Specification in Mahavier Systems via Closed Relations

TL;DR

This paper extends the classical specification property to Mahavier systems governed by closed relations, revealing new behaviors and structural differences in dynamical systems beyond traditional continuous functions.

Contribution

It introduces generalized specification properties for CR-dynamical systems, particularly Mahavier systems, highlighting their differences from classical systems and extending the existing theory.

Findings

Generalized specification properties differ from classical ones in CR-dynamical systems.

New properties reduce to standard specification when restricted to continuous functions.

Structural differences in dynamical behavior are revealed through these generalizations.

Abstract

We study two fundamental properties of topological dynamical systems, the specification property and the initial specification property, and explore their generalizations to the broader setting of CR-dynamical systems, where the dynamics are governed by closed relations rather than continuous functions. While these two properties are equivalent for many classical systems, we demonstrate that their generalizations to CR-dynamical systems often lead to distinct behaviors. Applying them to Mahavier dynamical systems, we introduce new specification-type properties. These generalized notions extend the classical theory and reveal rich structural differences in dynamical behavior. Moreover, each of the new properties reduces to the standard specification property when restricted to continuous functions.