June  2006, 16(2): 383-392. doi: 10.3934/dcds.2006.16.383

Every ergodic measure is uniquely maximizing

1. 

School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS

Received  April 2005 Revised  November 2005 Published  July 2006

Let $M_{\phi}$ denote the set of Borel probability measures invariant under a topological action $\phi$ on a compact metrizable space $X$. For a continuous function $f:X\to\R$, a measure $\mu\in\M_{\phi}$ is called $f$-maximizing if $\int f\, d\mu = s u p\{\int f dm:m\in\M_{\phi}\}$. It is shown that if $\mu$ is any ergodic measure in $\M_{\phi}$, then there exists a continuous function whose unique maximizing measure is $\mu$. More generally, if $\mathcal E$ is a non-empty collection of ergodic measures which is weak$^*$ closed as a subset of $\M_{\phi}$, then there exists a continuous function whose set of maximizing measures is precisely the closed convex hull of $\mathcal E$. If moreover $\phi$ has the property that its entropy map is upper semi-continuous, then there exists a continuous function whose set of equilibrium states is precisely the closed convex hull of $\mathcal E$.
Citation: Oliver Jenkinson. Every ergodic measure is uniquely maximizing. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 383-392. doi: 10.3934/dcds.2006.16.383
[1]

Oliver Jenkinson. Ergodic Optimization. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 197-224. doi: 10.3934/dcds.2006.15.197

[2]

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

[3]

Liyuan Wang, Zhiping Chen, Peng Yang. Robust equilibrium control-measure policy for a DC pension plan with state-dependent risk aversion under mean-variance criterion. Journal of Industrial and Management Optimization, 2021, 17 (3) : 1203-1233. doi: 10.3934/jimo.2020018

[4]

Ian D. Morris. Ergodic optimization for generic continuous functions. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 383-388. doi: 10.3934/dcds.2010.27.383

[5]

Yunmei Chen, Jiangli Shi, Murali Rao, Jin-Seop Lee. Deformable multi-modal image registration by maximizing Rényi's statistical dependence measure. Inverse Problems and Imaging, 2015, 9 (1) : 79-103. doi: 10.3934/ipi.2015.9.79

[6]

Jon Chaika, Howard Masur. There exists an interval exchange with a non-ergodic generic measure. Journal of Modern Dynamics, 2015, 9: 289-304. doi: 10.3934/jmd.2015.9.289

[7]

Jialu Fang, Yongluo Cao, Yun Zhao. Measure theoretic pressure and dimension formula for non-ergodic measures. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2767-2789. doi: 10.3934/dcds.2020149

[8]

Nuno Luzia. On the uniqueness of an ergodic measure of full dimension for non-conformal repellers. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5763-5780. doi: 10.3934/dcds.2017250

[9]

Tomasz Downarowicz, Benjamin Weiss. Pure strictly uniform models of non-ergodic measure automorphisms. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 863-884. doi: 10.3934/dcds.2021140

[10]

Yufei Sun, Grace Aw, Kok Lay Teo, Guanglu Zhou. Portfolio optimization using a new probabilistic risk measure. Journal of Industrial and Management Optimization, 2015, 11 (4) : 1275-1283. doi: 10.3934/jimo.2015.11.1275

[11]

Zhiyuan Wen, Meirong Zhang. On the optimization problems of the principal eigenvalues of measure differential equations with indefinite measures. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3257-3274. doi: 10.3934/dcdsb.2020061

[12]

Xi Chen, Zongrun Wang, Songhai Deng, Yong Fang. Risk measure optimization: Perceived risk and overconfidence of structured product investors. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1473-1492. doi: 10.3934/jimo.2018105

[13]

Jianxin Zhou. Optimization with some uncontrollable variables: a min-equilibrium approach. Journal of Industrial and Management Optimization, 2007, 3 (1) : 129-138. doi: 10.3934/jimo.2007.3.129

[14]

Chunyang Zhang, Shugong Zhang, Qinghuai Liu. Homotopy method for a class of multiobjective optimization problems with equilibrium constraints. Journal of Industrial and Management Optimization, 2017, 13 (1) : 81-92. doi: 10.3934/jimo.2016005

[15]

Guirong Pan, Bing Xue, Hongchun Sun. An optimization model and method for supply chain equilibrium management problem. Mathematical Foundations of Computing, 2022, 5 (2) : 145-156. doi: 10.3934/mfc.2022001

[16]

Lluís Alsedà, David Juher, Deborah M. King, Francesc Mañosas. Maximizing entropy of cycles on trees. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3237-3276. doi: 10.3934/dcds.2013.33.3237

[17]

Benedetto Piccoli. Optimal syntheses for state constrained problems with application to optimization of cancer therapies. Mathematical Control and Related Fields, 2012, 2 (4) : 383-398. doi: 10.3934/mcrf.2012.2.383

[18]

Eduardo Casas, Fredi Tröltzsch. State-constrained semilinear elliptic optimization problems with unrestricted sparse controls. Mathematical Control and Related Fields, 2020, 10 (3) : 527-546. doi: 10.3934/mcrf.2020009

[19]

Evrad M. D. Ngom, Abdou Sène, Daniel Y. Le Roux. Global stabilization of the Navier-Stokes equations around an unstable equilibrium state with a boundary feedback controller. Evolution Equations and Control Theory, 2015, 4 (1) : 89-106. doi: 10.3934/eect.2015.4.89

[20]

Leonid Shaikhet. Stability of a positive equilibrium state for a stochastically perturbed mathematical model of glassy-winged sharpshooter population. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1167-1174. doi: 10.3934/mbe.2014.11.1167

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (101)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]