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

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


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