We show that each power bounded operator with spectral radius equal to one on a reflexive Banach space has a nonzero vector which is not supercyclic. Equivalently, the operator has a nontrivial closed invariant homogeneous subset. Moreover, the operator has a nontrivial closed invariant cone if 1 belongs to its spectrum. This generalizes the corresponding results for Hilbert space operators. Fo...

We study dynamical notions lying between $$\mathcal{U}$$ -frequent hypercyclicity and reiterative by investigating weighted upper densities the unweighted density Banach density. While chaos implies hypercyclicity, we show that does not imply with respect to any Moreover, if T is -frequently hypercyclic (resp. reiteratively hypercyclic) then n-fold product of still U-frequently this implicatio...