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arXiv · 1703.09172

Frequent hypercyclicity and piecewise syndetic recurrence sets

Yunied Puig
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Abstract

Motivated by a question posed by Sophie Grivaux concerning the regularity of the orbits of frequently hypercylic operators, we show the following: for any operator $T$ on a separable metrizable and complete topological vector space $X$ which is both frequently hypercyclic and piecewise syndetic hypercyclic, the lower density and upper Banach density of the recurrence set $\{n\geq 1: T^n x\in U\}$ are different, for any hypercyclic vector $x\in X$ for $T$, and a certain collection of non-empty open sets $U\subseteq X$. As an immediate consequence we got a sufficient condition for a chaotic operator to be non frequently hypercyclic.

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Yunied Puig. 2019-06-22. Frequent hypercyclicity and piecewise syndetic recurrence sets. https://arxiv.org/abs/1703.09172

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