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.
Feldman, N. S. (2002). Perturbations of hypercyclic vectors. Journal of Mathematical Analysis and Applications, 273(1), 67–74. https://doi.org/10.1016/S0022-247X(02)00207-X