Chain recurrent shifts on trees

TL;DR

This paper characterizes chain recurrence of weighted backward shifts on tree-based sequence spaces using divergence conditions on weights, generalizing classical results for symmetric trees.

Contribution

It provides a new characterization of chain recurrence for weighted shifts on trees, extending classical sequence space results to more general tree structures.

Findings

Characterization in terms of divergence conditions on weights

Conditions reduce to classical cases for symmetric trees

Applicable to $ ext{ell}^p$ and $c_0$ spaces on directed trees

Abstract

We characterize when a weighted backward shift is chain recurrent on the $\ell^p$ ($1\leq p<\infty$) and $c_0$ spaces of a directed tree. The characterization is given in terms of two divergence conditions on the weights: a forward condition on the descendants of each vertex and, in the unrooted case, a backward condition on the descendants of each ancestor. The conditions reduce, in the case of symmetric weighted shifts on symmetric trees, to the classical characterizations of chain recurrence on the sequence spaces $\ell^p(\mathbb{N})$, $\ell^p(\mathbb{Z})$, $c_0(\mathbb{N})$, and $c_0(\mathbb{Z})$.