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arXiv · 2507.19752

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

Jian Li·Xinsheng Wang·Jianjie Zhao
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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.

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Jian Li, Xinsheng Wang, Jianjie Zhao. 2025-07-26. Density properties of orbits for a hypercyclic operator on a Banach space. https://doi.org/10.4153/s0008439525100908

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