Ergodic Theory


Convergence in L^p, in measure and a.e.; integrability; Fatou Lemma, Monotone Convergence and Dominated Convergence; isomorphism in measure. Periodic orbit, recurrent, non-wandering, dense; conjugacy, stability. Invariant measures by a dynamic. Invariants by isomorphism. Existence of invariant probability measures for continuous dynamics on compact metric spaces. Ergodicity, mixing, uniquely ergodicity; dynamical and spectral interpretations of these metrical properties. Examples: homeomorphism of R; qx(mod1); Gauss map; shifts; powers of z and rotation in S^1; linear maps in R^n; Anosov diffeomorphisms; Smale horseshoe; pedal triangle; coupled systems; Markov chains; search engines. Poincaré, Birkhoff and Kac Theorems; connection with the second law of Thermodynamics. Applications: Borel Theorem on normal numbers; Multiple Recurrence Theorem; van der Waerden Theorem. Ergodic decomposition. Topological and measure-theoretic entropy. Variational principle and equilibrium states.

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Defended Theses

  • Structure, algorithmics and dynamics of endomorphisms for certain classes of groups
      André da Cruz Carvalho (April 2023)
      Pedro V. Silva
  • A point-free study of z-embeddings, more general classes of localic maps, and uniform continuity
      Ana Belén Avilez García (April 2023)
      Jorge Picado
  • Analysis of equations of motion of inextensible strings and networks
      Ayk Telciyan (April 2023)
      Dmitry Vorotnikov
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