Density properties of orbits for a hypercyclic operator on a Banach space

Abstract We study density properties of orbits for a hypercyclic operator T on a separable Banach space X , and show that exactly one of the following four cases holds: (1) every vector in X is asymptotic to zero with density one; (2) generic vectors in X are distributionally irregular of type $1$ ; (3) generic vectors in X are distributionally irregular of type $2\frac {1}{2}$ and no hypercyclic vector is distributionally irregular of type $1$ ; (4) every hypercyclic vector in X is divergent to infinity with density one. We also present some examples concerned with weighted backward shifts on $\ell ^p$ to show that all the above four cases can occur. Furthermore, we show that similar results hold for $C_0$ -semigroups.