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We prove that reiteratively hypercyclic operators have perfect spectrum.Consequently, it follows that there exist separable infinite dimensional Banachspaces that do not support any reiteratively hypercyclic operator. For this, westudy $\mathcal F$-recurrence and almost $\mathcal {F}$-recurrence of operatorsfor general families and in particular for a special class of families, calledblock families.

Frequently recurrence properties and block families