STARTING DATE: 01 OF SEPTEMBER.
Tuesday: 16:00 - 18:00
Thursday: 14:00 - 16:00
Virtual Room: write to me.
Program: Revision of prerequisites (measure theory). Measure-preserving transformations. Poincaré recurrence theorem. Ergodicity. Examples. Markov Chains and torus' automorphisms. Birkhoff's ergodic theorem and applications. Mixing and weak mixing. Kolmogorov-Sinai entropy, partition's entropy and conditional entropy. Entropy of some examples and Ornstein Theory. Topological and Symbolical dynamics. Existence invariant probability measures. Compactness of the invariant probability measures set. Ergodic Decomposition Theorem. Topological Entropy (definition and examples). Shannon-McMillan-Breiman theorem. Variational Principles and Pressure. Measures of Maximal entropy. Thermodynamic Formalism. Theory of Ruelle-Perron-Frobenius. Gibbs Measures. Ergodic Optimization. Multifractal Analysis.
Evaluation: 1/3 x [Homework's + (2 interviews) + (presentation of an article or topic)]
Lista de Exercícios (1/3 da nota)
Survey articles written by students and slides presentation (many of them are in Portuguese)
On non-regular g-measures. Sandro Gallo and Frédéric Paccaut. Nonlinearity. (2013). [Julia Codas e Cayo Douglas] Survey article slides
The space of invariant measures for countable Markov shifts (Godofredo Iommi and Aníbal Velozo). Journal d'Analyse Mathematique (2020+). [Danrlei Vaz Oliveira] Survey article slides
Suspension flows over countable Markov shifts. L. Barreira and G. Iommi. J. Stat. Phys. 124(1), (2006). [Alan Ramer] Survey article slides
On the relation between Gibbs and g-measures. Steven Berghout, Roberto Fernández, Evgeny Verbitskiy. ETDS (2019) [João Maia e Leonardo Teramatsu] Survey article slides
An introduction to geometric Gibbs theory. Yunping Jiang. Dynamics, Games and Science, CIM Series in Mathematical Sciences, (2015). [Geovane Cardoso] Survey article slides
Induced topological pressure for countable state Markov shifts. J. Jaerisch, M. Kesseböhmer and S. Lamei. Stoc. and Dynamics 14 (2), (2014). [Henrique Corsini] Survey article slides
Differentiability of the pressure in non-compact spaces (Godofredo Iommi and Mike Todd). [João Rodrigues Pereira e Kelvyn Welsch] Survey article slides
The Chowla and the Sarnak conjectures from ergodic theory point of view. El Houcein El Abdalaoui, Joanna Kułaga-Przymus, Mariusz Lemańczyk and Thierry de la Rue. Discrete & Continuous Dynamical Systems - A. 37 (2017). [Julian Lazaro Aguirre] Survey article slides
Ergodic theorem involving additive and multiplicative groups of a field and {x + y, xy} patterns. Ergodic theorem involving additive and multiplicative groups of a field and {x + y, xy} patterns. Ergod. Th. & Dynam. Sys. (2015) [Iuri Grangeiro e Débora de Oliveira]. Survey article slides
Suggested papers to present at the course as part of the evaluation:
Multiple phase transitions on compact symbolic systems - Tamara Kucherenko, Antony Quas, Christian Wolf - arXiv:2006.13988 (2020).
Uncountably many maximizing measures for a dense subset of continuous functions - M. Shinoda - Nonlinearity (2018).
Thermodynamic formalism for suspension flows over countable Markov shifts. T. Kempton. Nonlinearity 24, (2011).
The Chowla and the Sarnak conjectures from ergodic theory point of view. El Houcein El Abdalaoui, Joanna Kułaga-Przymus, Mariusz Lemańczyk and Thierry de la Rue. Discrete & Continuous Dynamical Systems - A. 37 (2017).
Sarnak's Conjecture -- what's new. S. Ferenczi, J. Kułaga-Przymus, M. Lemańczyk. arXiv:1710.04039
Ergodic theory: Nonsingular transformations. Alexandre I Danilenko, Cesar E Silva. 2008.
Weak mixing for nonsingular Bernoulli actions of countable amenable groups. Alexandre I. Danilenko. Proc. Amer. Math. Soc. 147 (2019).
Proving ergodicity via divergence of ergodic sums. Zemer Kosloff. Studia Math. 248 (2019).
Positive topological entropy implies chaos DC2. Tomasz Downarowicz. Proc. Amer. Math. Soc. (2013).
Topological entropy zero and asymptotic pairs T. Downarowicz & Y. Lacroix. Israel Journal of Mathematics. 189, (2012)
Quenched and annealed equilibrium states for random Ruelle expanding maps and applications. Manuel Stadlbauer, Paulo Varandas, Xuan Zhang. arXiv:2004.04763
Sequence entropy and rigid σ-algebras. D. Coronel, A. Maass and S. Shao. Studia Mathematica (2009).
On the relation between finite range potentials and subshifts of finite type. Olle Häggström, PTRF (1995).
Asymptotic decoupling and weak Gibbs measures for finite alphabet shift spaces. C-E Pfister and W G Sullivan. Nonlinearity (2020).
volume
Bibliography:
1. Teoria Ergódica - Um curso introdutório - Krerley Oliveira e Marcelo Viana.
2. Uma Introdução à Teoria Ergódica - Carlos Bocker, Krerley Oliveira and Marcelo Viana. V Bienal da SBM. (2010)
3. Um primeiro curso sobre teoria ergódica com aplicações. Krerley Oliveira. 25° Colóquio Brasileiro de Matemática. (2005)
4. Introdução à teoria ergódica. Mañé, R. Rio de Janeiro. IMPA.(Projeto Euclides) (1982)
5. An Introduction to Ergodic Theory. P. Walters. (GTM-Springer). (1982)
6. Ergodic Theory. K. Petersen. (Cambridge Studies in Advances Mathematics 2). (1983)
7. Topics in Ergodic Theory. W. Parry. (Cambridge Tracts in Mathematics) (1983)
8. Invitation to Ergodic Theory. C. E. Silva. (Student Mathematical Library vol 42) (AMS). (2007)
9. Introduction to Dynamical Systems. M. Brin and G. Stuck. (Cambridge) (2002)
10. Entropy. A. Greven, G. Keller and G. Warneck Editors. (Princeton Series in Applied Mathematics) (2003)
11. Entropy in Dynamical Systems. T. Downarowicz. (Cambridge University Press). (2011)
12. Gibbs measures and phase transitions. Hans-Otto Georgii. (2 edition). (2011)
13. Probability Measures on Metric Spaces. K. R. Parthasarathy (AMS Chelsea Publishing) (new edition) (2005)
14. Probability Measures on Metric Spaces. Onno van Gaans. (2003)
15. An outline of Ergodic Theory. S. Kalikow and R. McCutcheon. (Cambridge Studies in Advances Mathematics). (2010)
16. Introduction à la théorie ergodique. Thierry de la Rue.
17. Notes on Ergodic Theory. J. Steif.
18. Lectures on Choquet's theorem. R.R. Phelps. (Lecture Notes in Mathematics) (2ed., Springer). (2001)
19. A short proof of the Ornstein theorem. T. Downarowicz and J. Serafin. Ergodic Theory and Dynamical Systems. (2011)
20. Introdução à Teoria das Medidas de Gibbs. R. Bissacot and Leandro Cioletti.
21. Lecture Notes on Ergodic Theory. O. Sarig. (2008)
22. Elemental Methods in ergodic Ramsey Theory. R. McCutcheon (1999)
23. Ergodic Theory and Information. P. Billingsley. (1965)
24. An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs vol 50) (AMS). Jon Aaronson.(1997).
25. Ergodic Theory: with a view towards Number Theory. Manfred Einsiedler and Thomas Ward. Springer (GTM Vol. 259) (2011).
26. Markov partitions for surface diffeomorphisms. Krerley Oliveira and Omri Sarig. Mini-course lecture Notes at ICTP-ESF School and Conference in Dynamical Systems.(2012)
27. Dynamical Systems and Ergodic Theory. Lectures Notes of Corinna Ulcigrai (University of Bristol). (2011)
Page of mathematicians where you can find more material related to this course: