Finite type as fundamental objects even non-single-valued and non-continuous

TL;DR

This paper explores the structure of multivalued relations on totally disconnected spaces, linking finite graphs, shifts of finite type, and shadowing properties through inverse limits and the Mittag-Leffler condition.

Contribution

It establishes a representation of closed relations as inverse limits of finite graphs and characterizes shadowing in terms of the Mittag-Leffler condition, extending dynamical systems theory.

Findings

Closed relations are represented as inverse limits of finite directed graphs.

Shadowing property is characterized by the Mittag-Leffler condition.

Multivalued systems exhibit more diverse behaviors than single-valued systems.

Abstract

In this paper, inspired by the elegant work of Good and Meddaugh \cite{GM} and the graph models for zero-dimensional systems developed by several authors, like Gambaudo and Martens \cite{GM06}, Shimomura \cite{Sh14}. We try to discover a connection among some objects, such as finite directed graph, shift of finite type and shadowing property by employing the Closed Graph Theorem for multivalued maps. From the perspective of structure theorems, we demonstrate that every closed relation (multivalued map) on a compact, totally disconnected space is represented as an inverse limit of finite directed graph homomorphisms satisfying the Mittag-Leffler condition. Moreover, from dichotomy-theorem point of view, we prove that an inverse limit of finite directed graph homomorphisms possesses the shadowing property if and only if its induced space of infinite graph walks (as a shift of finite type) satisfies the Mittag-Leffler condition. As an application, a question raised by Boro\'nski, Bruin and Kucharski \cite{BBK} is also concerned. Furthermore, we show that under a multivalued dynamical system, the resulting dynamical behaviors exhibit greater diversity and counterintuitively compared to those observed in single-valued continuous systems.