Frequently hypercyclic operators

By Fr'ed'eric Bayart, S. Grivaux. 2006

277 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/0d7f7bde650138144ec0313c0d07b4f44d3f803c

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In Algebra and Number Theory

In Applied Mathematics

Citations in the corpus

A new class of frequently hypercyclic operators (2010, background): “Then it was shown in [4] that if T has “sufficiently many eigenvectors associated to unimodular eigenvalues” (precise definitions will be given later on) then T is hypercyclic.”
Upper frequent hypercyclicity and related notions (2016, background): “On the other hand, a recent and by now central notion in linear dynamics, that of frequently hypercyclic operators as introduced by Bayart and Grivaux [2], [3], has to be studied without the use of Baire category.”
Linear dynamical systems on Hilbert spaces: typical properties and explicit examples (2017, background): “This closed subspace is spanned by unimodular eigenvectors of T (see [5] or [8] for details), from which it follows that T does not belong to NEVM (H).”
Unimodular eigenvalues, uniformly distributed sequences and linear dynamics (2007, methodology): “Operators with perfectly spanning set of eigenvectors associated to unimodular eigenvalues are always hypercyclic (see [3]) and when X is a Hilbert space, they are frequently hypercyclic (see [4], and [5] for a study of what happens in the general Banach space case).”
Distributional chaos for linear operators (2013, methodology): “Since a composition operator with a parabolic or hyperbolic automorphism in H(2)(D) has a perfectly spanning set of eigenvectors associated to unimodular eigenvalues with respect to the Lebesgue length measure on the unit circle [2], we obtain Corollary 24.”
Hypercyclic behaviour of operators in a hypercyclic C 0 -semigroup (2007, background): “1 The first and third authors were supported in part by MEC and FEDER, Project MTM2004-02262 and Research Net MTM2006-26627-E 2 The second author was partially supported by grant No. 201/06/0128 of GA ČR”
On the spectrum of frequently hypercyclic operators (2008, background): “Recently Bayart and Grivaux [5] have introduced the concept of frequently hypercyclic operators.”
Invariant measures for frequently hypercyclic operators (2014, background): “Frequent hypercyclicity, which was introduced in [2], is a strengthening of hypercyclicity which quantifies the frequency with which the orbit of a vector x ∈ X visits a given nonempty open set.”
Hilbertian Jamison sequences and rigid dynamical systems (2011, methodology): “Using the same kind of arguments as in [4] or [6, Ch.”
Recurrence properties of hypercyclic operators (2014, background): “Frequently hypercyclic operators were introduced by Bayart and Grivaux [2] while U-frequently hypercyclic operators were introduced by Shkarin [24].”
Mixing operators and small subsets of the circle (2011, methodology): “when the transformation T is a continuous linear operator acting on a topological vector space X ([25], [8], [7], [14], [28]).”
Linear subsets of nonlinear sets in topological vector spaces (2013, background): “In fact, the more popular operators (derivative operators, weighted backward shifts, composition operators) satisfy that their respective sets of frequent hypercyclic vectors are not residual; see [47] and [104].”
Difference sets and frequently hypercyclic weighted shifts (2013, background): “A necessary condition for Bw to be frequently hypercyclic was given in [2] and improved in [11].”
Frequently recurrent operators (2020, methodology): “y, positive upper Banach density, respectively). An operator that possesses such a vector is called frequently hypercyclic (upper frequently hypercyclic, reiteratively hypercyclic, respectively), see [3], [42], [9], [13], and the textbooks [5], [26]. Note that uniformrecurrence admits no hypercyclic”
Linear chaos and frequent hypercyclicity (2014, background): “er density, where the lower density of a set A of non-negative integers is given by dens(A) := liminf N→∞ #(A ∩[0,N]) N +1 . Several classical hypercyclic operators are in fact frequently hypercyclic [4]: the translation operators on H(C), the derivative operator on H(C),... We even know 2010 Mathem”