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

Previous Article
Existence and regularity of solutions in nonlinear wave equations
DCDS Home
This Issue
Next Article
On the Hausdorff dimension of the Sierpiński Julia sets

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
Related Papers

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 and 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.
[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.
[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.
[4]	G. Atkinson, Recurrence of cocycles and random walks, J. London Math. Soc., 13 (1976), 486-488.
[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.
[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.
[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.
[8]	J. P. Conze and Y. Guivarch, Croissance des sommes ergodiques et principe variationnel, 1993, Manuscript.
[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.
[10]	O. Jenkinson, Ergodic optimization, Discrete Contin. Dyn. Syst., 15 (2006), 197-224. doi: 10.3934/dcds.2006.15.197.
[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.
[12]	A. O. Lopes and P. Thieullen, Sub-actions for Anosov diffeomorphisms, Astérisque, xix, Geometric methods in dynamics. II (2003), 135-146.
[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.

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.
[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.
[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.
[4]	G. Atkinson, Recurrence of cocycles and random walks, J. London Math. Soc., 13 (1976), 486-488.
[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.
[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.
[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.
[8]	J. P. Conze and Y. Guivarch, Croissance des sommes ergodiques et principe variationnel, 1993, Manuscript.
[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.
[10]	O. Jenkinson, Ergodic optimization, Discrete Contin. Dyn. Syst., 15 (2006), 197-224. doi: 10.3934/dcds.2006.15.197.
[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.
[12]	A. O. Lopes and P. Thieullen, Sub-actions for Anosov diffeomorphisms, Astérisque, xix, Geometric methods in dynamics. II (2003), 135-146.
[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.

[1]	Ilie Ugarcovici. On hyperbolic measures and periodic orbits. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 505-512. doi: 10.3934/dcds.2006.16.505
[2]	Misha Bialy. Maximizing orbits for higher-dimensional convex billiards. Journal of Modern Dynamics, 2009, 3 (1) : 51-59. doi: 10.3934/jmd.2009.3.51
[3]	Krerley Oliveira, Marcelo Viana. Existence and uniqueness of maximizing measures for robust classes of local diffeomorphisms. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 225-236. doi: 10.3934/dcds.2006.15.225
[4]	Salvador Addas-Zanata, Fábio A. Tal. Support of maximizing measures for typical $\mathcal{C}^0$ dynamics on compact manifolds. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 795-804. doi: 10.3934/dcds.2010.26.795
[5]	Ana Cristina Mereu, Marco Antonio Teixeira. Reversibility and branching of periodic orbits. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 1177-1199. doi: 10.3934/dcds.2013.33.1177
[6]	Katrin Gelfert, Christian Wolf. On the distribution of periodic orbits. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 949-966. doi: 10.3934/dcds.2010.26.949
[7]	Jacky Cresson, Christophe Guillet. Periodic orbits and Arnold diffusion. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 451-470. doi: 10.3934/dcds.2003.9.451
[8]	Flávia M. Branco. Sub-actions and maximizing measures for one-dimensional transformations with a critical point. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 271-280. doi: 10.3934/dcds.2007.17.271
[9]	Michihiro Hirayama. Periodic probability measures are dense in the set of invariant measures. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1185-1192. doi: 10.3934/dcds.2003.9.1185
[10]	Alain Jacquemard, Weber Flávio Pereira. On periodic orbits of polynomial relay systems. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 331-347. doi: 10.3934/dcds.2007.17.331
[11]	Peter Albers, Jean Gutt, Doris Hein. Periodic Reeb orbits on prequantization bundles. Journal of Modern Dynamics, 2018, 12: 123-150. doi: 10.3934/jmd.2018005
[12]	Jan Sieber, Matthias Wolfrum, Mark Lichtner, Serhiy Yanchuk. On the stability of periodic orbits in delay equations with large delay. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 3109-3134. doi: 10.3934/dcds.2013.33.3109
[13]	Nicola Guglielmi, Christian Lubich. Numerical periodic orbits of neutral delay differential equations. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 1057-1067. doi: 10.3934/dcds.2005.13.1057
[14]	Jorge Rebaza. Bifurcations and periodic orbits in variable population interactions. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2997-3012. doi: 10.3934/cpaa.2013.12.2997
[15]	Răzvan M. Tudoran. On the control of stability of periodic orbits of completely integrable systems. Journal of Geometric Mechanics, 2015, 7 (1) : 109-124. doi: 10.3934/jgm.2015.7.109
[16]	Corey Shanbrom. Periodic orbits in the Kepler-Heisenberg problem. Journal of Geometric Mechanics, 2014, 6 (2) : 261-278. doi: 10.3934/jgm.2014.6.261
[17]	Luca Dieci, Timo Eirola, Cinzia Elia. Periodic orbits of planar discontinuous system under discretization. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2743-2762. doi: 10.3934/dcdsb.2018103
[18]	Rossella Bartolo. Periodic orbits on Riemannian manifolds with convex boundary. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 439-450. doi: 10.3934/dcds.1997.3.439
[19]	Piotr Oprocha, Xinxing Wu. On averaged tracing of periodic average pseudo orbits. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4943-4957. doi: 10.3934/dcds.2017212
[20]	Jeremias Epperlein, Vladimír Švígler. On arbitrarily long periodic orbits of evolutionary games on graphs. Discrete and Continuous Dynamical Systems - B, 2018, 23 (5) : 1895-1915. doi: 10.3934/dcdsb.2018187

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Tatiane C. Batista Juliano S. Gonschorowski Fábio A. Tal
on Google Scholar
Tatiane C. Batista Juliano S. Gonschorowski Fábio A. Tal

[Back to Top]