On spectral invariants in modern ergodic theory

Abstract

This is a short survey of recent developments in one of the oldest areas of ergodic theory, sometimes called the spectral theory of dynamical systems. We mainly discuss the spectral realization problem in the rich class of all invertible measure preserving dynamical systems, a “behavior” of different spectral invariants in natural subclasses of dynamical systems, and a complete solution of Rokhlin’s problem on homogeneous spectrum in ergodic theory.