论文标题
组成操作员的混乱和频繁的高度环境
Chaos and frequent hypercyclicity for composition operators
论文作者
论文摘要
混乱和频繁的高度环境的概念在线性动力学中享有亲密关系。确实，在一系列部分结果之后，Bayart和Rusza在2015年表明，对于$ \ ell_p（\ Mathbb {z}）$的向后加权转移，概念混乱和频繁的超循环性重合。正是通过一些努力表明这两个概念是不同的。 Bayart和Grivaux于2007年在$ C_0 $上构建了一种非偶然的高度加权转移。直到2017年，Menet才对每个混乱的操作员经常过度地定居。在本文中，我们表明，对于$ l^p $ space上的一大批作曲运算符，混乱的概念和频繁的超循环量重合。此外，在这个特定类中，可逆操作员经常是超环状的，并且仅当其逆时经常是超循环时。这与MENET的最新结果相反，MENET的最新结果经常在$ \ ell_1 $上进行可逆性操作员，其反向并不经常构建超循环。
The notions of chaos and frequent hypercyclicity enjoy an intimate relationship in linear dynamics. Indeed, after a series of partial results, it was shown by Bayart and Rusza in 2015 that for backward weighted shifts on $\ell_p(\mathbb{Z})$, the notions chaos and frequent hypercyclicity coincide. It is with some effort that one shows that these two notions are distinct. Bayart and Grivaux in 2007 constructed a non-chaotic frequently hypercyclic weighted shift on $c_0$. It was only in 2017 that Menet settled negatively whether every chaotic operator is frequently hypercylic. In this article, we show that for a large class of composition operators on $L^p$-spaces the notions of chaos and frequent hypercyclicity coincide. Moreover, in this particular class an invertible operator is frequently hypercyclic if and only if its inverse is frequently hypercyclic. This is in contrast to a very recent result of Menet where an invertible frequently hypercyclic operator on $\ell_1$ whose inverse is not frequently hypercyclic is constructed.
