Recurrence properties for linear dynamical systems: An approach via invariant measures

TL;DR

This paper explores recurrence properties of linear dynamical systems using ergodic theory, establishing equivalences between different recurrence notions and linking them to invariant measures and hypercyclicity.

Contribution

It introduces a novel approach connecting recurrence notions with invariant measures and provides new characterizations for operators on Banach and Hilbert spaces.

Findings

Reiterative recurrence coincides with frequent recurrence.

Uniform recurrence relates to unimodular eigenvectors.

Non-zero reiterative recurrence is typical in the Baire category sense.

Abstract

We study different pointwise recurrence notions for linear dynamical systems from the Ergodic Theory point of view. We show that from any reiteratively recurrent vector $x_0$, for an adjoint operator $T$ on a separable dual Banach space $X$, one can construct a $T$-invariant probability measure which contains $x_0$ in its support. This allows us to establish some equivalences, for these operators, between some strong pointwise recurrence notions which in general are completely distinguished. In particular, we show that (in our framework) reiterative recurrence coincides with frequent recurrence; for complex Hilbert spaces uniform recurrence coincides with the property of having a spanning family of unimodular eigenvectors; and the same happens for power-bounded operators on complex reflexive Banach spaces. These (surprising) properties are easily generalized to product and inverse dynamical systems, which implies some relations with the respective hypercyclicity notions. Finally we study how typical is an operator with a non-zero reiteratively recurrent vector in the sense of Baire category.