# American Institute of Mathematical Sciences

February  2006, 15(1): 197-224. doi: 10.3934/dcds.2006.15.197

## Ergodic Optimization

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

Received  December 2004 Revised  October 2005 Published  February 2006

Let $f$ be a real-valued function defined on the phase space of a dynamical system. Ergodic optimization is the study of those orbits, or invariant probability measures, whose ergodic $f$-average is as large as possible.
In these notes we establish some basic aspects of the theory: equivalent definitions of the maximum ergodic average, existence and generic uniqueness of maximizing measures, and the fact that every ergodic measure is the unique maximizing measure for some continuous function. Generic properties of the support of maximizing measures are described in the case where the dynamics is hyperbolic. A number of problems are formulated.
Citation: Oliver Jenkinson. Ergodic Optimization. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 197-224. doi: 10.3934/dcds.2006.15.197
 [1] 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 [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] 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 [4] 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 [5] Lin Xu, Rongming Wang, Dingjun Yao. On maximizing the expected terminal utility by investment and reinsurance. Journal of Industrial and Management Optimization, 2008, 4 (4) : 801-815. doi: 10.3934/jimo.2008.4.801 [6] 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 [7] 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 [8] 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 [9] Keiji Tatsumi, Masashi Akao, Ryo Kawachi, Tetsuzo Tanino. Performance evaluation of multiobjective multiclass support vector machines maximizing geometric margins. Numerical Algebra, Control and Optimization, 2011, 1 (1) : 151-169. doi: 10.3934/naco.2011.1.151 [10] Genggeng Huang, Congming Li, Ximing Yin. Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 935-942. doi: 10.3934/dcds.2015.35.935 [11] Yue Qi, Tongyang Liu, Su Zhang, Yu Zhang. Robust Markowitz: Comprehensively maximizing Sharpe ratio by parametric-quadratic programming. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2021235 [12] Welington Cordeiro, Manfred Denker, Xuan Zhang. On specification and measure expansiveness. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1941-1957. doi: 10.3934/dcds.2017082 [13] Welington Cordeiro, Manfred Denker, Xuan Zhang. Corrigendum to: On specification and measure expansiveness. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3705-3706. doi: 10.3934/dcds.2018160 [14] Petr Kůrka. On the measure attractor of a cellular automaton. Conference Publications, 2005, 2005 (Special) : 524-535. doi: 10.3934/proc.2005.2005.524 [15] Tomasz Downarowicz, Yonatan Gutman, Dawid Huczek. Rank as a function of measure. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2741-2750. doi: 10.3934/dcds.2014.34.2741 [16] 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 [17] Xing Liang, Lei Zhang. The optimal distribution of resources and rate of migration maximizing the population size in logistic model with identical migration. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2055-2065. doi: 10.3934/dcdsb.2020280 [18] Shaojun Zhang, Zhong Wan. Polymorphic uncertain nonlinear programming model and algorithm for maximizing the fatigue life of V-belt drive. Journal of Industrial and Management Optimization, 2012, 8 (2) : 493-505. doi: 10.3934/jimo.2012.8.493 [19] Zari Dzalilov, Iradj Ouveysi, Alexander Rubinov. An extended lifetime measure for telecommunication network. Journal of Industrial and Management Optimization, 2008, 4 (2) : 329-337. doi: 10.3934/jimo.2008.4.329 [20] M. Baake, P. Gohlke, M. Kesseböhmer, T. Schindler. Scaling properties of the Thue–Morse measure. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4157-4185. doi: 10.3934/dcds.2019168

2020 Impact Factor: 1.392