Temporal limit theorems for irrational rotations

Jeudi 25 avril 2019 14:00-15:00 - Omri Sarig - Weizmann Institue (Tel Aviv)

Résumé : A temporal limit theorem for a dynamical system T:X—>X, test function f:X—>R, and a fixed initial condition x_0 is a scaling limit for the distributions of S(n,x_0)=f(x_0)+f(T x_0)+...+f(T^n x_0) , when n is chosen randomly uniformly between 1 and N, and N—> infinity.
Such laws are interesting, because they sometimes hold for low complexity maps T, for which the more traditional limit theorems, where the initial condition x_0 is sampled randomly, fail. They are also interesting because they can exhibit different qualitative behavior for different initial conditions, for uniquely ergodic maps. Examples include irrational rotations, horocycle flows and some substitution systems.
The talk will focus on the following question : Which irrational rotations satisfy temporal DLT for piecewise smooth, mean zero, f:X—>R ? The set of such angles has full Hausdorff dimension (Bromberg, Ulcigrai). We show it has zero Lebesgue measure.
Joint work with D. Dolgopyat.

Lieu : Institut de Mathématique d’Orsay, salle 2L8

Notes de dernières minutes : Café culturel à 13h par Damien Thomine.

Temporal limit theorems for irrational rotations  Version PDF