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

Pages: 795 - 804, Volume 26, Issue 3, March 2010

doi:10.3934/dcds.2010.26.795       Abstract        Full Text (148.5K)       Related Articles

Salvador Addas-Zanata - Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brazil (email)
Fábio A. Tal - Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brazil (email)

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.

Keywords:  periodic orbits, ergodic optimization, maximizing measures.
Mathematics Subject Classification:  Primary: 37A05; Secondary: 37A99, 37E10.

Received: December 2008;      Revised: August 2009;      Published: December 2009.