[Paper Review] Distributional chaos and Li-Yorke chaos in metric spaces

Marko Kostić|arXiv (Cornell University)|Jan 8, 2019

Holomorphic and Operator Theory45 references4 citations

TL;DR

This paper introduces and systematically analyzes new generalizations of distributional chaos and Li-Yorke chaos for sequences of binary relations in metric spaces, extending prior concepts to multivalued linear operators in Fréchet spaces. The key contribution is a unified framework that establishes connections between various chaos types and characterizes irregular vectors and manifolds through density-based criteria and operator restrictions.

ABSTRACT

In this paper, we introduce several new types and generalizations of the concepts distributional chaos and Li-Yorke chaos. We consider the general sequences of binary relations acting between metric spaces, while in a separate section we focus our attention to some special features of distributionally chaotic and Li-Yorke chaotic multivalued linear operators in Frechet spaces.

Motivation & Objective

• To extend the theory of distributional chaos and Li-Yorke chaos beyond single operators to general sequences of binary relations in metric spaces.
• To define and analyze new chaos types, including reiteratively $\tilde{X}$-distributionally chaotic, $\langle\tilde{X},i\rangle$-mixed, and $\tilde{X}$-Li-Yorke chaotic systems.
• To investigate the behavior of distributionally and Li-Yorke chaotic multivalued linear operators (MLOs) in Fréchet spaces, focusing on irregular vectors and invariant submanifolds.
• To establish connections between chaos in the full space and chaos restricted to subsets $\tilde{X} \subseteq X$ via induced sequences $({\mathbb{A}}_k)_{k \in \mathbb{N}}$.
• To provide a foundational, heuristic framework for future research on chaos in abstract dynamical systems, especially for unbounded and multivalued operators.

Proposed method

• Introduces generalized chaos concepts using lower and upper densities of sets of natural numbers to define reiterative and syndetic recurrence properties in metric spaces.
• Defines $\tilde{X}$-distributional chaos and $\tilde{X}$-Li-Yorke chaos by restricting dynamics to a non-empty subset $\tilde{X} \subseteq X$ and analyzing orbit distributions and separation properties.
• Applies the concept of distributional irregular vectors and manifolds, particularly by constructing $X' = \text{span}\{x\}$ from a single irregular vector $x$ to generate uniformly chaotic submanifolds.
• Uses the induced sequence $({\mathbb{A}}_k)_{k \in \mathbb{N}}$ defined by $D({\mathbb{A}}_k) = D({\mathcal{A}}_k) \cap \tilde{X}$ and ${{\mathbb{A}}_k}x = {\mathcal{A}}_k x$ to relate chaos in $\tilde{X}$ to chaos in the full space.
• Applies density-theoretic tools such as syndeticity, upper and lower density, and difference sets to characterize reiterative and mixed chaos types.
• Proposes a hierarchy of chaos types, including distributional chaos of type $s \in \{1,2,2\frac{1}{2},3\}$, and establishes equivalence results between chaos in $({\mathcal{A}}_k)$ and $({\mathbb{A}}_k)$ via Proposition 4.10.

Experimental results

Research questions

• RQ1 How can distributional chaos and Li-Yorke chaos be generalized beyond single operators to sequences of binary relations in metric spaces?
• RQ2 What is the relationship between $\tilde{X}$-distributional chaos and standard distributional chaos in the context of induced sequences $({\mathbb{A}}_k)$?
• RQ3 In what way do irregular vectors and their linear spans generate distributionally or Li-Yorke chaotic submanifolds in Fréchet spaces?
• RQ4 How do reiterative and mixed chaos types ($\langle\tilde{X},i\rangle$-mixed, $(\tilde{X},j)$-mixed) refine or extend classical chaos notions?
• RQ5 Can the framework be extended to unbounded or multivalued linear operators, and what are the implications for hypercyclicity and spectral theory?

Key findings

• The paper establishes that a vector $x$ is a (reiteratively) $\tilde{X}$-distributionally irregular vector for $({\mathcal{A}}_k)_{k \in \mathbb{N}}$ if and only if it is a (reiteratively) distributionally irregular vector for the induced sequence $({\mathbb{A}}_k)_{k \in \mathbb{N}}$, as formalized in Proposition 4.10.
• The linear span $X' = \text{span}\{x\}$ of a (reiteratively) $\tilde{X}$-distributionally irregular vector $x$ forms a uniformly (reiteratively) $\tilde{X}$-distributionally irregular manifold for $({\mathcal{A}}_k)_{k \in \mathbb{N}}$, provided $x \in \tilde{X} \cap \bigcap_{k=1}^\infty D({\mathcal{A}}_k)$.
• The paper demonstrates that distributional chaos and Li-Yorke chaos can occur even in finite-dimensional spaces, as shown by simple counterexamples, challenging the intuition that such chaos requires infinite-dimensional settings.
• For multivalued linear operators in Fréchet spaces, the existence of irregular vectors and their associated submanifolds is linked to the structure of the induced operator sequences $({\mathbb{A}}_k)$, which inherit the chaotic properties of the original system.
• The framework allows for the definition of $\tilde{X}$-Li-Yorke chaos and $\tilde{X}$-distributional chaos of type $s$ for $s \in \{1,2,2\frac{1}{2},3\}$, extending earlier results from Banach spaces to general metric spaces.
• The paper shows that $\tilde{X}$-Li-Yorke irregular manifolds are equivalent to $\tilde{X}$-scrambled sets, and that the same holds for mixed chaos types, establishing a hierarchy of chaotic behaviors.