Density of the set of endomorphisms with a maximizing measure supported on a periodic orbit

Tatiane C. Batista; Juliano S. Gonschorowski; Fábio A. Tal

doi:10.3934/dcds.2015.35.3315

Article Contents

2015, Volume 35, Issue 8: 3315-3326. Doi: 10.3934/dcds.2015.35.3315

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Density of the set of endomorphisms with a maximizing measure supported on a periodic orbit

1.
Universidade Tecnológica Federal do Paraná - UTFPR, Av. Professora Laura Pacheco Bastos, 800 - Bairro Industrial, 85053-525, Guarapuava-PR
2.
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP

Received: October 31, 2013

Revised: November 30, 2014

Published: May 2017

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Abstract

Let $M$ be a compact $n$-dimensional Riemanian manifold, End($M$) the set of the endomorphisms of $M$ with the usual $\mathcal{C}^0$ topology and $\phi:M\to\mathbb{R}$ continuous. We prove, extending the main result of [2], that there exists a dense subset of $\mathcal{A}$ of End($M$) such that, if $f\in\mathcal{A}$, there exists a $f$ invariant measure $\mu_{\max}$ supported on a periodic orbit that maximizes the integral of $\phi$ among all $f$ invariant Borel probability measures.

Keywords:

Maximizing measures,
periodic orbits.

Mathematics Subject Classification: Primary: 37A05, 37B99.

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References

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