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Density properties of orbits for a hypercyclic operator on a Banach space
- Part of
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- Journal:
- Canadian Mathematical Bulletin , First View
- Published online by Cambridge University Press:
- 18 July 2025, pp. 1-16
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- Article
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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.
