Bayart and Ruzsa [Difference sets and frequently hypercyclic weighted shifts. Ergod. Th. & Dynam. Sys.35 (2015), 691–709] have recently shown that every frequently hypercyclic weighted shift on $\ell ^p$ is chaotic. This contrasts with an earlier result of Bayart and Grivaux [Frequently hypercyclic operators. Trans. Amer. Math. Soc.358 (2006), 5083–5117], who constructed a non-chaotic frequently hypercyclic weighted shift on $c_0$ . We first generalize the Bayart–Ruzsa theorem to all Banach sequence spaces in which the unit sequences form a boundedly complete unconditional basis. We then study the relationship between frequent hypercyclicity and chaos for weighted shifts on Fréchet sequence spaces, in particular, on Köthe sequence spaces, and then on the special class of power series spaces. We obtain, rather curiously, that every frequently hypercyclic weighted shift on $H(\mathbb {D})$ is chaotic, while $H(\mathbb {C})$ admits a non-chaotic frequently hypercyclic weighted shift.

中文翻译：

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加权移位的混沌和频繁超循环

Bayart 和 Ruzsa [差异集和频繁的超循环加权移位。埃尔戈德。钍。&动态。系统。35(2015), 691–709] 最近表明，每个频繁的超循环加权移位$\ell ^p$是混乱的。这与 Bayart 和 Grivaux [Frequently hypercyclic operator。反式。阿米尔。数学。社会党。358(2006), 5083–5117]，他在$c_0$. 我们首先将 Bayart-Ruzsa 定理推广到所有 Banach 序列空间，其中单位序列形成有界完全无条件基。然后，我们研究了 Fréchet 序列空间，特别是 Köthe 序列空间，然后是特殊类的幂级数空间上的频繁超循环和混沌之间的关系。我们得到，相当奇怪的是，每一个频繁的超循环加权移位$H(\mathbb {D})$是混乱的，而$H(\mathbb {C})$承认非混沌频繁超循环加权移位。