Quantifying Noninvertibility in Discrete Dynamical Systems

Abstract

Given a finite set $X$ and a function $f:X\to X$, we define the \emph{degree of noninvertibility} of $f$ to be $\displaystyle\deg(f)=\frac{1}{|X|}\sum_{x\in X}|f^{-1}(f(x))|$. 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 $f:X\to X$ with that of its iterate $f^k$, we prove that \[\max_{\substack{f:X\to X\\ |X|=n}}\frac{\deg(f^k)}{\deg(f)^\gamma}=\Theta(n^{1-1/2^{k-1}})\] for every real number $\gamma\geq 2-1/2^{k-1}$. We end with several conjectures and open problems.

The expected degree of noninvertibility of compositions of functions and a related combinatorial identity

“…That Theorem 2.3 is a strengthening of [2,Theorem 3.4] in the case Y = Z = X follows from Lemma 3.9 but the extent of this strengthening may be appreciated by comparing the order of the expectation of [4, p. 500] or the references in OEIS A208250).…”

“…Only a small modification of the proof of [2,Theorem 3.4] in necessary to prove Theorem 2.3. We give the full proof for completeness:…”

“…Recently, [2] defined the degree of noninvertibility of a function f : X → Y between two finite set X and Y by…”

“…as a measures of how far f is from being injective. The authors of [2] then computed the degrees of noninvertibility of several specific functions and studied, from an extremal point of view, the connection between the degrees of noninvertibility of functions and those of their iterates.…”

Fertilitopes

“…West's map has now become the most vigorously-studied form of stack-sorting; an exploration of this map and its close relatives winds through analytic combinatorics (see [4,15,24,25] and the many references therein); combinatorial dynamics [17,27]; considerations of symmetric, unimodal, logconcave, and real-rooted polynomials [5,6,9,15,23,24,26,47]; special partially ordered sets [14,20]; and even noncommutative probability theory [24,26]. The goal of this paper is to introduce concepts from discrete convexity theory and polyhedral geometry into the theory of the stack-sorting map.…”