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

Zhengyu Yin

TL;DR

The paper develops a unified framework linking multivalued dynamics to finite-type graph models, shadowing, and shifts of finite type via inverse limits under the Mittag-Leffler condition. It proves that every closed relation on a compact totally disconnected space is an ML inverse limit of finite graph homomorphisms, and that a multivalued system has shadowing precisely when its orbit space is conjugate to an ML inverse limit of shifts of finite type. The work extends the Good–Meddaugh paradigm to multivalued maps, showing richer dynamical behavior and density of shadowing within upper semicontinuous maps, and connects graph covers to Shimomura’s graph-cover framework. It also furnishes a general method to approximate multivalued dynamics by finite-type models and extends shadowing results from zero-dimensional to general spaces via inverse-limit constructions, yielding broad structural insight into multivalued shadowing phenomena.

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 Boroński, 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.

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

TL;DR

The paper develops a unified framework linking multivalued dynamics to finite-type graph models, shadowing, and shifts of finite type via inverse limits under the Mittag-Leffler condition. It proves that every closed relation on a compact totally disconnected space is an ML inverse limit of finite graph homomorphisms, and that a multivalued system has shadowing precisely when its orbit space is conjugate to an ML inverse limit of shifts of finite type. The work extends the Good–Meddaugh paradigm to multivalued maps, showing richer dynamical behavior and density of shadowing within upper semicontinuous maps, and connects graph covers to Shimomura’s graph-cover framework. It also furnishes a general method to approximate multivalued dynamics by finite-type models and extends shadowing results from zero-dimensional to general spaces via inverse-limit constructions, yielding broad structural insight into multivalued shadowing phenomena.

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 Boroński, 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.

Paper Structure

This paper contains 23 sections, 41 theorems, 141 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Multivalued maps, orbits and graph walks
  4. Multivalued maps
  5. Orbit spaces and graph walks
  6. Semiconjugacy
  7. Inverse systems and Mittag-Leffler condition
  8. Inverse limit of multivalued systems
  9. Tracking orbits of multivalued maps
  10. Expansiveness and generators for multivalued maps
  11. Shifts of finite type and multivalued shadowing property
  12. Shifts of finite type and multivalued shadowing property
  13. multivalued shadowing on compact Hausdorff spaces
  14. Multivalued of finite type, graph cover and totally disconnected space
  15. Graph cover
  16. ...and 8 more sections

Key Result

Theorem 1

Let $(X, f)$ be an expansive compact totally disconnected multivalued system. If $f$ has the shadowing property, then its orbit system is conjugate to a subshift of finite type.

Figures (2)

  • Figure 1: Record orbit
  • Figure 2: The finite directed graph representation of golden mean shift

Theorems & Definitions (106)

  • Theorem
  • Theorem
  • Theorem
  • Corollary
  • Proposition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • ...and 96 more