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Chain recurrent shifts on trees

Andrew Mortensen, David Walmsley

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.

Chain recurrent shifts on trees

Abstract

We characterize when a weighted backward shift is chain recurrent on the , and spaces of a directed tree.
Paper Structure (5 sections, 10 theorems, 41 equations)

This paper contains 5 sections, 10 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. General background and definitions
  3. Directed trees
  4. Sequence spaces and weighted shifts on directed trees
  5. Characterization of recurrent backward shifts on trees

Key Result

Lemma 2.1

Let $(V,E)$ be an unrooted directed leafless tree, and fix $v_0\in V$ so as to enumerate the generations of the tree. If $k<n$, then Consequently, $|\mathop{\mathrm{Gen}}\nolimits_k|\leq |\mathop{\mathrm{Gen}}\nolimits_n|$ whenever $k<n$.

Theorems & Definitions (22)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • ...and 12 more