Yunied Puig de Dios – DISJOINT HYPERCYCLICITY ALONG FILTERS.

Imatge

Dimecres 19 de Febrer, 12:30 hores, Saló de Graus.

Disjoint hypercyclicity was introduced independently by Bernal G. and Bes-Peris in 2007. We extend and study this notion to some filter classes frequently used in Ramsey theory. In 2012 Bes-Martin-Peris-Shkarin have shown the following: a weighted shift Bw is mixing if and only if Bw, …, Bw^r is d-mixing, for any r>0. Now, we show a more general result by stating: Bw is F operator if and only if Bw, …, Bw^r is d-F operator for any r>0 and F=Δ*, IP*, PS*, S.

We point out that this phenomenon does not occur beyond the weighted 
shift frame by showing a mixing linear operator T such that the tuple T, T^2 is not d-S. Concerning the non-filter classes, we show that this result does not hold even for weighted shifts by exhibiting a weakly mixing weighted shift Bw such that the tuple Bw, Bw^2 is not d-topologically transitive.

On the other hand, all of this is very connected with a notion introduced by Salas in 2013, namely, the strong disjoint Blow-up/Collapse property. Now, in this sense an extension to filters cast more light on the nature of this property allowing to make some progress in order to answer some of the questions posed by Salas in his paper.

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