数理解析研究所講究録
第 1872 巻 2014 年 122-131

122

The Burnside ring and the universal zeta functionof finite dynamical systems
YOSHIDA, Tomoyuki (Hokkaido univ.)

yoshidat@math.sci.hokudai.ac.jp
2013/01/0913:30–14:30
lowing product formula:

0. Introduction$Z_{G}(t):= \exp(\sum_{m=1}^{\infty}\frac{a_{m}}{m}t^{m})=\prod_{n=1}^{\infty}(\frac{1}{1-t^{n}})^{b_{n}/n}(2)$0.1 Zeta functions of finite groups
There are some series associated to finite (or infinite) groups which remind us of zeta functions.

We start with an example from elementary grouptheory ([Yo92]).
Let be a finite group. Define two series$\{a_{n}\}$

$G$

and

$\{b_{n}\}$In particular, when $G=C_{N}$ , a cyclic group oforder $N$ , we have$a_{n}=(N, n)$ ,$b_{n}=[Matrix]$

Thus

by
$Z_{C_{N}}(t)= \exp(\sum_{n=1}^{\infty}\frac{(N,n)}{n}t^{n})$$a_{n}:=\#\{g\in G|g^{n}=1\}=|Hom(C_{n}, G)|.$$b_{n}:=\#\{g\in G||g|=n\},$$= \prod_{n|N}(\frac{1}{1-t^{n}})^{\varphi(n)/n}$denote the order of $g\in G$ . We call
where
Frvbenius numbers after Frobenius’s theorem(1903):

(3)

$|g|$

$a_{n}$$a_{n}\equiv 0$

mod $gcd(n, |G|)$

0.2 Absolute zeta function“Absolute mathematics” means mathematics
over the field with one element (see [KOW03],[KK10] . This curious idea goes back to Tits
(1956). He stated that a Weyl group is a simple$F_{1}$$:=b_{n}/\varphi(n)$ , whereis the EuNote thatler function, is equal to the number of cyclic subgroups of order . By the trivial formula $a_{n}=$, we have$\varphi(n)$$c_{n}$$n$$\sum_{d|n}b_{d}$$\zeta(z)$$:= \sum_{n=1}^{\infty}\frac{1}{n^{z}}$algebraic group with corresponding Dynkin diagram, for example, $GL(n, F_{1})=S_{n-1}.$(1)

During the last twenty years, this imaginary
field has been studied mainly in algebra, especiallyis the Riemann zeta func-

in algebraic number theory and algebraic geometry. At present time, Kurokawa and Koyama’sbook [KKIO] is a unique and nice hterature on ab-$\sum_{m=1}^{\infty}\frac{a_{m}}{m^{z}}=\zeta(z)\sum_{g\in G}\frac{1}{|g|^{z}}=\zeta(z)\sum_{n\geq 1}\frac{b_{n}}{n^{z}}$

where

$)$

tion. This equation can be presented by the fol-

solute mathematics.

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In this book, we find many interesting state-ments.

0.4 Dynamical zeta functions(Tits 1956) The $N$ -dimensional projective space$\mathbb{P}^{N-1}(F_{1})$over$F_{1}$

is an $N$ -point set. Note that

Let $X$ is a dynamical system, that is, $X$ is a finite set with permutation : $Xarrow X$ . Then theArtin-Mazur zeta function of $X$ is defined by$\sigma$$|\mathbb{P}^{N-1}(F_{q})|=1+q+q^{2}+\cdots+q^{N-1}arrow N$(Manin 1993) The zeta functionof$(qarrow 1)$$\mathbb{P}^{N-1}(F_{1})$$Z_{X}^{AM}(t):= \exp(\sum_{m=1}^{\infty}\frac{|Fix_{X}(\sigma^{m})|}{m}t^{m})$should be

Let $\mathbb{R}\cross Xarrow X;(t,x)\mapsto f^{t}(x)$ be a flow (or$\zeta(s,\mathbb{P}^{N}(F_{1})=s(s-1)(s-2)\cdots(s-N)$(Soul\’e 1999) An extension field of$\dot{u}$-dynamical system), and so $f^{0}=id_{X},$ $f^{s+t}=$$\mathbb{R}$$F_{1}$

defined to beof degree$N$$f^{S}of^{t}$. Then the Ruelle zeta function is definedby$F_{1^{N}}:=\{0\}\cup\mu_{N}$where$F_{1^{N}}$$\zeta_{X}(s)=\prod_{\gamma}(1-e^{-sT(\gamma)})^{-1}$is the group of $N$ -th roots of unity. Thus$\mu_{N}$is a multiplicative monoid with zero element.

In absolute mathematics, an -algebra is defined to be a multiplicative monoid with zeroelement ([KK10]).
$F_{1}$$F_{q}$$T(\gamma 9$$\gamma$$\gamma$

0.5 The universal zeta function

0.3 Absolute Weil zeta function
The Weil zeta function of an

where runs over periodic orbits and whereis the period of . The Artin-Mazur zeta functions
is a special case of the Ruelle zeta functions.

-algebra $A$ is de-

The universal zeta function (UZF) of a category is defined by$\mathscr{C}$

fined by$Z_{\mathscr{C}}(t):= \mathscr{C}(t):=\sum_{N\in \mathscr{C}/\cong}\frac{1}{|Aut(N)|}t^{N}$$Z_{A}^{Wei1}(t):= \exp(\frac{|Hom_{F_{q}}(A,F_{q^{m}})|}{m}t^{m})$Thus the absolute Weil zeta function of analgebra should be defined by formally replacingby
In particular, the absolute zeta function of $A=$
is given by$F_{1^{-}}$$F_{q}$$F_{1}.$$F_{1^{N}}$Here the formal summation on the right hand side
is taken over all isomorphism classes of objects of. The symbols
denotes a variable associatedto the object. Of course, every automorphism group Aut $(N)$ must be a finite group. The
author called such a series as an exponential function of in the paper $[YoO1]$
Assume that has any finite coproduct. In thiscase, we usually assume that the symbols
satisfy the relations$t^{N}$$\mathscr{C}$$N\in \mathscr{C}$$\mathscr{C}$$Z_{F_{1^{N}}}^{Wei1}(t)=\exp(\sum_{n=1}^{\infty}\frac{(N,n)}{n}t^{n})$$\mathscr{C}$because an$F_{1}$-algebra homomorphism from $F_{1^{N}}=$

to $F_{1^{m}}=\{0\}\cup\mu_{m}$ is uniquely given bya group homomorphism from, that is,to$\{0\}\cup\mu_{N}$$\mu_{m}$$\mu_{N}$$Z_{F_{1^{N}}}^{Wei1}(t)=Z_{C_{N}}(t)$$t^{N\prime}s$$t^{M+N}=t^{M}\cdot t^{N},$ $t^{\emptyset}=1,$ $t^{N}=t^{N’}$Thusbelongs to a complete semigroup algebra. Furthermore, the Krull-Schmidt
property for induces the exponential formula$Z_{\mathscr{C}}(t)$As a consequence, the absolute Weil zeta function of the-algebra

is nothing but the”zetafunction” of a cyclic group defined by (3).
$F_{1}$$F_{1^{N}}$if $N\cong N’$$\mathbb{Q}[[\mathscr{C}^{op}/\cong]]$$\mathscr{E}$$\mathscr{E}=\exp(J(t))$

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where $J$ is the subcategory of connected objects.

Dynamical systems and their zeta

1

0.6 Zeta properties

1.1 Almost finite dynamical systems ( $DS$ )As is well-known, the Riemann zeta function
has has some remarkable properties called thezeta properties (of course, Riemann hypothesis is$\zeta(z)$still open).

or simply $X$
A dynamical system ( $DS$ )Sym(X).
is a set equipped with a permutation
Thus a $DS$ is algebraically nothing but a $C$-set,
is a infinite cyclic group. Such a
where$(X, \sigma)$$\sigma\in$$C=\langle\sigma\rangle$(M)$\zeta(z)$is a meromorphic function on$\mathbb{C}$with sim-ple pole $z=1.$( $FE$ ) $\zeta(1-z)=2^{1-z}\pi^{-z}\cos(\pi z/2)\Gamma(z)\zeta(z)$ .
($EP$

set is often called a cyclic set([DS89]).
Any $DS$ is a disjoint union of orbits. An orbit
is transitive (or often connected) as a -set.
. Then $C(n):=C/C^{n}$ isLet$:=1$ , thea dynamical system of size . Let$:=C/C^{\infty}$ istrivial subgroup of . Then
an infinite and transitive -set and is called to be
free. Conversely, any transitive $DS$ of size$C$

) Euler product: $\zeta(z)=\prod_{p}(1-p^{-z})^{-1}$$C^{n}=\langle\sigma^{n}\rangle\leq C$$n$( $SV$ ) $\zeta(-n)=\frac{-B_{n+1}}{n+1},$ $\zeta(-2n)=0,$ $n\geq 1.$( $RH$ ) Non-trivial zeros of( $PF$ )$\zeta(z)$lie on $\Re(z)=1/2.$$\zeta(z)=\frac{-\pi^{z/2}}{z(1-z)\Gamma(z/2)}\prod_{\rho}(\begin{array}{l}1-\underline{z}\rho\end{array})(p$$C^{\infty}$$C(\infty)$

$C$

$C$

runs

$n\leq\infty$

over nontrivial zeros).

is isomorphic to $C(n)$ .
Now, a $DS$ $X$ is called to be almost flnite

0.7 Zeta properties for UZF([DS89])if

We are interested in a”category” whose universal zeta function (UZF) satisfy “zeta properties”.
However, a UZF is not a series with single variable despite its appearance, but it is a series with
infinitely many variables in general. Zeta properties for UZF might be written by the language ofcategory theory. Perhaps, such a categorical zeta
property for UZF gives, for example, Riemann hy-(a) $X$ has only a finite number of orbits of given

length(b)

Example. A UZF give some classical zeta functions by specializations. Let $DS$ be the cate$:=$gory of finite $DS$ ’s. Then the specialization$|Hom(N, X)|u^{|N|}$ gives the Artin-Mazur zeta func$t^{N}$for any $n<\infty.$
has no free orbits.
$n$

The condition (a) is equivalent to(a’)$N_{m}$$:=Fix_{X}(\sigma^{m})|<\infty$

for any $m=1,2,$$\cdots.$

A DS satisfying (a) or (a’) is called to be essentially flnite.
Furthermore, the condition (b) is equivalent topothesis for a classical zeta function by applying asuitable functor.

$X$(b’) Any element of $X$ is periodic, that is, it is contained in an orbit of finite length, or equivalently,

for any , their exists $m\geq 1$ such that $\sigma^{m}x=x.$$x$

For any infinite series of non-negative integers$b=(b_{1}, b_{2}, \cdots)$ ,tion:$X(b) :=b_{1}C(1)+b_{2}C(2)+\cdots$$Z_{DS}(t)=Z_{X}^{AM}(u)$The main purpose of this study is to find categorical zeta properties for categorical Artin-Mazur

zeta function $Z_{DS}(t)$is almost finite. Here $bnC(n)$ is a disjoint union of
copies of $DS$ isomorphic to $C(n)$ .
Conversely, any almost finite $DS$ $X$ is isomor-$n$phic to $X(b)$ for a uniquely determined

$b.$

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Let

$X^{per}$(resp. $X^{aper}$ ) be the set of periodic(resp. aperiodic) of a $DSX$ . Then any $DS$ $X$ isthe disjoint union of its periodic part

its aperiodic part
finite $DSX$ , we have$X^{aper}$and. Thus for any essentially$X\cong X(b)+b_{\infty}C(\infty)$

1.3 Zeta properties of AMZ of finite $DS$

$X^{per}$,for some integral vector and cardinal number$b$(4)$b_{\infty}.$

AMZ of a finite $DS$ satisfies some zeta properties
and they are proved easily.
Theorem. Let
following hold:$(X, \sigma)$( $EP$ ) $Z_{X}^{AM}(u)= \prod_{P}(\frac{1}{1u^{|P|}})$ ,where $P$ runs over

1.2 The Artin-Mazur zeta functions
Let$(X, \sigma)$be a finite $DS$. Then the$AMZ$ is a rational$peri_{0}^{-}dic$

orbits. In particular,function.

( $DE$ ) $Z_{X}^{AM}(u)=\det(I_{X}-A_{\sigma}u)^{-1},$

be an essentially finite $DS$ . Since

where
tri associated to$A_{\sigma}=(\delta_{x,\sigma y})_{x,y\in X}$$N_{m}:=|Fix_{X}(\sigma^{m})|=\#\{x\in X|\sigma^{m}x=x\}<\infty,$

is the permutation ma-$\sigma.$( $FE$ ) $Z_{X}^{AM}(1/u)=(-u)^{|X|}\det(\sigma)Z_{X}^{AM}(u)$ .

the Artin-Mazur zeta function (AMZ)( $RH$ ) $Z_{X}^{AM}(u)$ has no zero on C. Its poles lie on$|u|=1$ . Furthermore, $u=e^{-s}$ is a pole if and$Z_{X}^{AM}(u):= \exp(\sum_{m=1}^{\infty}\frac{N_{m}}{m}u^{m})$only if $\Re(s)=0.$

is well-defined.

The theory of AMZ of almost finite but not fi-

Lemma. For AMZ’s of essentially finite $DS$ ’s, thefollowing hold:(i) $Z_{X}^{AM}(u)=Z_{X^{per}}^{AM}(u),$ $Z_{X^{aper}}^{AM}(u)=1.$(ii)$Z_{\emptyset}^{AM}(u)=1,$$Z_{X+Y}^{AM}(u)=Z_{X}^{AM}(u)\cdot Z_{Y}^{AM}(u)$.

nite $DS$ ’s becomes extremely difficult. The zeta
properties except for ( $EP$ ) do not hold in general.
There are some examples of well-known $DS$ whichshow the difficulty.

(iii) $Z_{C(n)}^{AM}(u)=\overline{1-u^{n}}.$

1.4 lhara zeta functions(iv)

Let $(V, E)$ be a finite simple graph and $X^{I}(V, E)$
the set of walks with no backtrak:$Z_{X(b)}^{AM}(u)= \prod_{n=1}^{\infty}(\frac{1}{1-t^{n}})^{b_{n}}$PROOF. (i) and (ii) are trivial. (iii) For $C(n)=$$X^{I}(V, E)$ $:=\{(x_{i})\in V^{Z}|(x_{i}, x_{i+1})\in E,x_{i+2}\neq x:\}$$\langle\sigma\rangle/\langle\sigma^{n}\rangle,$

makes an essentially finite $DS$ together with rightshift operator:$N_{m}=|Fix_{C(n)}(\sigma^{m})|=\{\begin{array}{l}n if n|m0 else\end{array}$$\sigma:(x_{i})_{i}\mapsto(x_{i-1})_{i}$ThusThen the Ihara zeta function: is defined by$Z_{C(n)}^{AM}(u)= \exp\sum_{m\equiv 0(n)}\frac{n}{m}u^{m}=\exp\sum_{k\geq 1}\frac{1}{k}u^{kn}$$Z_{(V,E)}^{I}(u)=Z_{X^{I}(V,E)}^{AM}(u)= \prod_{P}(1-u^{|P|})^{-1}$$= \frac{1}{1-u^{n}}$(iv) follows from (4).

where $P$ runs over prime walks with no backtrack.
See [GIL08]

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If $G=(V, E)$ is $(q+1)$ -regular, then

Let $F:Xarrow X$ be the Frobenius automorphism). Then $(X, F)$ is essentially(induced byfinite $DS$ . Since $Fix_{X}(F^{m})=X(F_{p^{m}})$ , the Weilzeta function is equal to the AMZ of this $DS$ :$x\mapsto x^{p}$$Z_{G}^{I}(u)=(1-u^{2})^{\chi(G)}\det(I-uA+qu^{2})^{-1},$

where $\chi(G)$ $:=|E|-|V|+1$ and$A$

is the incidence

matrix.

$\zeta(X, u)=Z_{(X,F)}^{AM}(u)$A symbolic $DS$ on alphabet
defined by$Q$

Weil zeta functions
called Weil conjectures were proved mainly byGrothendieck and Deligne.
The Weil zeta function is essentially same asof -algebra appeared in section 0.3.

with $|Q|=q$ isforproperties

Zeta

1.5 Symbolic dynamical systems$Z_{A}^{Wei1}(t)$$A$$F_{p}$$Q^{Z}:=\{(x_{i})_{i\in Z}|x, \in Q\}$

with shift operator
tially finite because$x=(x_{i})\in$$N_{m}=|$$\sigma$Fix: $(x_{i})\mapsto(x_{i-1})$ . It is essen-

2 The Burnside ring$\Omega(C)$

2.1 The category of dynamical systems$(\sigma^{m})\Leftrightarrow x_{i+m}=x_{i}(\forall i\in Z)$

Fix $(\sigma^{m})|=q^{m}<\infty$

be an infinite cyclic group. Since aLetfinite $DS$ is nothing but a finite $C$-set, we denoteby set the category of finite $DS$ ’s. Furthermore,
we denote by afse the category of almost finite$dDS’ s$ . These categories are like to set , the category of finite -sets, where is a finite group.
$X\cross Y,$ $X+Y$ denote a direct product and a
disjoint union of two $DS$ ’, respectively. In partic$C=\langle\sigma\rangle$

Thus AMZ has the following form:$c$$t^{C}$$Z_{Q^{z}}^{AM}(u)= \exp\sum_{m=1}^{\infty}\frac{q^{m}}{m}u^{m}=\frac{1}{1qu}$$G$$furtherm^{-}ore$$Q^{z}$$G$$G$

we haveis essentially finite and$(Q^{z})^{per}\cong_{n=1}II^{M(q,n)C(n)}\infty$ular,$X^{n}$(resp. $nX$ ) denotes the direct product(resp. disjoint union) of -copies of $X.$$n$

$Z_{Q^{z}}^{AM}( u)=\frac{1}{1-qu}=\prod_{n=1}^{\infty}(\frac{1}{1t^{n}})^{M(q,n)}$denotes the set of $C$-maps between
two $DS$ ’s $X$ , Y. Note that for $m=1,2,$$Map_{C}(X, Y)$

where $M(q, n)$ is the Necklace$pol^{-}ynomia1$:$\cdots,$$Map_{C}(C(m), X)\cong Fix_{X}(\sigma^{m});f\mapsto f(C^{m})$(5)$M(q, n):= \frac{1}{n}\sum_{k|n}\mu(\frac{n}{k})q^{k}$$Map_{C}(C(\infty), X)\cong X;f\mapsto f(C^{\infty})$$Map_{C}(C(m), C(\infty))=\emptyset$

1.6 Weil zeta functions as AMZ

be the set of maps between from $X$ to $Y,$
is a $DS$ with -action defined byso thatLet$Y^{X}$$Y^{X}$The Weil zeta function (or congruence zetaisfunction) of a variety $X$ over a finite field$C$$\sigma f:Xarrow Y;x\mapsto\sigma f(\sigma^{-1}x)$$F_{p}$

isis almost finite, then
If $X$ is finite and$X$
almost finite. If is finite and if is essentiallyis essentially finite iff $X$ has only
finite, thenfinite number of aperiodic orbits. In particular,isis a finite set with trivial $C$-action, then$Y$defined by$Y^{X}$$Y$$Y^{X}$$\zeta(X, u)=\exp\sum_{m=1}^{\infty}\frac{|X(F_{p^{m}})|}{m}u^{m}$$Y$

where $X(F_{p^{m}})$ is the (flnite) set ofpoints.

$F_{p^{m}}$-rational$Y^{C(\infty)}$a symbolic $DS.$127
ring which is called a Necklace algebra and is

2.2 The Burnside ring
The Burnside ring

denoted by Nr(Z).
The multiplication is defined byfinite groupis the Grothendieck ring of set , the category offinite -sets, that is, the abelian group generated
by the symbols [X], where $X$ is any finite -set,

respectively. This formula gives the multiplication

with relation

formula in the Necklace algebra$\Omega(G)$of a$pre\succ$$[C(m)]\cdot[C(n)]=(m, n)[C([m, n])]$

$G$

$G$

$G$

$G$

where $(m, n)$ and $[m, n]$ denote the $gcd$ and lcm,

if $X\cong X’$ and$[X+Y]=[X]+[Y]$$[X]=[X’]$$Z^{N}$

as follows:$a \cdot b=c, c_{l}=\sum_{[m,n]=l}(m, n)a_{m}b_{n}$where $X+Y$ denotes the disjoint union. The mul-where the summation is taken over non-negative
integers such that $[m, n]=l.$$l$tiplication on$\Omega(G)$is defined by $[X]\cdot[Y]=[X\cross Y].$For each subgroups$S$, the map

The k-th component of the (complete) Burnsidehomomorphism
gh $(G)$ is given by$\hat{\varphi}:\hat{\Omega}(C)arrow$$\varphi_{S}:\Omega(G)arrow Z;[X]\mapsto|X^{S}|$where$:=\{x\in X|Sx=x\}$ , defines a homomorphism called a Burnside homomorphism:$\hat{\varphi}_{k}:=\varphi_{C^{k}}:[C(n)]\mapsto\{\begin{array}{l}n if n|k0 if n\int k\end{array}$$X^{S}$

Thus we have$\varphi=\prod_{(S)}\varphi_{S}:\Omega(G)arrow gh(G):=\prod_{(S)}Z$

where $(S)$ runs over all conjugacy classes of subgroups of . The Burnside homomorphism is
an injective ring homomorphism, and soisviewed as a subring of the ghost ring Gh $(G)$ .
The complete Burnside ringis the closure ofin the product space gh $(G)$ , where$G$$\hat{\varphi}_{k}(X(b))=d_{k}:=\sum_{i|k}ib_{i}$This relation between
is equivalentandto the following cyclotomic identity.
$(d_{k})$$(b_{i})$$\Omega(G)$$\hat{\Omega}(G)$$\Omega(G)$2.3 The complete Burnside ring
The complete Burnside ring$\hat{\Omega}(C)$of an infinite
is the Grothendieck ring ofalmost finite $DS$ ’s. Thus its element is presented
by an infinite sum
cyclic group$\hat{\Omega}(C)$$C=\langle\sigma\rangle$$X(b):= \sum_{n=1}^{\infty}b_{n}[C(n)]$for some integral vector $b=(b_{1}, b_{2}, \cdots)\in Z^{N}$ . The
map $b\mapsto X(b)$ induces a bijection$X$:$\prod_{n=1}^{\infty}(\frac{1}{1-t^{n}})^{b_{n}}=\exp\sum_{n=1}^{\infty}\frac{d_{n}}{n}t^{n}(=Z_{X(b)}^{AM}(t))$

Note that two almost finite $DS$ $sX$ andisomorphic if and only if
.
)

$Y$

2.4

Dress-Siebeneicher’s theory

In their paper in [DS89], Dress and Siebeneicher
stated an important and surprising result whichconnects the Burnside ringof infinite cyclic
group $C,$ -rings and Witt vectors Their result is$\hat{\Omega}(C)$$A$

presented by the following diagram:
Nr(Z) $=Z^{N}$$\cong\downarrow X$$Z^{N}arrow^{\underline{}\simeq}\hat{\Omega}(C);b\mapsto X(b)$$WZ–gh(C)\iota_{N}^{z)_{\underline{\simeq}}^{\tau}}\Phiarrow\hat{\Omega}r_{\hat{\varphi}})arrow\Lambda-tZ[[t]]\underline{\simeq\underline{s_{l}\simeq}}7_{L})$Since
is a commutative ring, by pulling back$X$
along , we have that
becomes a commutative$\hat{\Omega}(C)$$Z^{N}$

are$\hat{\varphi}(X)=\hat{\varphi}(Y)$128
Here, we briefly explain the rings and maps ap-

peared in the diagram([DS89], [YalO], [Ha09]).

3 The universal zeta functions(1-a) Nr(Z) $:=Z^{N}$ : Necklace algebra. Its elementof integers.
is an infinite vector

3.1 The UZF of categories and functors$b=(b_{n})_{n=1}^{\infty}$Let(1-b) $W(Z)$ $:=Z^{N}$ : the ring of universal Witt vec-

tors. Its element is a sequence$q=(q_{n})_{n=1}^{\infty}$of integers.
(1-c): the complete Burnside ring of an infinite cyclic group $C$ . Its element is an inflnite$\hat{\Omega}(C)$$\Lambda(Z)$is locally flnite, that is, $|Hom(X, Y)|<\infty.$of objects is a(ii) The isomorphism classes(i)$\mathscr{C}$$\mathscr{C}/\cong$

sum $\sum_{n=1}^{\infty}b_{n}[C(n)]=:X(b)$ .
(1-d)

be a category. We assume that$\mathscr{C}$small set.
The universal zeta function (UZF) or the exis
ponential function $([YoO1])$ of a category
defined by$\mathscr{E}$$:=1+tZ[[t]]$ : the universal -ring. The$\lambda$

addition is defined by the multiplication offormal power series.
(2-a) $X$ : $b \mapsto X(b)=\sum_{n=1}^{\infty}b_{n}[C(n)]$is the peri, where:
odic part of the (virtual) symbolic $DS$ , and: $[C(m)]\mapsto[C(mnr)].$: $[X] \mapsto 1+\sum\varphi_{n}(S^{n}(X))t^{n}$ , where $S^{n}(X)$
(2-c)

(2-b)

$q \mapsto\sum ind_{n}q_{n}^{(C)}$$\tau$$q^{(C)}$$ind_{n}$$s_{t}$

is the n-th symmetric power.
$\hat{\varphi}$$t^{N}$

ject $N$ such that $t^{N}=t^{N’}$ if $N\cong N’.$be the category of dynamical systems
Let
whose underlying set is an object of . Then$\mathscr{C}^{c}$(2-d) $\Phi:(q_{n})\mapsto(\sum_{k|n}kq_{k}^{n/k})_{n}$(2-e)

Here the summation is taken over a complete
representatives of isomorphism classes of objects.
Furthermore,is a variable associated to an ob-$\mathscr{C}$: the complete Burnside homomorphism.
$\mathscr{C}^{c}(t)=\sum_{N\in \mathscr{C}/^{\underline{\simeq}}}t^{N}$

(2-f)

$L$

:

$a(t) \mapsto t\frac{d}{dt}\log a(t)$The maps$\Phi,\hat{\varphi},$$L$are injective ring homomor-

is the geometric series type UFZ of

are ring isomorphisms.
Assume that $b\in$ Nr(Z), $q=(q_{n})\in W(Z)$ ,phisms.

$\tau,$$\mathscr{C}.$$s_{t}$$a(t)=1+ \sum a_{n}t^{n}\in\Lambda(Z),$$d=(d_{n})\in$gh(Z)are corresponding each other by these maps:$( \mathscr{C}^{C})^{op}(t)=\sum_{N\in \mathscr{C}/\cong}N^{t}$is the Dirichlet type UFZ of
The UZF of a functor$\mathscr{C}.$$F:\mathscr{C}arrow \mathscr{S}$is defined by$\tau(q)=X(b),$ $s_{t}(X(b))=a(t),\hat{\varphi}(X(b))=d$$(q_{n})rightarrow(b_{n})rightarrow(a_{n})rightarrow(d_{n})$Then the above diagram implies the following
identities which remind us of dynamical zeta func-$F(t);= \sum_{A\in \mathscr{C}/\simeq}\frac{1}{|Aut(A)|}t^{F(A)}$is faithful and that for any
Assume that, there exists only finite number of isosuch that $F(A)\cong N.$
morphism classes of, a -structure on $N$ is defined
For anand an isomorphismsofto be a pairand
: $F(A)\cong N$ . Two -structure
on $N$ is called to be isomorphic if there exists anautomorphism : $A\cong B$ such that $\beta oF(f)=\alpha.$$F$$N\in \mathscr{S}$tions:$A\in \mathscr{C}$$N\in \mathscr{S}$$(A, \alpha)$$\prod_{n=1}^{\infty}\frac{1}{1-q_{n}t^{n}}=\prod_{n=1}^{\infty}(\frac{1}{1t^{n}})^{b_{n}}$$\mathscr{C}$$A\in \mathscr{C}$$\mathscr{C}$$\alpha$$=1+ \sum a_{n}t^{n}=e^{-}xp\sum_{n=1}^{\infty}\frac{d_{n}}{n}t^{n}$$f$$(A, \alpha)$$(B, \beta)$

129
Let denote by Strmorphism classes ofthe (finite) set of iso-stmctures on $N$ . Then$(\mathscr{C}/N)/\cong$$\mathscr{C}$$\sum_{n=0}^{\infty}\frac{|Hom(G,S_{n})|}{n!}t^{n}=\exp(\sum_{H\leq G}\frac{t^{(G..\cdot H)}}{(GH)})$

3.4 The UZF of finite dynamical systems

Assume that a locally finite category has any
finite coproducts. In this case, we can think that

Since the category of finite dynamical systems
satisfies the strict Krull-Schmidt property, its UZF
satisfies the exponential formula. The automorphism group of $C(n)$ is a cyclic group of order

the UZF

Furthermore, by the $KS$ -property,

3.2 The exponential formula$\mathscr{E}$$\mathscr{E}(t)=Z_{\mathscr{E}}(t)$semigroup algebra

belongs to the complete$\mathbb{Q}[[\mathscr{E}^{op}/\cong]]$$n.$, and so we intro-

duce the following $mle$ :Aut$( X(b))\cong\prod_{n\geq 1}$Aut$(b_{n}C(n)) \cong\prod_{n\geq 1}C_{n}tS_{b_{n}}$$t^{M+N}=t^{M}\cdot t^{N},t^{\emptyset}=1$

Assume further thatsatisfies the strict KrullSchmidt property, that is, that any object hasa unique decomposition into a coproduct of connected objects. Then the exponential formula
holds:$\mathscr{E}$

Thus the universal zeta function of finite $DS$ ’s isgiven by

Let $X$ be an essentially finite $DS$ . Then by thewhere $J=$ Con

is the subcategory of connected objects. Under some technical conditions,the exponential formula implies the $KS$ -property,
that is, $KS$ -property is a categorification of ex$(\mathscr{E})$specialization $t^{N}arrow|Hom(N, X)|u^{|N|}$ , we have theArtin-Mazur zeta function$Z_{X}^{AM}(u)= \exp(\sum_{m^{1}}^{\infty}\frac{|Fix_{X}(\sigma^{m})|}{m}u^{m})$ponential formulas.

3.3 Wohlfahrt formula
Let

$G$

be a finitely generated group. Then theHere, note that $Hom(C(m), X)$$=$$Fix_{X}(\sigma^{m})$Thus we have a group homomorphism

exponential formula has the form:$Z^{AM}:\hat{\Omega}(C)arrow \mathbb{Q}(u)^{\cross};[X]\mapsto Z_{X}^{AM}(u)$$\sum_{X\in set^{G}/\cong}\frac{t^{X}}{|Aut(X)|}=\exp(\sum_{H\leq G}\frac{t^{G/H}}{(G\cdot H)})$(Here we used that $Aut(G/H)\cong N_{G}(H)/H$ andthat the number of subgroups conjugate to $H$ isequal to $(G:N_{G}(H)))$ .

Applying this formula to the forgetful functor,
we have the Wohlfahrt formula:

4 Problems
4.1

Zeta properties for UZF

Let $Z_{DS}(t)$ be the universal zeta function of finite dynamical systems. It is mapped by the Burn-.

130side homomorphism$\varphi_{k}$to the following series:$\varphi_{k}(Z_{DS}(t))=\exp(\sum_{n=1}^{\infty}\frac{1}{n}\varphi_{k}(t^{C(n)})$

It induces many formula, e.g. the Wohlfahrt formula for one-variable series. Unfortunately, exponential formulas are not called one of zeta properties.

$= \exp(\sum_{n=1}^{\infty}\frac{\langle C(k)\cross C(n),t\rangle}{n})$4.2 AMZ of non-invertible $DS’s$$= \exp(\sum_{n=1}^{\infty}\frac{\langle C([k,n]),t\rangle^{(k,n)}}{n})$A rooted forest $F$ can be viewed as a non, where maps
invertible dynamical systemany vertex to its unique child, such that any pe$(F_{)}\sigma)$$= \exp(\sum_{n=1}^{\infty}\frac{1}{n}(t_{[k,n]})^{(k,n)})$$\sigma$riodic point is a fixed point of . Let $RF$ be thebe the catcategory of rooted forests. Let
egory of non-invertible dynamical systems, whose, where $X$ is a set with selfobject is a pair$\sigma$Thus $Z_{DS}(t)$ can be viewed as a series with infinite
number of variables.
areNote that two almost finite $DS$ ’s $X$ andisomorphic if and only if they have the same AMZ.
$Y$

The theory of almost flnite $DS$ is extremely difficult. whereas it is rather strange that almost finite$DS$ ’s are controlled by only finite $DS$ ’s.

Question. Are there any zeta properties for UZF
of finite $DS$ ’s which induce those for usual zeta
properties? For example, is there a universal func-$set^{N_{0}}$$(X, \sigma)$

map $\sigma:Xarrow X.$
Then the category $RF$ of rooted forests is equivalent to the indexed category of the category $RT$set( $RT$ ), that is, $RF$ isof rooted trees. $RF$$\cong$a strict Krull-Schmidt category whose connected
objects are rooted trees.
$set^{N_{0}}arrow^{forest}RF$tional equation$\downarrow$$Z_{DS}(1/t)=\gamma(t)Z_{DS}(t)$where the gamma factor$\gamma(t)$$?$?

is a group homomor-

phism

se

PerPB
$\underline{forget}$$t^{}$$\downarrow root$

set

dynamical system

The pullback diagram in
the 2-category CAT, thecategory of categories, Fora non-invertible finite$(X, \sigma)$,

Per(X) $:=\{x\in X|\sigma^{m}x=x$ for$\gamma(t):\hat{\Omega}(C)\mapsto \mathbb{Q}(t)^{\cross}$

If such a formula holds, then applying functors
we have a usual function equation. However, therea $m\geq 1\},$

Root $(F)$ : the set of roots.
Then we have$\sum_{(X,\sigma)}\frac{t^{F(X,\sigma)}}{|Aut(X,\sigma)|}=(1-\sum_{T}\frac{t^{T}}{|Aut(T)|})^{-1}$

is no possibility in this form, because

runs over rooted trees,
where
invertible finite $DS$ ’s.
gives
Specialization:$T$$\varphi_{1}(Z_{DS}(t))=\exp(\sum_{m}\frac{1}{n}t_{n})$$(X,\sigma)$non-$t^{T}\mapsto t^{|V(T)|}$

does not satisfy any functional equation.

Remark. There is a categorical exponential formula for UZF$Z_{DS}(t)= \sum_{N}\frac{t^{N}}{|Aut(N)|}=\exp\sum_{m=1}^{\infty}\frac{t^{C(m)}}{m}$$\sum_{n=0}^{\infty}\frac{n^{n}}{n!}t^{n}=(1-\sum_{n=1}^{\infty}\frac{n^{n-1}}{n!}t^{n})^{-1}$

Problem. Build the theory of non-invertible flnite dynamical systems (NIFDS).

131
Let
be a semigroup
generating only one element . Then a NIFDS
is nothing but finite $D$ -set. A NFDS $X$ is ir$D=\langle\sigma\rangle=\{1, \sigma, \sigma^{2}, \cdots\}$$\sigma$reducible if $X=A\cup B$ implies $A=X$ or$B=X$ . Any irreducible NIFDS has the form
$D(h, n)=D/(\sigma^{h}=\sigma^{h+n})$.

Let $G_{0}(C)$ be a abelian group generated by [X],
where $X$ is satisfying the relation:(i) $[X]=[Y]$ ffi $|Hom(D[h, n)$ for any $h\geq$ Oand$n\geq 1.$(ii) $X[cupY]+[X\cap Y]=[X]+[Y],$$D$$[\emptyset]=0$

We call $G_{0}(C)$ the Burnside ring of a semigroup. The Burnside homomorphism is defined by

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$C$

$*$

$\varphi:G_{0}(D)arrow gh(Z)=\prod_{(h,n)}Z$; $[X]\mapsto Hom(D(h, n),$ $X]$
By this way, we have the theory of Burnside rings.

4.3 Rational$DS^{l}s$?

solute derivations and zeta functions, Docu-

Remember the equation (2). In this case, $b=$$(b_{n}/b)$ is notcorresponding to any dynamical system. Since
is the number of elements of of$b_{n}$$G$

order $n,$ $b_{n}/n$ is not an integer in general. But
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565-584.
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In particular, we can not treat absolute Weilzeta function as AMZ of $DS$ . Is there any idea
by which we involve it in the theory of dynamicalzeta functions.
2421-431.
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$4t\vdash$