Ergodic Optimization in the Expanding Case

1. Front Matter

   Pages i-viii

   PDF

2. Introduction

   • Eduardo Garibaldi

   Pages 1-7

3. Duality

   • Eduardo Garibaldi

   Pages 9-12

4. Calibrated Sub-actions

   • Eduardo Garibaldi

   Pages 13-20

5. Aubry Set

   • Eduardo Garibaldi

   Pages 21-25

6. Mañé Potential and Peierls Barrier

   • Eduardo Garibaldi

   Pages 27-36

7. Representation of Calibrated Sub-actions

   • Eduardo Garibaldi

   Pages 37-40

8. Separating Sub-actions

   • Eduardo Garibaldi

   Pages 41-45

9. Further Properties of Sub-actions

   • Eduardo Garibaldi

   Pages 47-51

10. Relations with the Thermodynamic Formalism

    • Eduardo Garibaldi

    Pages 53-63

11. Back Matter

    Pages 65-73

    PDF

This book focuses on the interpretation of ergodic optimal problems as questions of variational dynamics, employing a comparable approach to that of the Aubry-Mather theory for Lagrangian systems. Ergodic optimization is primarily concerned with the study of optimizing probability measures. This work presents and discusses the fundamental concepts of the theory, including the use and relevance of Sub-actions as analogues to subsolutions of the Hamilton-Jacobi equation. Further, it provides evidence for the impressively broad applicability of the tools inspired by the weak KAM theory.

• ergodic optimization
• weak KAM
• sub-actions
• Aubry set
• Mañé potential
• thermodynamics

• University of Campinas - IMECC, Campinas, Brazil

  Eduardo Garibaldi

Eduardo Garibaldi holds a PhD in Mathematics from the Federal University of Rio Grande do Sul, Brazil, and an MA from the IMPA – National Institute for Pure and Applied Mathematics, Brazil, having pursued postdoctoral studies at the Université Bordeaux, France. He is currently affiliated with the University of Campinas, Brazil, and his research efforts chiefly focus on dynamical systems and ergodic theory, motivated by problems from equilibrium statistical mechanics and solid state physics.

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