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.

Research and Events


  • There is no information available on this topic.
More Events

Defended Theses

  • Statistical instability in chaotic dynamics
      Muhammad Ali Khan (March 2018)
      José Ferreira Alves
  • Profinite topologies on the free group
      Khadijeh Alibabaei (March 2018)
      Jorge Almeida
  •   Rui Manuel Tavares Pinto de Sá Pereira (January 2018)
      Evelina Shamarova
More Theses