The construction of \(P\)-expansive maps of regular continua: A geometric approach (Q1566931)

In this paper a graph is a continuum that can be written as a finite union of arcs that intersect at most in their endpoints. Any finite set \(P\) splits the graph into components, hence a map \(F:G\to G\) defines the itinerary \(I(x)\) for points in \(G\) (one assumes \(F(P)\subset P)\). \(F\) is called \(P\)-expansive if the itineraries separate points. The authors show an extension of a theorem of \textit{S. Baldwin} [Int. J. Bifurcation Chaos Appl. Sci. Eng. 5, 1307-1318 (1995; Zbl 0886.58018)]. Any graph map \(F:G\to G\) as above admits a regular continuum as a factor \(\pi:G\to Z\subset\mathbb{R}^3\) which is \(\pi(P)\)-expansive and such that \[ \forall p,q\in P,\;Q\subset P\ni\text{arc} \{p,q\}\cap Q\neq \emptyset\Rightarrow \forall\text{arc} \bigl\{F(p),F(q) \bigr\}: \text{arc} \bigl\{F(p), F(q)\bigr\} \cap\pi(Q) \neq\emptyset. \] The construction of the factor is carried out geometrically.