Lefschetz numbers and fixed point theory in digital topology

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.