On the Banach mapping theorem and a related conjecture

Abstract

We extend, using an elementary method, results of Banach (1924), Fan (1952), and Sanders (1961), which concern a finite collection ${\{f_i:A_i\to A_{i+1}\}}_{i=1}^n$ of mappings with $A_{n+1}=A_1$ which is decomposable as $f_i(B_i)=A_{i+1}\setminus B_{i+1}$, where $B_i\subseteq A_i$ for all $i$ and $B_{n+1}=B_1$. Our theorem determines when such a collection is decomposable. We also show that such a set $B_1$ is unique up to an addition of a certain set, which was conjectured by Sanders.