Dynamical Systems Seminar

The variational principle states that the topological entropy of a compact dynamical system is a supremum of measure-theoretic entropies of invariant measures supported on this system. Therefore, one may ask whether we can get a similar formula for the topological entropy of a dynamical system restricted to the level sets, which are usually not compact. In several cases it was then possible to prove the so-called restricted variational principle formula: For every possible value $\alpha$ of the Lyapunov exponent, the topological entropy of the set of points with the Lyapunov exponent $\alpha$ is equal to the supremum of measure-theoretic entropies of invariant measures with Lyapunov exponent $\alpha$. In this talk, I will investigate the structure of the level sets of all Lyapunov exponents for typical cocycles. I will show that the restricted variational principle formula for a vector of Lyapunov exponents holds for typical cocycles. This generalizes the works of Barreira-Gelfert and Feng-Huang.

Suppose we have a collection of objects such as such as the group of homeomorphisms or diffeomorphisms. We choose an object from this collection at “random”. What dynamical properties does this object have? The usual notion of a “random” or a “large” collection is that of generic or comeager, i.e., we show that the collection of all objects with certain dynamical properties contain a countable intersection of dense open sets. There is an alternate natural notion of bigness called prevalence and its complement called shy sets or Haar null sets. This notion of smallness often highlights different properties than the usual notion of generic. In this introductory talk, we give a background on prevalence, discuss a variety of results involving prevalence and analysis, prevalence and automorphism groups and state some open problems concerning prevalence and dynamics.