g$.
We show that if $X$ is a $n$-dimensional connected Riemaniann manifold, with $n \geq 2$, then the set of homeomorphisms for which there is a maximizing measure supported on a periodic orbit is meager.
We also show that, if $X$ is the circle, then the "topological size'' of the set of endomorphisms for which there are $g$ maximizing measures with support on a periodic orbit depends on properties of the function $g.$ In particular, if $g$ is $\mathcal{C}^1$, it has interior points.

Keywords: periodic orbits, ergodic optimization, maximizing measures..

Mathematics Subject Classification: Primary: 37A05; Secondary: 37A99, 37E1.

Citation: Salvador Addas-Zanata, Fábio A. Tal. Support of maximizing measures for typical $\mathcal{C}^0$ dynamics on compact manifolds. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 795-804. doi: 10.3934/dcds.2010.26.795

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