On several dynamical properties of shifts acting on directed trees

TL;DR

The paper develops an $ abla F$-transitivity framework for backward shifts on weighted $ abla$ell^p$ and $c_0$ spaces on directed trees, linking recurrence, hypercyclicity, and topological recurrence through both rooted and unrooted tree geometries. A central $ abla F$-transitivity criterion is established, enabling precise characterizations in rooted versus unrooted settings, including explicit weight-sum conditions and index-set intersections that govern transitivity. The work extends prior results on hypercyclicity to tree-indexed spaces and introduces $ abla extGamma$-supercyclicity, yielding unified criteria and revealing nuanced behaviors, such as the possibility of non-hypercyclic operators possessing nonzero limit-point orbits. These findings advance the understanding of linear dynamics on non-Euclidean index sets and provide concrete tools for assessing transitivity and recurrence in tree-structured operator settings.

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.

Theorems & Definitions (25)

• Definition 2.1: Directed trees
• Proposition 2.2
• Theorem 2.3
• Lemma 2.4: GrPa
• Theorem 3.1: $\mathcal{F}$-Transitivity Criterion
• proof
• Theorem 4.1
• proof
• Example 4.2
• Corollary 4.3