Lefschetz numbers and fixed point theory in digital topology

Abstract

In this paper, we present two types of Lefschetz numbers in the topology of digital images. Namely, the simplicial Lefschetz number $L(f)$ and the cubical Lefschetz number $\bar L(f)$. We show that $L(f)$ is a strong homotopy invariant and has an approximate fixed point theorem. On the other hand, we establish that $\bar L(f)$ is a homotopy invariant and has an $n$-approximate fixed point result. In essence, this means that the fixed point result for $L(f)$ is better than that for $\bar L(f)$ while the homotopy invariance of $\bar L(f)$ is better than that of $L(f)$. Unlike in classical topology, these Lefschetz numbers give lower bounds for the number of approximate fixed points. Finally, we construct some illustrative examples to demonstrate our results.