Chains without regularity

TL;DR

This paper demonstrates that chain-recurrence and chain-transitivity persist in compact dynamical systems even when the evolution map $f$ is completely arbitrary and discontinuous. It proves three core results: existence of a nonempty chain-recurrent set $CR_f$, a closed invariant chain-transitive subsystem, and that every point is chain-related to such a subsystem or to a chain-recurrent point; it also confirms $GR(f)$ is nonempty. The key methodological advance is a Choice-free, constructive transfinite construction that does not rely on continuity, highlighting the robustness of chain relations under compactness. The work thus extends foundational recurrence notions to highly irregular dynamics, with implications for understanding long-term behavior and invariant structures in broad settings.

Abstract

We study chain-recurrence and chain-transitivity in compact dynamical systems without any regularity assumptions on the map. We prove that every compact system has a chain-recurrent point and a closed, invariant, chain-transitive subsystem. The proofs do not use any form of the Axiom of Choice.

Figures (2)

• Figure 1: Topological dynamical relations. It is graphically emphasized that, if we replace $\mathcal{N}$ by $\widetilde{\mathcal{N}}$, additional $\epsilon$-corrections are allowed whenever one goes down in the table (right).
• Figure 2: Graphic representation of (a sample case of) the reasoning in the proof of Theorem \ref{['thmsubsyst']}.

Theorems & Definitions (27)

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