Ergodic optimization in dynamical systems

Abstract

Ergodic optimization is the study of problems relating to maximizing orbits, maximizing invariant measures and maximum ergodic averages. An orbit of a dynamical system is called f-maximizing if the time average of the real-valued function f along the orbit is larger than along all other orbits, and an invariant probability measure is called f-maximizing if it gives f a larger space average than does any other invariant probability measure. In this survey we consider the main strands of ergodic optimization, beginning with an influential model problem, and the interpretation of ergodic optimization as the zero temperature limit of thermodynamic formalism. We describe typical properties of maximizing measures for various spaces of functions, the key tool of adding a coboundary so as to reveal properties of these measures, as well as certain classes of functions where the maximizing measure is known to be Sturmian.

Survey, to appear in Ergodic Theory & Dynamical Systems

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Cited by in corpus (14)

• Ergodic optimization theory for a class of typical maps
• Ergodic optimization for hyperbolic flows and Lorenz attractors
• Fractal Weyl law for the Ruelle spectrum of Anosov flows
• Applications of Variable Discounting Dynamic Programming to Iterated Function Systems and Related Problems
• Ergodic optimization theory for Axiom A flows
• Entropy, pressure, ground states and calibrated sub-actions for linear dynamics
• Optimal minimax bounds for time and ensemble averages for the incompressible Navier-Stokes equations
• Equilibrium states of intermediate entropies
• Typical periodic optimization for dynamical systems: symbolic dynamics
• Pressure at infinity on countable Markov shifts
• Zero-Temperature Chaos in Bidimensional Models with Finite-Range Potentials
• Topological structures on saturated sets, optimal orbits and equilibrium states
• Ergodic optimization for continuous functions on non-Markov shifts
• Lipschitz sub-actions for locally maximal hyperbolic sets of a $C^1$ maps