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 $\mathcal{F}$-recurrence under very weak assumptions on the Furstenberg family $\mathcal{F}$. This allows us, as a by-product, to deduce Ansari- and Léon-Müller-type theorems for $\mathcal{F}$-hypercyclicity.

References in corpus (1)

• Linear dynamical systems on Hilbert spaces: typical properties and explicit examples

Cited by in corpus (11)

• Recurrence properties for linear dynamical systems: An approach via invariant measures
• Recurrent subspaces in Banach spaces
• Two remarks on the set of recurrent vectors
• On Shadowing and Chain Recurrence in Linear Dynamics
• On several dynamical properties of shifts acting on directed trees
• Invariant measures from locally bounded orbits
• Recurrence in collective dynamics: From the hyperspace to fuzzy dynamical systems
• Disjoint hypercyclicity, Sidon sets and weakly mixing operators
• Shifts on trees versus classical shifts in chain recurrence
• Zero-one law of orbital limit points for weighted shifts
• Frequently hypercyclic composition operators on the little Lipschitz space of a rooted tree