Tranched graphs: consequences for topology and dynamics

TL;DR

This paper introduces tranched graphs, compares them to quasi-graphs and generalized sine-type continua, and explores how their topological structure influences possible dynamics.

Contribution

It unifies quasi-graphs and sine-type continua into tranched graphs and analyzes the impact of their topology on dynamics.

Findings

Neither quasi-graphs nor sine-type continua are subsets of each other.

Tranched graphs encompass both classes and relate to known concepts.

Topological restrictions of tranched graphs limit possible dynamics.

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.