Quantifying Noninvertibility in Discrete Dynamical Systems

Given a finite set X and a function f : X -> X, we define the degree of non-invertibility of f to be deg(f)=1 vertical bar X vertical bar Sigma(x is an element of X)vertical bar f(-1)(f(x))vertical bar. This is a natural measure of how far the function f is from being bijective. We compute the degrees of noninvertibility of some specific discrete dynamical systems, including the Carolina solitaire map, iterates of the bubble sort map acting on permutations, bubble sort acting on multiset permutations, and a map that we call "nibble sort." We also obtain estimates for the degrees of noninvertibility of West's stack-sorting map and the Bulgarian solitaire map. We then turn our attention to arbitrary functions and their iterates. In order to compare the degree of noninvertibility of an arbitrary function f:X -> X with that of its iterate f(k), we prove that max(f:X -> X vertical bar X vertical bar=n) deg(f(k))/deg(f)gamma=Theta(n(1-1)/2(k-1)) for every real number gamma >= 2 - 1/2(k-1). We end with several conjectures and open problems.