Ergodic optimization is the study of problems relating maximizing invariant measures and maximum ergodic averages. In ergodic optimization theory, one important problem is the typically periodic optimization (TPO) conjecture. This conjecture was proposed by Mañé, Hunt, Ott and Yuan[5,9] in 1990s, which reveals the principle of least action in dynamical system settings. To be more precise, TPO indicates that when the dynamical system is suitably hyperbolic and the observable is suitably regular, then the maximizing measure is ``genetically’’ supported on a periodic orbit with relatively low period. In the setting of uniformly open expanding maps with Lipschitz/Holder observables, TPO was obtained in topological genetic sense by Contreras  in 2016.
In this series of talks, I will report several recent progresses on understanding TPO conjecture both in probabilistic and topological sense, and for more general hyperbolic systems. These are joint works with Jairo Bochi, and Huang Wen, Zeng Lian, Xiao Ma, Leiye Xu . The series of talks are potentially composed with four parts:
1. Preliminaries and background of ergodic optimization and its relationship with thermodynamics formalism.
2. History and developments of TPO conjecture.
Reference for part 1 and 2 will be [1,5,7,8,9]
3. A new proof of Contreras theorems.
Reference for part 3 will be [2,4,6,9]
4. Prevalence in TPO conjectures, and its relationships with probabilistic percolation.
Reference for part 4 will be [1,3].
1. Bochi. J. Ergodic optimization of Birkhoff averages and Lyapunov exponents,Proceedings of the International Congress of Mathematicians 2018, Rio de Janeiro, vol. 2, pp. 1821-1842.
2. Bochi. J. Genericity of periodic maximization: proof of Contreras' theorem following Huang, Lian, Ma, Xu, and Zhang, lecture note, available at http://www.mat.uc.cl/~jairo.bochi/
3. Bochi, J and Zhang. Y. Ergodic optimization of prevalent super-continuous functions,International Mathematics Research
Notices 2016 (2016), no. 19, pp. 5988-6017.
4. Contreras. G. Ground states are generically a periodic orbit. Invent. Math. 205 (2016), no. 2, 383–412.
5. Hunt, B.R.; Ott, E. Optimal Periodic Orbits of Chaotic Systems. Phys. Rev. Letter 54 (1996), no. 76, 2254.
6. Huang, W.; Lian, Z.; Ma, X.; Xu, L.; Zhang, Y. Ergodic optimization theory for a class of typical maps. arXiv:1904.01915
7. Jenkinson, O. Ergodic optimization in dynamical systems. Ergodic Theory Dynam. Systems 2018, to appear.
8. Mañé, R. Generic properties and problems of minimizing measures of Lagrangian systems. Nonlinearity 9 (1996), no. 2, 273–310.
9. Yuan, G.; Hunt, B.R. Optimal orbits of hyperbolic systems. Nonlinearity 12 (1999), no. 4, 1207–1224.