Dense Lineable Criterion for Linear Dynamics

TL;DR

We address Li-Yorke chaos for sequences of continuous linear operators from an $F$-space to a normed space by introducing the D-phenomenon to unify recurrence, universality, and Li-Yorke chaos under a common dense-lineable framework. Irregular vectors for a sequence $(T_n)$ are defined via $\liminf_{n\to\infty} \|T_n x\| = 0$ and $\limsup_{n\to\infty} \|T_n x\| = \infty$, and several equivalences are established: dense irregular vectors, residual irregular vectors, and densely Li-Yorke chaotic behavior are equivalent under separability. The D-phenomenon, including $\Psi^{\mathrm{Rec}}$, $\Psi^{\mathrm{HC}}$, $\Psi^{h}$, and the Li–Yorke variant $\Psi^{\mathrm{LY}}$, provides a unified lens to study recurrence, universality, and chaos via Furstenberg families, with Mycielski’s theorem enabling the construction of large, dense scrambled sets. A main result yields a common dense-lineable criterion for countable families of D-phenomena, yielding negative answers to open questions (e.g., Grivaux et al.) by showing that densely Li-Yorke chaotic sequences can fail to possess a dense irregular manifold. The work also demonstrates the existence of pathological examples, such as a recurrent operator whose set of recurrent vectors is not dense-lineable and a sequence with a dense set of irregular vectors but no dense irregular manifold, highlighting limits of dense-lineability as a universal tool.

Abstract

We study Li-Yorke chaos for sequences of continuous linear operators from an \(F\)-space to a normed space. We introduce the \emph{D-phenomenon} to establish a common dense lineable criterion that encompasses properties such as recurrence, universality, and Li-Yorke chaos. We show that in every infinite-dimensional separable complex Banach space, there exists a sequence of operators with a dense set of irregular vectors but without a dense irregular manifold, and we exhibit a recurrent operator whose set of recurrent vectors is not dense-lineable. This resolves in the negative a question posed by Grivaux et al.

Figures (1)

• Figure 1: Diagram of the partial order of D-phenomena

Theorems & Definitions (50)

• Theorem 1.2: Theorem 2.2 and Remark 2.3, bernal2003hypercyclicity
• Definition 2.1
• Definition 2.3
• Proposition 2.5
• proof
• Theorem 2.6
• Remark 2.7
• Theorem 2.8: Mycielski Theorem
• proof : Proof of Theorem \ref{['equiv-den-LY']}
• Remark 2.9