# 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 & Continuous Dynamical Systems - A, 2006, 15 (1) : 197-224. doi: 10.3934/dcds.2006.15.197
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