Frequently recurrent operators

Abstract

Motivated by a recent investigation of Costakis et al. on the notion of recurrence in linear dynamics, we study various stronger forms of recurrence for linear operators, in particular that of frequent recurrence. We study, among other things, the relationship between a type of recurrence and the corresponding notion of hypercyclicity, the influence of power boundedness, and the interplay between recurrence and spectral properties. We obtain, in particular, Ansari-and Léon-Müller-type theorems for F -recurrence under very weak assumptions on the Furstenberg family F . This allows us, as a by-product, to deduce Ansari-and Léon-Müller-type theorems for Fhypercyclicity.

Multiple recurrence and hypercyclicity

“…For more F -hypercyclicity see [9,14,15,17,24,35,31,32]. The following general result was proved in [14].…”

“…Recall that, given a family F of natural numbers, a vector x is said to be F -recurrent for T provided that for every open set U containing x the set of hitting times N T (x,U ) ∈ F (see [25,15]). If the set of F -recurrent vectors is dense, then the operator is said to be F -recurrent.…”

“…Shkarin in [35,Section 5] proved that for any right shift invariant Ramsey family F , if an operator is Fhypercyclic then so are its powers and its rotations (see also [15]). Thus by van Waerden's theorem we have the following.…”

“…T is not weakly mixing). Over the last years much of the attention was driven to special types of hypercyclicity like frequent hypercyclicity [3], upper frequent hypercyclicity [35] and more recently to F -hypercyclicity [9,10,14,15], for more general families F of subsets of N.…”

On super-rigid and uniformly super-rigid operators

“…and we have that T is recurrent if and only if Rec(T ) is dense in X. For more information about this classe of operators, see [1,7,8,9,11,12,14,18,19,21].…”