John Baez Zeta functions of Z-sets


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 SS equipped with a bijection f:SSf: S \to S, and its Artin–Mazur zeta function is

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

where fix(f n)fix(f^n) is the set of fixed points of the nnth iterate of ff.

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 SS gives a species Z SZ_S for which a Z XZ_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 SS.

In the next section we show that the Artin–Mazur zeta function ζ S\zeta_S is the generating function of the species Z SZ_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 SS, or in other words, a set equipped with a bijection

f:SS. f\colon S \to S \, .

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

We say a \mathbb{Z}-set SS is tame if |fix(f n)||fix(f^n)| is finite for all nn \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 SS 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 SS. And, we have:

Proposition: The power series

n>0|fix(f n)|x nn \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 SS”.

Proof: Given an nn-element set, there are (n1)!(n-1)! ways to cyclically order it if n>0n \gt 0, and none if n=0n = 0. After having chosen a cyclic ordering, to specify a \mathbb{Z}-equivariant map to SS we simply map a chosen point to any element of fix(f n)fix(f^n). So there are |fix(f n)|(n1)!|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 SS” is as described.

Proposition: The power series

Z(S,t)=exp( n>0|fix(f n)|t nn) 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 SS”.

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

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

So, by the previous proposition,

exp( n>0|fix(f n)|x nn) \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 SS”. 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 SS — or in category-theoretic jargon, ”being a \mathbb{Z}-set over SS“.

Orbits and multisubsets

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

  • Let orb S(n)orb_S(n) be the set of ff-orbits of SS having cardinality nn.

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

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

Any subset XSX \subseteq S gives a multi-subset where ψ=χ X\psi = \chi_X is the characteristic function of XX, and then the weight of this multi-subset is the cardinality of XX. XSX \subseteq S is an ff-invariant subset of SS iff χ X\chi_X is invariant. Every ff-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 SS of finite weight can be written

n0S nn! \sum_{n \ge 0} \frac{S^n}{n!}

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


Note that if SS is tame, the cardinalities of orb S(n)orb_S(n) and z S(n)z_S(n) are finite for all n0n \ge 0.


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

Proof: Suppose xSx \in S is a fixed point of f nf^n. Then there is some least m1m \ge 1 such that xx is a fixed point of f mf^m, and mm divides nn. In this case xx lies in a (necessarily unique) ff-orbit of cardinality mm. Conversely if xx lies in an ff-orbit of cardinality mm we have xfix(f m)x \in fix(f^m) and thus xfix(f n)x \in fix(f^n) if mm divides nn. In short, fix(f n)fix(f^n) is the disjoint union of all orbits of cardinality mm where m|nm | n. The formula follows.

Last revised on August 16, 2022 at 07:12:10. See the history of this page for a list of all contributions to it.