Commuting Contractive Families

2016/01/26 by Milićević, Luka · 3 citations
#Combinatorics (math.CO) #FOS: Mathematics #General Topology (math.GN) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1602.00725

Abstract

A family f₁,...,f_n of operators on a complete metric space X is called contractive if there exists a positive λ< 1 such that for any x,y in X we have d(f_i(x),f_i(y)) ≤ λd(x,y) for some i. Austin conjectured that any commuting contractive family of operators has a common fixed point, and he proved this for the case of two operators. Our aim in this paper is to show that Austin's conjecture is true for three operators, provided that λ is sufficiently small.

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