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

Salvador Addas-Zanata; Fábio A. Tal

doi:10.3934/dcds.2010.26.795

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2010, Volume 26, Issue 3: 795-804. Doi: 10.3934/dcds.2010.26.795

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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

Received: November 30, 2008

Revised: July 31, 2009

Published: August 2010

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Abstract

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.

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Mathematics Subject Classification: Primary: 37A05; Secondary: 37A99, 37E10.

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