Commuting Contractive Families

TL;DR

This paper proves Austin's conjecture that three commuting contractive operators on a complete metric space have a common fixed point, extending the known result from two operators under certain contraction conditions.

Contribution

The paper establishes the validity of Austin's conjecture for three operators with sufficiently small contraction factor, advancing the understanding of fixed points in contractive families.

Findings

Proves the conjecture for three operators with small contraction constant

Extends the fixed point result from two to three commuting operators

Shows the importance of the contraction factor in fixed point existence

Abstract

A family $f_1,...,f_n$ of operators on a complete metric space $X$ is called contractive if there exists a positive $\lambda < 1$ such that for any $x,y$ in $X$ we have $d(f_i(x),f_i(y)) \leq \lambda d(x,y)$ 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 $\lambda$ is sufficiently small.