On the self-similarity problem for ergodic flows

Abstract

Given an ergodic flow (Tt)t ∈ ℝ we study the problem of its self-similarities, that is, we want to describe the set of s ∈ ℝ for which the original flow is isomorphic to the flow (Tst)t ∈ ℝ. The problem is examined in some classes of special flows over irrational rotations and over interval exchange transformations. In particular, translation flows on translation surfaces are considered: we prove that under the weak mixing condition the set of self-similarities has Lebesgue measure zero. For von Neumann special flows over irrational rotations given by Diophantine numbers, this set is shown to be equal to {1}, while for horocycle flows a weak convergence in case of some singular (with respect to the volume measure) measures is shown to give rise to some new equidistribution result. The problem of self-similarity is also studied from the spectral point of view, especially in the class of Gaussian systems.