New spectral multiplicities for ergodic actions

Abstract

Let G be a locally compact second countable Abelian group. Given a measure preserving action T of G on a standard probability space (X, µ), let M(T ) denote the set of essential values of the spectral multiplicity function of the Koopman representation U T of G defined in L 2 (X, µ) ⊖ C by U T (g)f := f •T −g . In the case when G is either a discrete countable Abelian group or R n , n 1, it is shown that the sets of the form {p, q, pq}, {p, q, r, pq, pr, qr, pqr} etc. or any multiplicative (and additive) subsemigroup of N are realizable as M(T ) for a weakly mixing G-action T .

Spectral theory of dynamical systems

“…See [54] and [59], where some of the results below proved for A = Z have been extended to (weakly mixing) flows. See also the case of so called product Z d -actions [86] and [275] for general countable Abelian group actions.…”

Spectral Theory of Dynamical Systems

“…See Danilenko and Lemańczyk (2013) and Danilenko and Solomko (2010), where some of the results below proved for  ¼ ℤ have been extended to (weakly mixing) flows. See also the case of so-called product ℤ d -actions (Filipowicz 1997) and (Solomko 2012) for general countable Abelian group actions.…”

О наборе спектральных кратностей $\{2,4,…,2^n\}$ для вполне эргодических $\mathbb{Z}^2$-действий

“…. , 2 } для эргодических Z 2 -действий вытекает также из более общих фактов, полученных в [8], [9]. Результат этой заметки анонсирован в 2009 г. в [6].…”