Nielsen coincidence theory of $(n,m)$-valued pairs of maps

TL;DR

This paper refines a Nielsen-type invariant for pairs of multi-valued maps to accurately estimate their coincidence points, providing exact bounds for circle maps.

Contribution

It offers a corrected construction of the Nielsen invariant for $(n,m)$-valued maps, improving upon previous flawed approaches.

Findings

The new invariant correctly bounds the number of coincidence points.

For circle maps, the invariant yields a sharp, exact lower bound.

The construction is based on intersection points of graph images.

Abstract

We consider pairs of maps $(f,g)$, where $f$ is an $n$-valued map and $g$ is an $m$-valued map, defined on connected finite polyhedra. A point $x$ such that $f(x)\cap g(x)\neq \emptyset$ is called a coincidence point of $f$ and $g$. A useful device for studying coincidence points would be a Nielsen-type invariant which provides a lower bound for the number of coincidence points of all $(n, m)$-valued pairs of maps homotopic to $(f,g)$. The construction of such an invariant $N(f:g)$ was proposed in [J. Fixed Point Theory Appl. 14, 309--324 (2013)]. Unfortunately, this approach has some flaws. In this paper, we present a modified construction that yields a corrected form of the invariant, defined in terms of the intersection points of the graphs of $f$ and $g$. In the case of $(n, m)$-valued pairs of maps of the circle our invariant provides a sharp lower bound, which we precisely determine.