2023/03/07 by Karl‐G. Grosse‐Erdmann, Grosse-Erdmann, Karl-G., Dimitris Papathanasiou +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · #05C05 #11A55 #46A45 #47A16 #47B37 #Approximation Theory and Sequence Spaces #FOS: Mathematics #Fibroblast Growth Factor Research #Functional Analysis (math.FA)
paper · pdf · doi:10.48550/arxiv.2303.03980
openalex publication_date 2023/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the dynamical behaviour of weighted backward shift operators defined on sequence spaces over a directed tree. We provide a characterization of chaos on very general Fréchet sequence spaces in terms of the existence of a large supply of periodic points, or of fixed points. In the special case of the space ℓp, 1≤ p<∞, or the space c0 over the tree, we provide a characterization directly in terms of the weights of the shift operators. It has turned out that these characterizations involve certain generalized continued fractions that are introduced in this paper. Special attention is given to weighted backward shifts with symmetric weights, in particular to Rolewicz operators. In an appendix, we complement our previous work by characterizing hypercyclic and mixing weighted backward shifts on very general Fréchet sequence spaces over a tree. Also, some of our results have a close link with potential theory on flows over trees; the link is provided by the notion of capacity, as we explain in an epilogue.