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Shifts on trees versus classical shifts in chain recurrence

Antoni López-Martínez, Dimitris Papathanasiou

TL;DR

The paper tackles whether the chain-recurrence property is preserved under restriction to $CR(T)$ for continuous linear operators on Banach or Hilbert spaces. It develops a tree-based shift framework and constructs explicit counterexamples: a non-invertible and an invertible weighted backward shift on directed trees whose restriction to $CR(T)$ is not chain recurrent, with $CR(T)$ infinite-dimensional in the non-invertible case and controlled in the invertible one. It also shows that classical unilateral or bilateral weighted backward shifts cannot yield such a counterexample, highlighting a fundamental difference between shifts on trees and classical shifts in chain recurrence. The results advance the understanding of how chain recurrence interacts with invariant subspaces and demonstrate the power of tree-based operator models in Linear Dynamics.

Abstract

We construct continuous (and even invertible) linear operators acting on Banach (even Hilbert) spaces whose restrictions to their respective closed linear subspaces of chain recurrent vectors are not chain recurrent operators. This construction completely solves in the negative a problem posed by Nilson C. Bernardes Jr. and Alfred Peris on chain recurrence in Linear Dynamics. In particular: we show that the non-invertible case can be directly solved via relatively simple weighted backward shifts acting on certain unrooted directed trees; then we modify the non-invertible counterexample to address the invertible case, but falling outside the class of weighted shift operators; and we finally show that this behaviour cannot be achieved via classical (unilateral neither bilateral) weighted backward sifts (acting on $\mathbb{N}$ and $\mathbb{Z}$ respectively) by noticing that a classical shift is a chain recurrent operator whenever it admits a non-zero chain recurrent vector.

Shifts on trees versus classical shifts in chain recurrence

TL;DR

The paper tackles whether the chain-recurrence property is preserved under restriction to for continuous linear operators on Banach or Hilbert spaces. It develops a tree-based shift framework and constructs explicit counterexamples: a non-invertible and an invertible weighted backward shift on directed trees whose restriction to is not chain recurrent, with infinite-dimensional in the non-invertible case and controlled in the invertible one. It also shows that classical unilateral or bilateral weighted backward shifts cannot yield such a counterexample, highlighting a fundamental difference between shifts on trees and classical shifts in chain recurrence. The results advance the understanding of how chain recurrence interacts with invariant subspaces and demonstrate the power of tree-based operator models in Linear Dynamics.

Abstract

We construct continuous (and even invertible) linear operators acting on Banach (even Hilbert) spaces whose restrictions to their respective closed linear subspaces of chain recurrent vectors are not chain recurrent operators. This construction completely solves in the negative a problem posed by Nilson C. Bernardes Jr. and Alfred Peris on chain recurrence in Linear Dynamics. In particular: we show that the non-invertible case can be directly solved via relatively simple weighted backward shifts acting on certain unrooted directed trees; then we modify the non-invertible counterexample to address the invertible case, but falling outside the class of weighted shift operators; and we finally show that this behaviour cannot be achieved via classical (unilateral neither bilateral) weighted backward sifts (acting on and respectively) by noticing that a classical shift is a chain recurrent operator whenever it admits a non-zero chain recurrent vector.
Paper Structure (12 sections, 7 theorems, 43 equations, 4 figures)

This paper contains 12 sections, 7 theorems, 43 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation and general background
  3. Chain recurrence in Linear Dynamics
  4. Fréchet sequence spaces and weighted shifts on directed trees
  5. A bilateral backward shift with some extra branches
  6. Calculating $CR(B_{\boldsymbol{\lambda}})$: the first inclusion
  7. Calculating $CR(B_{\boldsymbol{\lambda}})$: the second inclusion
  8. The subspace of chain recurrent vectors can be finite-dimensional
  9. The invertible case
  10. Calculating $CR(T)$: the first inclusion
  11. Calculating $CR(T)$: the second inclusion
  12. Classical backward shifts are not enough

Key Result

Theorem 1.2

There exist a Banach (even Hilbert) space $X$ and a continuous (and even invertible) linear operator $T:X\longrightarrow X$ such that the restriction of $T$ to its closed $T$-invariant subspace of chain recurrent vectors $T\vert_{CR(T)}:CR(T)\longrightarrow CR(T)$ is not a chain recurrent operator.

Figures (4)

  • Figure 1: Graphic representation of the directed tree $V$ from Section \ref{['Sec_3:non-invertible']}.
  • Figure 2: Graphic representation of the backward shift $B_{\boldsymbol{\lambda}}$ from Section \ref{['Sec_3:non-invertible']}.
  • Figure 3: Graphic representation of the directed tree $V'$ from Subsection \ref{['SubSec_3.3:finite']}.
  • Figure 4: Graphic representation of the backward shift $B_{\boldsymbol{\lambda'}}$ from Subsection \ref{['SubSec_3.3:finite']}.

Theorems & Definitions (17)

  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Theorem 3.1
  • proof
  • proof
  • proof
  • proof
  • ...and 7 more