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Questions in linear recurrence: From the $T\oplus T$-problem to lineability

Sophie Grivaux, Antoni López-Martínez, Alfred Peris

TL;DR

This paper studies recurrence for linear operators on infinite-dimensional spaces, introducing quasi-rigidity as the recurrence analogue of weak-mixing and linking it to the recurrence of all finite direct sums via $T_{(N)}$. It then constructs recurrent but not quasi-rigid operators to give a negative answer to the $T\oplus T$-recurrence problem, showing recurrence stability can fail under direct sums. The authors develop an $\\mathcal{F}$-recurrence framework grounded in Furstenberg families, proving that the set of $\\mathcal{F}$-recurrent vectors is (dense) lineable and that these properties extend to $N$-fold direct sums, including a generalized Herrero–Bourdon phenomenon for $\\mathcal{F}$-hypercyclicity. Overall, the work unifies recurrence and lineability in Linear Dynamics, introduces robust tools for studying $\\mathcal{F}$-recurrence under quasi-conjugacies and commutants, and raises several open problems about recurrence behavior under broader families.

Abstract

We study, for a continuous linear operator $T$ on an F-space $X$, when the direct sum operator $T\oplus T$ is recurrent on $X\oplus X$. In particular: we establish, for recurrence, the analogous notion to that of (topological) weak-mixing for transitivity/hypercyclicity, namely quasi-rigidity; and we construct a recurrent but not quasi-rigid operator on each infinite-dimensional Banach space, solving the $T\oplus T$-recurrence problem in the negative way. The quasi-rigidity notion is closely related to the dense lineability of the set of recurrent vectors, and using similar conditions we study the lineability and dense lineability properties for the set of $\mathcal{F}$-recurrent vectors. This document has been split into two already published papers: Part I - Questions in linear recurrence I: The $T\oplus T$-recurrence problem. Analysis and Mathematical Physics, Volume 15, article number 1, (2025). https://doi.org/10.1007/s13324-024-00999-8 Part II - Questions in linear recurrence II: Lineability properties. Banach Journal of Mathematical Analysis, Volume 19, article number 61, (2025). https://doi.org/10.1007/s43037-025-00448-z

Questions in linear recurrence: From the $T\oplus T$-problem to lineability

TL;DR

This paper studies recurrence for linear operators on infinite-dimensional spaces, introducing quasi-rigidity as the recurrence analogue of weak-mixing and linking it to the recurrence of all finite direct sums via . It then constructs recurrent but not quasi-rigid operators to give a negative answer to the -recurrence problem, showing recurrence stability can fail under direct sums. The authors develop an -recurrence framework grounded in Furstenberg families, proving that the set of -recurrent vectors is (dense) lineable and that these properties extend to -fold direct sums, including a generalized Herrero–Bourdon phenomenon for -hypercyclicity. Overall, the work unifies recurrence and lineability in Linear Dynamics, introduces robust tools for studying -recurrence under quasi-conjugacies and commutants, and raises several open problems about recurrence behavior under broader families.

Abstract

We study, for a continuous linear operator on an F-space , when the direct sum operator is recurrent on . In particular: we establish, for recurrence, the analogous notion to that of (topological) weak-mixing for transitivity/hypercyclicity, namely quasi-rigidity; and we construct a recurrent but not quasi-rigid operator on each infinite-dimensional Banach space, solving the -recurrence problem in the negative way. The quasi-rigidity notion is closely related to the dense lineability of the set of recurrent vectors, and using similar conditions we study the lineability and dense lineability properties for the set of -recurrent vectors. This document has been split into two already published papers: Part I - Questions in linear recurrence I: The -recurrence problem. Analysis and Mathematical Physics, Volume 15, article number 1, (2025). https://doi.org/10.1007/s13324-024-00999-8 Part II - Questions in linear recurrence II: Lineability properties. Banach Journal of Mathematical Analysis, Volume 19, article number 61, (2025). https://doi.org/10.1007/s43037-025-00448-z
Paper Structure (17 sections, 29 theorems, 101 equations)

This paper contains 17 sections, 29 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Quasi-rigid dynamical systems
  3. Quasi-rigidity: definition and equivalences
  4. Quasi-rigid operators
  5. Existence of recurrent but not quasi-rigid operators
  6. Necessary prerequisites
  7. The projection $P$ and the sequences $(w_k^*)_{k\geq N+2}$, $(m_k)_{k\in\mathbb{N}}$ and $(\lambda_k)_{k\in\mathbb{N}}$
  8. The operators $R$ and $T$
  9. Furstenberg families for pointwise recurrence
  10. Definitions, examples and $\mathcal{F}(A)$-recurrence
  11. Appropriate Furstenberg families for $\mathcal{F}$-recurrence
  12. Quasi-conjugacies and commutants in $\mathcal{F}$-recurrence
  13. Infinite-dimensional vector spaces in $\mathcal{F}\textup{Rec}(T)$
  14. Lineability
  15. Dense Lineability
  16. ...and 2 more sections

Key Result

Lemma 2.3

Let $(X,T)$ be a dynamical system on a second-countable space $X$. The following are equivalent:

Theorems & Definitions (78)

  • Definition 2.1
  • Definition 2.2: Quasi-rigidity
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • proof
  • Remark 2.6
  • Proposition 2.7
  • ...and 68 more