Hilbertian Jamison sequences and rigid dynamical systems

Abstract

A strictly increasing sequence (n k ) k 0 of positive integers is said to be a Hilbertian Jamison sequence if for any bounded operator T on a separable Hilbert space such that sup k 0 T n k < +∞, the set of eigenvalues of modulus 1 of T is at most countable. We first give a complete characterization of such sequences. We then turn to the study of rigidity sequences (n k ) k 0 for weakly mixing dynamical systems on measure spaces, and give various conditions, some of which are closely related to the Jamison condition, for a sequence to be a rigidity sequence. We obtain on our way a complete characterization of topological rigidity and uniform rigidity sequences for linear dynamical systems, and we construct in this framework examples of dynamical systems which are both weakly mixing in the measure-theoretic sense and uniformly rigid.

Recurrent Linear Operators

“…In the context of topological dynamical systems the notions of rigidity and uniform rigidity have been introduced by Glasner and Maon, [27]. These notions have also been studied in linear dynamics for example in [21,22]. The corresponding definitions are as follows.…”

Kazhdan constants, continuous probability measures with large Fourier coefficients and rigidity sequences

“…, k r ≥ 0}. In other words, we show that certain large subsets of this set form are, when taken in a strictly increasing order, rigidity sequences in the sense of [8] or [19]. Recall that a dynamical system (X, B, m; T ) on a Borel probability space is called rigid if there exists a strictly increasing sequence of integers (n k ) k≥1 such that ||U n k T f − f || → 0 as k → +∞ for every f ∈ L 2 (X, B, m), where U T denotes as usual the Koopman operator f → f • T associated to T on L 2 (X, B, m).…”

“…Using Gaussian dynamical systems, one can show that (n k ) k≥1 is a rigidity sequence if and only if there exists a measure µ ∈ P c (T) such that µ(n k ) → 1 as k → +∞. The study of rigidity sequences was initiated in [8] and [19]. Further works on this topic include the papers [1], [3], [2], [25], [22] [21] and [24] among others.…”

“…As mentioned in Section 2, this statement is well-known: it suffices to consider the classical Riesz product associated to the sequence (p k ) k≥0 . One can also show, either as in [8] or [19], or as an application of Corollary 2.5, that (p k ) k≥0 is a rigidity sequence, so that there exists µ ∈ P c (T) with µ(p k ) → 1 as k → +∞.…”

Kazhdan sets in groups and equidistribution properties

“…Using the ideas of [12], this was generalized by Devinck in [10] to a larger class of Banach spaces. We recall here that a separable Banach space X admits an unconditional Schauder decomposition if there exists a sequence pX ℓ q ℓě1 of closed subspaces of X (different from t0u) such that any vector x of X can be written in a unique way as an unconditionally convergent series ř ℓě1 x ℓ , where x ℓ belongs to X ℓ for all ℓ ě 1.…”