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On limit sets and equicontinuity in the hyperspace of continua in dimension one

Domagoj Jelić, Piotr Oprocha

Abstract

The paper studies the structure of $ω$-limit sets of map $\tilde{f}$ induced on the hyperspace $C(G)$ of all connected compact sets, by dynamical system $(G,f)$ acting on a topological graph $G$. In the case of the base space being a topological tree we additionally show that $\tilde{f}$ is always almost equicontinuous and characterize its Birkhoff center.

On limit sets and equicontinuity in the hyperspace of continua in dimension one

Abstract

The paper studies the structure of -limit sets of map induced on the hyperspace of all connected compact sets, by dynamical system acting on a topological graph . In the case of the base space being a topological tree we additionally show that is always almost equicontinuous and characterize its Birkhoff center.
Paper Structure (12 sections, 36 theorems, 51 equations, 2 figures)

This paper contains 12 sections, 36 theorems, 51 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Topological graphs
  4. Dynamical systems
  5. $\omega$-limit sets in graph maps
  6. $\alpha$-limit sets in graph maps
  7. Hyperspaces
  8. Recurrence in the induced system of continua of topological graphs
  9. The proof of Theorem \ref{['thm:main']}
  10. Bound on periods
  11. center
  12. Almost equicontinuity

Key Result

Theorem 1.1

Let $G$ be a topological graph and let $\mathop{\mathrm{C}}\nolimits(G)$ be the hyperspace of all subcontinua of $G$. For any $A\in\mathop{\mathrm{C}}\nolimits(G)$ (at least) one of the following properties holds:

Figures (2)

  • Figure 1: Continua $C_i$ from Lemma \ref{['lem:alpha_solenoid']}.
  • Figure 2: The sketch represents graph $D$ in blue with its boundary in $G$ indicated. Set $N(D,\delta_{2m})\setminus D$ is colored red and its closure contains each arc $L_i$, $i=1,2,\ldots,2m$.

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4: Blokh, B1B2
  • Theorem 2.5
  • Remark 2.6
  • Theorem 2.7
  • Theorem 3.1: Auslander-Ellis
  • ...and 54 more