Spectral Properties of Generic Dynamical Systems

Generic spectral properties of measure-preserving maps and applications

“…Proof of Lemma 1.2. This is a strengthening of Lemma 3.9 of [K] (see also [St,Theorem 3 / z"'dp -* ak{l) ■ ■ ■ ak(i) and / z"'dp -* ak,{l) ■ ■ ■ ak,{¡l),…”

A note on the isomorphism of Cartesian products of ergodic flows

“…He proved that automorphisms T and S are isomorphic provided that their Cartesian powers T ×d and S ×d (for some d ≥ 1) are isomorphic and T is α-weakly mixing. This result, taking into account that α-weak mixing is generic [17], gives therefore the positive answer to (1) for a typical automorphism. In [16] Ryzhikov and Troitskaya strengthened the result from [14] by replacing α-weak mixing with the existence of a polynomial in the weak closure of time automorphisms.…”

Stochastic intertwinings and multiple mixing of dynamical systems

“…The first observation from [24] can be formulated in the following form: if mixing Z-action is not multiple mixing, then its spectral measure is not disjoint from the convolution a * a. This shows a connection of Kolmogorov's problem on group property of spectrum (see [6]- [8], [19]) with the multiple mixing problem.…”