We characterize chaotic linear operators on reflexive Banach spaces in terms of the existence long arithmetic progressions sets return times. also show that this characterization does not hold for arbitrary spaces. To achieve this, we study $$\mathcal {F}$$ -hypercyclicity a family subsets natural numbers associated to arbitrarily progressions.

In the paper, we study various countably generated algebras of entire analytic functions on complex Banach spaces and their homomorphisms. Countably often appear as symmetric with respect to a group symmetries, are interesting for possible applications. Some conditions existence topological isomorphisms between such obtained. We construct class algebras, where all normalized algebraic bases equ...

We show that, under suitable conditions, an operator acting like a shift on some sequence space has frequently hypercyclic random vector whose distribution is strongly mixing for the operator. This result will be applied to chaotic weighted shifts. also apply it every satisfying Frequent Hypercyclicity Criterion, recovering of Murillo and Peris.

We establish that if it is consistent that there is a supercompact cardinal, then it is consistent that every locally compact, hereditarily normal space which does not include a perfect pre-image of ω1 is hereditarily paracompact. This is the fifth in a series of papers ([LTo], [L2], [FTT], [LT], [T1] being the logically previous ones) that establish powerful topological consequences in models ...