Article Contents
Article Contents

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

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

 Citation:

•  [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.