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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

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, 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), 795-804. 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), 2347-2359. 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), 145-159. doi: 10.4064/fm200-2-3.  Google Scholar

[4]

G. Atkinson, Recurrence of cocycles and random walks, J. London Math. Soc., 13 (1976), 486-488.  Google Scholar

[5]

T. Bousch, La condition de Walters, Ann. Sci. École Norm. Sup. (4), 34 (2001), 287-311. 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), 1379-1409. 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), 155-196. 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), 563-611. doi: 10.1088/0951-7715/24/2/008.  Google Scholar

[10]

O. Jenkinson, Ergodic optimization, Discrete Contin. Dyn. Syst., 15 (2006), 197-224. 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), 1849-1860. doi: 10.1017/S0143385708000333.  Google Scholar

[12]

A. O. Lopes and P. Thieullen, Sub-actions for Anosov diffeomorphisms, Astérisque, xix, Geometric methods in dynamics. II (2003), 135-146.  Google Scholar

[13]

I. D. Morris, Ergodic optimization for generic continuous functions, Discrete Contin. Dyn. Syst., 27 (2010), 383-388. 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), 795-804. 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), 2347-2359. 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), 145-159. doi: 10.4064/fm200-2-3.  Google Scholar

[4]

G. Atkinson, Recurrence of cocycles and random walks, J. London Math. Soc., 13 (1976), 486-488.  Google Scholar

[5]

T. Bousch, La condition de Walters, Ann. Sci. École Norm. Sup. (4), 34 (2001), 287-311. 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), 1379-1409. 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), 155-196. 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), 563-611. doi: 10.1088/0951-7715/24/2/008.  Google Scholar

[10]

O. Jenkinson, Ergodic optimization, Discrete Contin. Dyn. Syst., 15 (2006), 197-224. 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), 1849-1860. doi: 10.1017/S0143385708000333.  Google Scholar

[12]

A. O. Lopes and P. Thieullen, Sub-actions for Anosov diffeomorphisms, Astérisque, xix, Geometric methods in dynamics. II (2003), 135-146.  Google Scholar

[13]

I. D. Morris, Ergodic optimization for generic continuous functions, Discrete Contin. Dyn. Syst., 27 (2010), 383-388. doi: 10.3934/dcds.2010.27.383.  Google Scholar

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