The full delocalization of eigenstates for the quantized cat map

Abstract:

A central theme in quantum chaos is the study of spectral problems arising in the settings of hyperbolic dynamics. One of the most well known examples for such dynamics is compact hyperbolic manifolds with constant negative curvature, on which one considers the Laplace-Beltrami operator. In our talk we recall a chaotic toy example of this model living on the 2-dimensional torus T2 and called “quantum cat map”. Then we present an analogue of a result originally proved by Dyatlov and Jin in the settings of compact hyperbolic surfaces. Roughly speaking our result means that semiclassical measures, a measure-theoretic invariant of the cat map, cannot concentrate on a proper open set of the torus. The proof relies on semi-classical methods and on the fractal uncertainty principle proved by Bourgain and Dyatlov in 2016.