Digital Lefschetz numbers and related fixed point theorems

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 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 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 L¯( f ) while the homotopy invariance of 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.