Chain recurrent shifts on trees

Abstract

We characterize when a weighted backward shift is chain recurrent on the ${\ell }^p$ ($1\le p<\infty$) 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 $c_0(\mathbb{Z})$.