Contractive families on compact spaces

Abstract

A family f1,fn of operators on a complete metric space X is called contractive if there exists λ < 1 such that for any x,y in X we have d(fi(x),f-i(y)) ≤ λ d(x,y) for some i. Stein conjectured that for any contractive family there is some composition of the operators fi that has a fixed point. Austin gave a counterexample to this, and asked whether Stein's conjecture is true if we restrict to compact spaces. Our aim in this paper is to show that, even for compact spaces, Stein's conjecture is false.