Frequently hypercyclic operators

Abstract

Abstract. We investigate the subject of linear dynamics by studying the notion of frequent hypercyclicity for bounded operators T on separable complex F -spaces: T is frequently hypercyclic if there exists a vector x such that for every nonempty open subset U of X, the set of integers n such that T n x belongs to U has positive lower density. We give several criteria for frequent hypercyclicity, and this leads us in particular to study linear transformations from the point of view of ergodic theory. Several other topics which are classical in hypercyclicity theory are also investigated in the frequent hypercyclicity setting.

Frequently hypercyclic operators: Recent advances and open problems

“…However, in some cases one may obtain frequently hypercyclic orbits by a (countable) constructive procedure. Such a construction is feasible if the operator satisfies the so-called Frequent Hypercyclicity Criterion, see [5], [6], [27].…”

“…Let us briefly mention that frequently hypercyclic operators are also found among other classes of operators. For example, on the space H(C) of entire functions the differentiation operator D : H(C) → H(C), Df = f ′ , and the translation operator T : H(C) → H(C), T f (z) = f (z + 1), are frequently hypercyclic [6]. More generally, any non-scalar operator T on H(C) that commutes with D is frequently hypercyclic [25].…”

“…For an introduction to linear dynamics we refer to the recent textbooks [8] and [43]. The concept of a frequently hypercyclic operator was introduced in 2004 by F. Bayart and S. Grivaux [5], [6]. The idea is to measure the size of the set N (x, U ) = {n ≥ 0 : T n x ∈ U } of powers of T that send a given vector x into a non-empty open set U ⊂ X.…”

A note on hypercyclicity sets

“…In particular, if m k = k, then T is frequently hypercyclic. See [1,4,3,7] and [2] for definitions of frequently hypercyclic operators and m k -hypercyclic operators.…”

Rate of growth of frequently hypercyclic functions

“…In [15], a Frequent Universality Criterion was obtained that generalizes the Frequent Hypercyclicity Criterion of Bayart and Grivaux [11]. We state it here only for Fréchet spaces.…”