Ergodic optimization, zero temperature limits and the max-plus algebra

Abstract

Lecture notes of a course at the Brazilian Mathematical Colloquium. We review some basic notions in ergodic theory and thermodynamic formalism, as well as introductory results in the context of max-plus algebra, in order to exhibit some properties of equilibrium measures when temperature goes to zero.

103 pages, misprints of the previous version fixed

References in corpus (2)

• Selection of measures for a potential with two maxima at the zero temperature limit
• Eigenfunctions of the Laplacian and associated Ruelle operator

Cited by in corpus (15)

• Ergodic optimization in dynamical systems
• Sensitive dependence of Gibbs measures at low temperatures
• Ergodic optimization theory for a class of typical maps
• Zero temperature limits of Gibbs states for almost-additive potentials
• Ergodic optimization theory for Axiom A flows
• Entropy, pressure, ground states and calibrated sub-actions for linear dynamics
• Existence of the zero temperature limit of equilibrium states on topologically transitive countable Markov shifts
• Ergodic optimization of prevalent super-continuous functions
• Ground States are generically a Periodic Orbit
• Entropy, Pressure and Duality for Gibbs plans in Ergodic Transport
• On involution kernels and large deviations principles on $\beta$-shifts
• The KMS Condition for the homoclinic equivalence relation and Gibbs probabilities
• Robust sensitive dependence of geometric Gibbs states for analytic families of quadratic maps
• Lyapunov optimization for non-generic one-dimensional expanding Markov maps
• Ergodic Maximizing Measures of Non-Generic, Yet Dense Continuous Functions