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 invariant subset of SS iff χ X\chi_X is invariant. Every invariant multi-subset is a (possibly infinite) sum of such invariant characteristic functions.

Further, XSX \subseteq S is an invariant subset of SS iff it a union of ff-orbits.

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 21, 2019 at 10:00:05. See the history of this page for a list of all contributions to it.