Classification of maps on a finite set under permutation

Abstract

We characterize when two maps f,g:X→X on a finite set are conjugated, that is, when there is a permutation σ:X→X such that f=σ−1gσ. We build a signature sgn (f) for every map f such that f and g are conjugated if and only if sgn (f)=sgn (g). This signature is an array that includes information about the cycles of the map and the noncyclic elements, called transients, besides data about the insertion of the transients in the cycle. The transient elements form several tree shape graphs. Our characterization is a generalization of the known fact that two permutations are conjugated if and only if they have the same cycle structure. We use elementary facts about finite dynamical systems and about the canonical labeling of unordered trees. The signature can be built by an algorithm of complexity O(n), being n the cardinality of X.