Constant-norm scrambled sets for hypercyclic operators

Abstract

Let T:X→X be a hypercyclic operator of a Banach space X, let D(T)={x∈X:x has a dense T-orbit}, and let Xr={x∈X:||x||=r} for r > 0. We show that there is a linearly independent subset S⊂D(T) with the following properties: (i) for any r > 0, and any nonempty, relatively open subset U of Xr, the intersection S∩U is uncountable, (ii) S-S⊂D(T)∪{0}; and in particular, liminfn→∞||Tna-Tnb||=0 and limsupn→∞||Tna-Tnb||=∞ for any two distinct a, b∈S. © 2011 Elsevier Inc.