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

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August 2015, 35(8): 3315-3326. doi: 10.3934/dcds.2015.35.3315

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

Tatiane C. Batista ^1, , Juliano S. Gonschorowski ^1, and Fábio A. Tal ^2,

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

Received November 2013 Revised December 2014 Published February 2015

Abstract
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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, 37B9.

Citation: Tatiane C. Batista, Juliano S. Gonschorowski, Fábio A. Tal. Density of the set of endomorphisms with a maximizing measure supported on a periodic orbit. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3315-3326. doi: 10.3934/dcds.2015.35.3315

References:

[1]	S. Addas-Zanata and F. A. Tal, Support of maximizing measures for typical $\mathcalC^0$ dynamics on compact manifolds,, Discrete Contin. Dyn. Syst., 26 (2010), 795. doi: 10.3934/dcds.2010.26.795. Google Scholar
[2]	S. Addas-Zanata and F. A. Tal, Maximizing measures for endomorphisms of the circle,, Nonlinearity, 21 (2008), 2347. doi: 10.1088/0951-7715/21/10/008. Google Scholar
[3]	S. Addas-Zanata and F. A. Tal, On maximizing measures of homeomorphisms on compact manifolds,, Fund. Math., 200 (2008), 145. doi: 10.4064/fm200-2-3. Google Scholar
[4]	G. Atkinson, Recurrence of cocycles and random walks,, J. London Math. Soc., 13 (1976), 486. Google Scholar
[5]	T. Bousch, La condition de Walters,, Ann. Sci. École Norm. Sup. (4), 34 (2001), 287. doi: 10.1016/S0012-9593(00)01062-4. Google Scholar
[6]	G. Contreras, A. O. Lopes and P. Thieullen, Lyapunov minimizing measures for expanding maps of the circle,, Ergodic Theory Dynam. Systems, 21 (2001), 1379. doi: 10.1017/S0143385701001663. Google Scholar
[7]	G. Contreras, J. Delgado and R. Iturriaga, Lagrangian flows: The dynamics of globally minimizing orbits. II,, Bol. Soc. Brasil. Mat. (N.S.), 28 (1997), 155. doi: 10.1007/BF01233390. Google Scholar
[8]	J. P. Conze and Y. Guivarch, Croissance des sommes ergodiques et principe variationnel, 1993,, Manuscript., (). Google Scholar
[9]	E. Garibaldi and P. Thieullen, Minimizing orbits in the discrete Aubry-Mather model,, Nonlinearity, 24 (2011), 563. doi: 10.1088/0951-7715/24/2/008. Google Scholar
[10]	O. Jenkinson, Ergodic optimization,, Discrete Contin. Dyn. Syst., 15 (2006), 197. doi: 10.3934/dcds.2006.15.197. Google Scholar
[11]	O. Jenkinson and I. D. Morris, Lyapunov optimizing measures for $C^1$ expanding maps of the circle,, Ergodic Theory Dynam. Systems, 28 (2008), 1849. doi: 10.1017/S0143385708000333. Google Scholar
[12]	A. O. Lopes and P. Thieullen, Sub-actions for Anosov diffeomorphisms,, Astérisque, II (2003), 135. Google Scholar
[13]	I. D. Morris, Ergodic optimization for generic continuous functions,, Discrete Contin. Dyn. Syst., 27 (2010), 383. doi: 10.3934/dcds.2010.27.383. Google Scholar

show all references

References:

[1]	S. Addas-Zanata and F. A. Tal, Support of maximizing measures for typical $\mathcalC^0$ dynamics on compact manifolds,, Discrete Contin. Dyn. Syst., 26 (2010), 795. doi: 10.3934/dcds.2010.26.795. Google Scholar
[2]	S. Addas-Zanata and F. A. Tal, Maximizing measures for endomorphisms of the circle,, Nonlinearity, 21 (2008), 2347. doi: 10.1088/0951-7715/21/10/008. Google Scholar
[3]	S. Addas-Zanata and F. A. Tal, On maximizing measures of homeomorphisms on compact manifolds,, Fund. Math., 200 (2008), 145. doi: 10.4064/fm200-2-3. Google Scholar
[4]	G. Atkinson, Recurrence of cocycles and random walks,, J. London Math. Soc., 13 (1976), 486. Google Scholar
[5]	T. Bousch, La condition de Walters,, Ann. Sci. École Norm. Sup. (4), 34 (2001), 287. doi: 10.1016/S0012-9593(00)01062-4. Google Scholar
[6]	G. Contreras, A. O. Lopes and P. Thieullen, Lyapunov minimizing measures for expanding maps of the circle,, Ergodic Theory Dynam. Systems, 21 (2001), 1379. doi: 10.1017/S0143385701001663. Google Scholar
[7]	G. Contreras, J. Delgado and R. Iturriaga, Lagrangian flows: The dynamics of globally minimizing orbits. II,, Bol. Soc. Brasil. Mat. (N.S.), 28 (1997), 155. doi: 10.1007/BF01233390. Google Scholar
[8]	J. P. Conze and Y. Guivarch, Croissance des sommes ergodiques et principe variationnel, 1993,, Manuscript., (). Google Scholar
[9]	E. Garibaldi and P. Thieullen, Minimizing orbits in the discrete Aubry-Mather model,, Nonlinearity, 24 (2011), 563. doi: 10.1088/0951-7715/24/2/008. Google Scholar
[10]	O. Jenkinson, Ergodic optimization,, Discrete Contin. Dyn. Syst., 15 (2006), 197. doi: 10.3934/dcds.2006.15.197. Google Scholar
[11]	O. Jenkinson and I. D. Morris, Lyapunov optimizing measures for $C^1$ expanding maps of the circle,, Ergodic Theory Dynam. Systems, 28 (2008), 1849. doi: 10.1017/S0143385708000333. Google Scholar
[12]	A. O. Lopes and P. Thieullen, Sub-actions for Anosov diffeomorphisms,, Astérisque, II (2003), 135. Google Scholar
[13]	I. D. Morris, Ergodic optimization for generic continuous functions,, Discrete Contin. Dyn. Syst., 27 (2010), 383. doi: 10.3934/dcds.2010.27.383. Google Scholar

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