On several dynamical properties of shifts acting on directed trees

Abstract

This paper explores the notions of $\mathcal{F}$-transitivity and topological $\mathcal{F}$-recurrence for backward shift operators on weighted ${\ell }^p$-spaces and $c_0$-spaces on directed trees, where $\mathcal{F}$ represents a Furstenberg family of subsets of ${\mathbb{N}}_0$. In particular, we establish the equivalence between recurrence and hypercyclicity of these operators on unrooted directed trees. For rooted directed trees, a backward shift operator is hypercyclic if and only if it possesses an orbit of a bounded subset that is weakly dense.

Cited by in corpus (2)

• Recurrence in collective dynamics: From the hyperspace to fuzzy dynamical systems
• Frequently hypercyclic composition operators on the little Lipschitz space of a rooted tree