A Birkhoff type transitivity theorem for non-separable completely metrizable spaces with applications to linear dynamics

Manoussos, Antonios

doi:10.7900/jot.2011may12.1971

Cited by 5 publications

(1 citation statement)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Historically, the underlying spaces considered in dynamics have been separable. This is the case of Linear Dynamics because of hypercyclicity (even though there are at least two remarkable exceptions, see [15] and [71]), but also for classical non-linear systems since the most usual set up is that of continuous maps acting on compact metrizable spaces, which are separable. Many references framed on the topic of compact dynamical systems have been used here (see [1,4,27,38,41,45,48,61,65]).…”

Section: Chaptermentioning

confidence: 99%

Combinatorial Number Theory, Recurrence of Operators and Linear Dynamics

López Martínez

View full text Add to dashboard Cite

Passem a les persones que més estime: la meua família. Sense ells no haguera pogut arribar on sóc, ni haguera gaudit tant del camí. Als meus pares, Toni i Mabel, pel seu incondicional suport, i pel gran deute que tinc amb ells i que mai podré pagar. A la meua germana, Isabel, per ser la meua companya i suportar-me tots aquests anys. A mi abuela, Maribel, por su paciencia y por haberme cuidado siempre. I als que ja no estan, que de segur, els hauria agradat celebrar amb mi aquesta etapa. Aquest treball està dedicat a tots ells.Per últim, i no menys important, a Clara. Per ser el pilar fonamental on m'he recolzat tot aquest temps. Per tots els moments viscuts fins ara, i per tots els que ens queden per viure junts. Gràcies de tot cor.València, 10 de juny de 2023 This memoir has been written at the "

show abstract

Section: Chaptermentioning

confidence: 99%

Combinatorial Number Theory, Recurrence of Operators and Linear Dynamics

López Martínez

View full text Add to dashboard Cite

show abstract

Coarse topological transitivity on open cones and coarsely J -class and D -class operators

Manoussos

2014

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

Abstract. We generalize the concept of coarse hypercyclicity, introduced by Feldman in [13], to that of coarse topological transitivity on open cones. We show that a bounded linear operator acting on an infinite dimensional Banach space with a coarsely dense orbit on an open cone is hypercyclic and a coarsely topologically transitive (mixing) operator on an open cone is topologically transitive (mixing resp.). We also "localize" these concepts by introducing two new classes of operators called coarsely J-class and coarsely D-class operators and we establish some results that may make these classes of operators potentially interesting for further studying. Namely, we show that if a backward unilateral weighted shift on l 2 (N) is coarsely J-class (or D-class) on an open cone then it is hypercyclic. Then we give an example of a bilateral weighted shift on l ∞ (Z) which is coarsely J-class, hence it is coarsely D-class, and not J-class. Note that, concerning the previous result, it is well known that the space l ∞ (Z) does not support J-class bilateral weighted shifts, see [10]. Finally, we show that there exists a non-separable Banach space which supports no coarsely D-class operators on open cones. Some open problems are added.

show abstract