Constant-norm scrambled sets for hypercyclic operators

Let T : X -> X be a hypercyclic operator of a Banach space X, let D(T) = {x e X: x has a dense T-orbit). and let X(r) = {x is an element of X: vertical bar vertical bar x vertical bar vertical bar = r} for r > 0. We show that there is a linearly independent subset S c D(T) with the following properties: (i) for any r > 0, and any nonempty, relatively open subset U of X(r), the intersection S boolean AND U is uncountable, (ii) S S c D(T) U (0); and in particular, lim inf(n ->infinity)) vertical bar vertical bar T(n)a - T(n)b vertical bar vertical bar = 0 and limsu(pn ->infinity)vertical bar vertical bar T(n)a - T(n)b vertical bar vertical bar = infinity for any two distinct a, b is an element of S. (C) 2011 Elsevier Inc. All rights reserved.