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 X r = { 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 X r , the intersection S ∩ U is uncountable, (ii) S − S ⊂ D ( T ) ∪ { 0 } ; and in particular, l i m i n f n → ∞ ‖ T n a − T n b ‖ = 0 and l i m s u p n → ∞ ‖ T n a − T n b ‖ = ∞ for any two distinct a , b ∈ S .