Contractive families on compact spaces (Q2874618)

\textit{J. D. Stein jun.} [Rocky Mt. J. Math. 30, No. 2, 735--754 (2000; Zbl 0977.54036)] stated the following conjecture: ``Let \(\mathcal{F}\) be a finite set of selfmaps on a complete metric space \((X,\rho)\) and let \(0<\lambda<1\). Assume that, for every pair \(x,y\in X\), there is an \(T\in\mathcal{F}\) such that \(\rho(Tx,Ty)\leq\lambda\rho(x,y)\). Then some composition of members of \(\mathcal{F}\) has a fixed point.'' This conjecture was disproved by \textit{T. D. Austin} [Mathematika 52, No. 1--2, 115--129 (2005; Zbl 1105.54020)] who asked if the conjecture is true for compact spaces. The present author shows that even for compact spaces the conjecture remains false. The construction of the counterexample is fairly complicated and the author remarks that instead of considering distances between points it is more convenient to consider diameters.NEWLINENEWLINELet \(X\) be non-empty set and \(\mathcal{D}\) a collection of subsets of \(X\). Then the pair \((X,\mathcal{D})\) is called a diametrizable space if any two points of \(X\) are contained in an element of \(\mathcal{D}\) and if, for any two elements of \(\mathcal{D}\) with non-empty intersection, the union belongs to \(\mathcal{D}\). A diameter on \(\mathcal{D}\) is a non-negative function \(\operatorname {diam}\) on \(\mathcal{D}\) such that \(\operatorname {diam} U+\operatorname {diam}V\geq\operatorname {diam}(U\cup V)\) whenever \(U\cap V\not=\emptyset\). If one defines a non-negative function \(d\) on \(X\times X\) by \(d(x,x):=0\) and \(d(x,y):=\inf\operatorname {diam}U\) for \(x\not=y\), where the infimum is taken over all \(U\in\mathcal{D}\) containing \(x\) and \(y\), then we get a pseudometric on \(x\). If, in addition, each point belongs to only finitely many members of \(\mathcal{D}\), then \(d\) is indeed a metric.NEWLINENEWLINEThe counterexample is constructed by letting \(X\) be the set of finite words on \(\{a,b\}\) and considering maps \(f,g: X\to X\) by prefixing \(a\) (resp., \(b\)) to a word \(u\in X\). By a meticulous choice of \(\mathcal{D}\), the author then constructs the required counterexample.