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Tranched graphs: consequences for topology and dynamics

Michał Kowalewski, Piotr Oprocha

TL;DR

The paper introduces tranched graphs as a unifying framework to compare quasi-graphs and generalized sin(1/x)-type continua, showing neither class contains the other and providing conditions that link the two notions. It proves that tranched graphs are quasi-graphs exactly when they are arcwise connected with finite depth, and that generalized sin(1/x)-type continua that are quasi-graphs share the same finite-depth, arcwise-connected, hereditary-tranched-graph structure, while constructing arcwise connected infinite-depth examples to show limits of these characters. A detailed construction yields an arcwise connected generalized sin(1/x)-type continuum that is a tranched graph of infinite depth, illustrating the necessity of finite depth for the quasi-graph intersection. The dynamics of tranched-graph maps are analyzed, revealing invariant tranche structures under onto maps and a spectrum of possible behaviors, from non-m-existing Leo to topologically mixing phenomena on spaces with infinite tranches.

Abstract

We compare quasi-graphs and generalized $\sin(1/x)$-type continua, which are two classes of continua that generalize topological graphs and contain the Warsaw circle as a nontrivial common element. We show that neither class is a subset of the other, provide some characterizations, and present illustrative examples. We unify both approaches by considering the class of tranched graphs, compare it to concepts known from the literature, and describe how the topological structure of its elements restricts possible dynamics.

Tranched graphs: consequences for topology and dynamics

TL;DR

The paper introduces tranched graphs as a unifying framework to compare quasi-graphs and generalized sin(1/x)-type continua, showing neither class contains the other and providing conditions that link the two notions. It proves that tranched graphs are quasi-graphs exactly when they are arcwise connected with finite depth, and that generalized sin(1/x)-type continua that are quasi-graphs share the same finite-depth, arcwise-connected, hereditary-tranched-graph structure, while constructing arcwise connected infinite-depth examples to show limits of these characters. A detailed construction yields an arcwise connected generalized sin(1/x)-type continuum that is a tranched graph of infinite depth, illustrating the necessity of finite depth for the quasi-graph intersection. The dynamics of tranched-graph maps are analyzed, revealing invariant tranche structures under onto maps and a spectrum of possible behaviors, from non-m-existing Leo to topologically mixing phenomena on spaces with infinite tranches.

Abstract

We compare quasi-graphs and generalized -type continua, which are two classes of continua that generalize topological graphs and contain the Warsaw circle as a nontrivial common element. We show that neither class is a subset of the other, provide some characterizations, and present illustrative examples. We unify both approaches by considering the class of tranched graphs, compare it to concepts known from the literature, and describe how the topological structure of its elements restricts possible dynamics.
Paper Structure (6 sections, 52 theorems, 37 equations, 12 figures)

This paper contains 6 sections, 52 theorems, 37 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Characterization of quasi-graphs that are generalized sin(1/x)-type continua
  4. Characterization of generalized sin(1/x)-type continua that are quasi-graphs
  5. Construction of an arcwise connected generalized sin(1/x)-type continuum that is a tranched graph of infinite depth
  6. Relation to other classes and dynamics of tranched graph maps

Key Result

Lemma 2.5

Let $X,Y$ be nondegenerate continua and let $\phi \colon X \rightarrow Y$ be a continuous monotone map. If the set $\phi^{-1} (D)$ is dense in $X$, where $D = \{y \in Y$ such that $\phi^{-1}(y)$ is degenerate in $X\}$, then $\phi^{-1}(y)$ is nowhere dense in $X$ for every $y\in Y$.

Figures (12)

  • Figure 1: The Warsaw circle $W$ and its image $\phi(W)$ under mapping $\phi$ from Definition \ref{['def_sin1/x']}. The points in topological graph $\phi(W)$ are colored in accordance to their preimage. Oscillatory quasi-arc required by Definition \ref{['def_quasigraph']} is marked in blue.
  • Figure 2: A quasi-arc with $4$-star as the limit set. If a map $\phi$ collapses the $4$-star to a point, then approximation property is violated, e.g. by subcontinuum marked in green.
  • Figure 3: A quasi-graph whose limit set is circle, but is not a generalized sin(1/x)-type continuum.
  • Figure 4: A quasi-graph which is generalized sin(1/x)-type continuum and contains 4-star as a tranche
  • Figure 5: An exemplary generalized sin(1/x)-type continuum whose set of tranches is not closed
  • ...and 7 more figures

Theorems & Definitions (118)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.7
  • proof
  • Remark 2.8
  • Remark 3.1
  • ...and 108 more