Constant-norm scrambled sets for hypercyclic operators (Q655525)

In this note, the author proves the following result on the scrambled set for a hypercyclic operator \(T\) on a separable Banach space \(X\): There is a linearly independent set \(S\) of hypercyclic vectors of \(T\) such that, for every \(r>0\) and for ever relatively open subset \(U\) of the sphere of radius \(r\) and center \(0\) in \(X\), the intersection \(S \cap U\) is nonempty, and every non-zero element of \(S-S\) is hypercyclic. The proof uses transfinite induction.