John Baez Zeta functions of Z-sets

Preface

This is work in progress by John Baez and James Dolan, based in part on material from week218 of This Week’s Finds.

In Dirichlet species and the Hasse-Weil zeta function, we discussed a way to categorify the Hasse–Weil zeta function of a scheme. Another class of mathematical objects that have zeta functions are dynamical systems. The simplest example is the Artin–Mazur zeta function of a discrete dynamical system. A discrete dynamical system is simply a set $S$ equipped with a bijection $f: S \to S$, and its Artin–Mazur zeta function is

$$\zeta_S(z)= exp\left(\sum_{n=1}^\infty |fix(f^n)| \frac {z^n}{n} \right)\, ,$$

where $fix(f^n)$ is the set of fixed points of the $n$th iterate of $f$.

Mathematically, we can also think of a discrete dynamical system as a $\mathbb{Z}$-set: a set equipped with an action of the additive group $\mathbb{Z}$. Any $\mathbb{Z}$-set $S$ gives a species $Z_S$ for which a $Z_X$-structure on a finite set is a way of making it into a $\mathbb{Z}$-set and equipping it with a $\mathbb{Z}$-set map to $S$.

In the next section we show that the Artin–Mazur zeta function $\zeta_S$ is the generating function of the species $Z_S$. This is meant to justify the rather funny-looking definition of the Artin–Mazur zeta function.

The zeta function of a Z-set

Suppose we have a $\mathbb{Z}$-set $S$, or in other words, a set equipped with a bijection

$$f\colon S \to S \, .$$

Let $fix(f^n)$ be the set of fixed points of $f^n: S \to S$, and let $|fix(f^n)|$ be the cardinality of this set.

We say a $\mathbb{Z}$-set $S$ is tame if $|fix(f^n)|$ is finite for all $n \in \mathbb{N}$.

We can “cyclically order” a finite set by drawing it as a little circle of dots with arrows pointing clockwise from each dot to the next. A cyclically ordered set is automatically a $\mathbb{Z}$-set in an obvious way, and indeed this gives a more precise definition: a cyclic ordering on a set $S$ is a making it into a $\mathbb{Z}$-set with a single orbit. Note that with this definition, the empty set admits no cyclic orderings.

So, here is a type of structure we can put on a finite set: cyclically ordering it and equipping the resulting $\mathbb{Z}$-set with a morphism to a fixed $\mathbb{Z}$-set $S$. And, we have:

Proposition: The power series

$$\sum_{n \gt 0} |fix(f^n)| \frac{x^n}{n}$$

is the generating function for the structure type “being cyclically ordered and equipped with a morphism to the $\mathbb{Z}$-set $S$”.

Proof: Given an $n$-element set, there are $(n-1)!$ ways to cyclically order it if $n \gt 0$, and none if $n = 0$. After having chosen a cyclic ordering, to specify a $\mathbb{Z}$-equivariant map to $S$ we simply map a chosen point to any element of $fix(f^n)$. So there are $|fix(f^n)| (n-1)!$ to do this, and the generating function of “being cyclically ordered and equipped with a morphism to the $\mathbb{Z}$-set $S$” is as described.

Proposition: The power series

$$Z(S,t) = exp \left( \sum_{n \gt 0} |fix(f^n)| \frac{t^n}{n} \right)$$

is the generating function for the structure type “being a $\mathbb{Z}$-set over $S$”.

Proof: For any structure type $F$ there is a structure type “being partitioned into nonempty parts, each equipped with an $F$-structure”, which is called $\exp(F)$ and satisfies

$$|\exp(F)| = \exp(|F|)$$

So, by the previous proposition,

$$\exp \left( \sum_{n \gt 0} |fix(f^n)| \frac{x^n}{n} \right)$$

is the generating function for “being partitioned into nonempty parts, each equipped with a cyclic ordering and a morphism to the $\mathbb{Z}$-set $S$”. But this is just a long way of saying: “being made into a $\mathbb{Z}$-set and equipped with a morphism to the $\mathbb{Z}$-set $S$ — or in category-theoretic jargon, ”being a $\mathbb{Z}$-set over $S$“.

Orbits and multisubsets

Again let $S$ be a $\mathbb{Z}$-set coming from a bijection $f \colon S \to S$.

• Let $orb_S(n)$ be the set of $f$-orbits of $S$ having cardinality $n$.

• Let $z_S(n)$ be the set of invariant multi-subsets of $S$ of weight $n$.

Here a multi-subset of a set $S$ is a function $\psi: S \to \mathbb{N}$, and its weight is $\sum_{x \in S} \psi(x)$, which could be infinite.

Any subset $X \subseteq S$ gives a multi-subset where $\psi = \chi_X$ is the characteristic function of $X$, and then the weight of this multi-subset is the cardinality of $X$. $X \subseteq S$ is an $f$-invariant subset of $S$ iff $\chi_X$ is invariant. Every $f$-invariant multi-subset is a (possibly infinite) sum of such invariant characteristic functions.

In fact, every invariant multi-subset of finite weight is a finite sum of characteristic functions of orbits in a unique way.

The collection of multi-subsets of $S$ of finite weight can be written

$$\sum_{n \ge 0} \frac{S^n}{n!}$$

where the quotient is by the action of the symmetric group on $n$ letters. This collection can heuristically be denoted $\exp(S)$.

RELATE THIS TO SET TO THE SETS $orb_S(n)$.

Note that if $S$ is tame, the cardinalities of $orb_S(n)$ and $z_S(n)$ are finite for all $n \ge 0$.

Proposition:

$$|fix(f^n)| = \sum_{m | n} \orb_S(m)$$

Proof: Suppose $x \in S$ is a fixed point of $f^n$. Then there is some least $m \ge 1$ such that $x$ is a fixed point of $f^m$, and $m$ divides $n$. In this case $x$ lies in a (necessarily unique) $f$-orbit of cardinality $m$. Conversely if $x$ lies in an $f$-orbit of cardinality $m$ we have $x \in fix(f^m)$ and thus $x \in fix(f^n)$ if $m$ divides $n$. In short, $fix(f^n)$ is the disjoint union of all orbits of cardinality $m$ where $m | n$. The formula follows.