Somewhere dense Cesàro orbits and rotations of Cesàro hypercyclic operators
Let T be a continuous linear operator acting on a Banach space X. We examine whether certain fundamental results for hypercyclic operators are still valid in the Cesàro hypercyclicity setting. In particular, in connection with the somewhere dense orbit theorem of Bourdon and Feldman, we show that if for some vector x ∈ X the set Tx,T²/2 x,T³/3 x, ... is somewhere dense then for every 0 < ε < 1 the set (0,ε)Tx,T²/2 x,T³/3 x,... is dense in X. Inspired by a result of Feldman, we also prove...