Approximation of chaotic operators

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Tian, Geng
Shi, Luoyi
Zhu, Sen
Hou, Bingzhe
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Abstract
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As well-known, the concept "hypercyclic" in operator theory is the same as the concept "transitive" in dynamical system. Now the class of hypercyclic operators is well studied. Following the idea of research in hypercyclic operators, we consider classes of operators with some kinds of chaotic properties in this article. First of all, the closures of the sets of all Li-Yorke chaotic operators or distributionally chaotic operators are discussed. We give a spectral description of them and prove that the two closures coincide with each other. Moreover, both the set of all Li-Yorke chaotic operators and the set of all distributionally chaotic operators have nonempty interiors which coincide with each other as well. The article also includes the containing relation between the closure of the set of all hypercyclic operators and the closure of the set of all distributionally chaotic operators. Finally, we get connectedness of the sets considered above.
Comment: 20 pages, 1 figure
Keywords
Mathematics - Functional Analysis, Mathematics - Dynamical Systems, 47A55, 47A53, 54H20, 37B99
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