Epsilon-hypercyclic operators

Abstract

For each fixed number " in (0, 1) we construct a bounded linear operator on the Banach space `1 having a certain orbit which intersects every cone of aperture ", but with every orbit avoiding a certain ball of radius d, for every d > 0. This answers a question from [8]. On the other hand, if T is an operator on the Banach space X such that for every " > 0 there is a point in X whose orbit under the action of T meets every cone of aperture ", then T has a dense orbit.