A spectral theory for combinatorial dynamics

Abstract

This article proposes a framework for the study of periodic maps $T$ from a (typically finite) set $X$ to itself when the set $X$ is equipped with one or more real- or complex-valued functions. The main idea, inspired by the time-evolution operator construction from ergodic theory, is the introduction of a vector space that contains the given functions and is closed under composition with $T$, along with a time-evolution operator on that vector space. I show that the invariant functions and 0-mesic functions span complementary subspaces associated respectively with the eigenvalue 1 and the other eigenvalues. Alongside other examples, I give an explicit description of the spectrum of the evolution operator when $X$ is the set of $k$-element multisets with elements in $\{0,1,\dots,n-1\}$, $T$ increments each element of a multiset by 1 mod $n$, and $g_i: X \rightarrow \mathbb{R}$ (with $1 \leq i \leq k$) maps a multiset to its $i$th smallest element.