  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.
