Perturbations of hypercyclic vectors

We show that a linear operator can have an orbit that comes within a bounded distance of every point, yet is not dense.We also prove that such an operator must be hypercyclic. This gives a more general form of the hypercyclicity criterion. We also show that a sufficiently small perturbation of a hypercyclic vector is still hypercyclic.  2002 Elsevier Science (USA). All rights reserved.