ENSPM2021 Talk

Title: Absolute continuity, Lyapunov exponents and rigidity

Abstract: Together with Artur Avila (IMPA/Zuich) and Amie Wilkinson (Chicago), we explore new connections between the dynamics of conservative partially hyperbolic systems and the geometric measure-theoretic properties of their invariant foliations.Those methods are applied to two main classes of volume preserving diffeomorphisms: fibered partially hyperbolic diffeomorphisms and center-fixing partially hyperbolic systems.When the center is 1-dimensional, assuming the diffeomorphism is accessible, we prove that the disintegration of the volume measure along the center foliation is either atomic or Lebesgue.Moreover, the latter case is rigid in dimension 3 (this does not require accessibility): the center foliation is actually smooth and the diffeomorphism is smoothly conjugate to an explicit rigid model.

Marcelo Viana (IMPA, Brazil)

Born in Rio de Janeiro, Brazil, his parents being Portuguese, grew up in Portugal, and his undergraduate studies were at the University of Porto. He received his Ph.D. degree from the IMPA in Rio de Janeiro, with Jacob Palis as advisor. Viana is now director at IMPA where he works in dynamical systems theory and so far has forty PhD students. He was a Guggenheim Fellow in 1993, received the TWAS Prize in 1998 and in 2005 he was awarded the inaugural ICTP Ramanujan Prize for his research achievements. Viana was vice-president of the International Mathematical Union in 2011–2014, and president of the Brazilian Mathematical Society (2013–2015). In 1998, he was a plenary speaker at the International Congress of Mathematicians, in Berlin.

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