PT EN

Ergodic Theory

Program

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

Events

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

Defended Theses

  • Some aspects of descent theory and applications
      Rui Rodrigues de Abreu Fernandes Prezado (January 2024)
      Maria Manuel Clementino
      Fernando Lucatelli Nunes
  • Comparability between different systems: star-shaped and convex transform orders
      Beatriz Ferreira Santos (December 2023)
      Paulo Eduardo Oliveira
      Idir Arab
  • On Lax Idempotent Monads in Topology
      Carlos Miguel Alves Fitas (December 2023)
      Maria Manuel Clementino
More Theses