On multivaled fixed-point free maps on R^n

TL;DR

This paper characterizes fixed-point free multivalued maps on ^n, showing their fixed-point freeness extends to compactifications and that such maps are always colorable with a number of colors depending only on dimension and range size.

Contribution

It establishes the equivalence of fixed-point freeness for multivalued maps on ^n with their extensions to compactifications and proves colorability with bounds depending on dimension and range size.

Findings

Fixed-point freeness extends to ^n compactifications.

Such maps are always colorable.

Number of colors depends only on dimension and range size.

Abstract

To formulate our results let $f$ be a continuous map from $\mathbb R^n$ to $2^{\mathbb R^n}$ and $k$ a natural number such that $|f(x)|\leq k$ for all $x$. We prove that $f$ is fixed-point free if and only if its continuous extension $\tilde f:\beta \mathbb R^n\to 2^{\beta \mathbb R^n}$ is fixed-point free. If one wishes to stay within metric terms, the result can be formulated as follows: $f$ is fixed-point free if and only if there exists a continuous fixed-point free extension $\bar f: b\mathbb R^n\to 2^{b\mathbb R^n}$ for some metric compactificaton $b\mathbb R^n$ of $\mathbb R^n$. Using the classical notion of colorablity, we prove that such an $f$ is always colorable. Moreover, a number of colors sufficient to paint the graph can be expressed as a function of $n$ and $k$ only. The mentioned results also hold if the domain is replaced by any closed subspace of $\mathbb R^n$ without any changes in the range.