Fast orbital convergence reveals more hypercyclic vectors

Let X be an infinite dimensional separable Banach space, T : X → X be a hypercyclic operator, and x ∈ X be a (frequently) hypercyclic vector of T. We show that if the terms from the T-orbit of x converge to a vector y sufficiently fast, then y is also a hypercyclic vector of T. As a corollary, we deduce that if T is a frequently hypercyclic operator with spectral radius r(T) = 1, then limn→ ∞ ∥Tnx∥1/n = 1 for every frequently hypercyclic vector x of T. Some related observations are also made.