Opposite power series

0.3 MB·English

by  Kyoji Saito

OPPOSITE POWER SERIES
KYOJI SAITO
2
1
Dedicated to Professor Antonio Mach`ı
0
2 on the occasion of his 70th birthday.

n
a
Abstract. In order to analyze the singularities of a power series
J
function P(t) on the boundary of its convergent disc, where P(t)
7
2 wasmainlythegrowthfunction(Poincar´eseries)forafinitelygen-
eratedgrouporamonoid[S1],weintroducedthespaceΩ(P)ofop-

] posite power series in the opposite variable s=1/t. In the present
A
paper,forgettingaboutthegeometricorcombinatorialbackground
C
onP(t), westudy thespaceΩ(P)abstractlyforanysuitablytame
h. power series P(t) C t . For the case when Ω(P) is a finite set
∈ { }
t and P(t) is meromorphic in a neighbourhood of the closure of its
a
m convergent disc, we show a duality between Ω(P) and the highest
order poles of P(t) on the boundary of its convergent disc.
[

1
v Contents
3
1 1. Introduction 2
7
2. The space of opposite series. 4
5
2.1. Tame power series 4
.
1
2.2. The space Ω(P) of opposite series 5
0
2 2.3. The τ -action on Ω(P) 6
Ω
1
2.4. Examples of τ -actions 6
: Ω
v 2.5. Stability of Ω(P) 6
i
X 3. Finite rational accumulation 8
r 3.1. Finite rational accumulation 8
a
3.2. τ -periodic point in Ω(P) 9
Ω
3.3. Example by Mach`ı [M] 11
3.4. Simply accumulating Examples 11
3.5. Miscellaneous Examples 12
4. Rational expression of opposite series 13
4.1. Rational expression 13
4.2. Coefficient matrix M of numerator polynomials 14
h
4.3. Linear dependence relations among opposite series 15
4.4. The module CΩ(P) 17
5. Duality theorem 18
5.1. Functions of class C t 18
r
{ }
5.2. The rational operator T 18
U 1
5.3. Duality theorem 19
5.4. Example by Mach`ı (continued) 24
References 25
2 KYOJISAITO
1. Introduction
There seems a remarkable “resonance” between oscillation behavior1
o(sfeaeseeqquuaetnicoen{(γ2n.}1n.2∈)Z)≥0aonfdcotmheplseinxgnuulmarbiteiresssoaftiistfsyignegnaertaatminegcofunndcittiioonn
P(t)= ∞n=0γntn on the boundary of the disc of convergence in C. The
idea was inspired by and strongly used in the study of growth functions
P
(Poincar´e series) for finitely generated groups and monoids [S1, 11].
§
Let us explain the “resonance” by a typical example due to Mach`ı
[M] (for details, see Examples in 3.3 and 5.4 of the present paper.
§ §
Other simple examples are given in 3.4 (see [C, S2, S3]) and 3.5). By
choosing generators of order 2 and§3 in PSL(2,Z), Mach`ı h§as shown
that the number γ of elements of PSL(2,Z) which are expressed in
n
words of length less or equal than n Z w.r.t. the generators is given
0
by γ =7 2k 6 and γ =10 2k∈6≥for k Z . On one hand, this
2k 2k+1 0
· − · − ∈ ≥
means that the sequence of ratios γ /γ (n=1,2, ) accumulates to
n 1 n
two distinct “oscillation” values 5, −7 according a·s··n is even or odd.
{7 10}
On the other hand, the generating function (or, so called, the growth
function) can be expressed as a rational function P(t)=(1+t)(1+2t) , and
(1 2t2)(1 t)
it has two poles at 1 on the boundary of its conve−rgent−disc of
{±√2}
radius 1 . We see that there is a “resonance” between the set 5, 7 of
√2 {7 10}
o“fostchiellafutinocntsio”noPf t(ht)e,sienqutheencwea{yγwn}en∈shZ≥a0llaenxdplathine isnetth{±e √p1r2e}seonft“ppaopleesr”.
In order to analyze these phenomena, in [S1, 11], we introduced a
§
space Ω(P) of opposite power series in the opposite variable s=1/t,
as a compact subset of C[[s]], where each opposite series is defined by
cuosminpgr“ehosecnislliavteioinnfso”rmofattihoensoefquoescniclleat{iγonn}sn(∈sZe≥e0 s2o.2thDaetfiΩni(tPio)nc(a2r.r2i.e2s))a.
§
On the other hand, the space Ω(P) has duality with the singularities of
thefunctionP(t)( 5Theorem). Thus, Ω(P)becomes abridgebetween
§
tShinectewtohesumbjeetchtosd: oissciinlldaetpioennsdeonft{oγfn}tnh∈eZ≥g0roaunpdtshineogrueltaircitbieasckogfrPou(nt)d.
and is extendable to a wider class of series (see 2.1 Example 2), which
§
we call tame, we separate the results and proofs in a self-contained way
in the present paper. We study in details the case when Ω(P) is finite,
where we have good understanding of the above mentioned resonance
by a use of rational subset explained in the following paragraph, and
Mach`ı’s example is understood in that frame.
One key concept in the present paper is a rational subset U ( 3),
whichisasubset ofthepositiveintegersZ suchthatthesum § tn
0 n U
≥ ∈
1Byanoscillationbehavior,wemeanthat,foreachfixedk Z 0calledPaperiod,
thesequenceoftherateγ /γ (n Z )hasseveraldifferent∈acc≥umulationvalues.
n k n >>0
− ∈
OPPOSITE POWER SERIES 3
is a rational function in t (i.e. U, up to finite, is a finite union of arith-
metic progressions). The concept is used twice in the present paper.
The first time it is used is in 3, where we show that, if the space of op-
positeseries Ω(P)isfinite, th§en thereisafinite partitionZ = U of
0 i i
Z into rational subsets so that there is no longer oscillat≥ion in∐side in
0
≥
each γ : n U . We call such phenomena “finite rational accumula-
n i
{ ∈ }
tion”( 3.2Theorem) (such phenomena alreadyappearedwhenwewere
§
studying the F-limit functions for monoids [S1, 11.5 Lemma]). The
§
secondtimeitisusedisin 5,whereweintroducearationaloperatorT
U
acting on a power series P§(t) C[[t]] by letting T P(t):= γ tn.
∈ U n U n
Therationaloperatorsformamachine that“manipulates” singu∈larities
P
of the power series P(t). In this way, rational subsets combine the os-
cfuilnlacttiioonnoPf(at)s:e=quen∞nc=e0{γγnntn}nf∈oZr≥t0haencdastehewshinenguΩla(rPit)ieissofifntihtee.generating
The contents of the present paper are as follows.
P
In 2, we introduce the space Ω(P) of opposite series as the accu-
mulat§ing subset in C[[s]] of the sequence X (P):= n γn ksk (n=
n k=0 γ−n
0,1,2, ) with respect to the coefficient-wise convergence topology,
··· P
where the kth coefficient describes an oscillation of period k. Dividing
byperiod-oneoscillation, we construct ashift actionτ ontheset Ω(P)
Ω
to itself, which shifts k-period oscillations to k 1-period oscillations.
−
In 3.1, we introduce the key concept: finite rational accumulation.
We show that if Ω(P) is a finite set, then Ω(P) is automatically a
finite rational accumulation set and the τ -action becomes invertible
Ω
and transitive. That is, τ is acting cyclically on Ω(P).
Ω
Starting with 4, we assume always finite rational accumulation for
§
Ω(P). In 4, we analyze in details of the opposite series in Ω(P) and
the modul§e CΩ(P) spanned by Ω(P), showing that the opposite series
become rationalfunctions withthe commondenominator ∆op(s)in 4.1,
and that the rank of CΩ(P) is equal to deg(∆op(s)) in 4.4.
§
In 5, we assume that the series P(t) defines a meromorphic function
§
in a neighbourhood of the closed convergent disc. Then we show that
∆op(s) is opposite to the polynomial ∆top(t) of the highest order part of
poles of P(t) (Duality Theorem in 5.3), and, in particular, the rank of
the space CΩ(P) is equal to the nu§mber of poles of the highest order of
P(t) on the boundary of the convergent disc. We get an identification
ofsome transitionmatrices obtainedins-side andint-side, which plays
a crucial role in the trace formula for limit F-function [S1, 11.5.6].
Problems. The space Ω(P) is new with respect to the study of the
singularities of a power series function P(t), and the author thinks the
following directions of further study may be rewarding.
4 KYOJISAITO
1. Generalize the space Ω(P) in order to capture lower order poles of
P(t) on the boundary of its convergent disc (c.f. [S1, 12, 2.]).
§
2. Generalize the duality for the case when Ω(P) is infinite. Some
probabilistic approach may be desirable (c.f. [S1, 12, 1.]).
§
2. The space of opposite series.
In this section, we introduce the space Ω(P) of opposite series for a
tame power series P C[[t]], and equip it with a τ -action.
Ω
∈
2.1. Tame power series.
Let us call a complex coefficient power series in t
(2.1.1) P(t) = ∞n=0γntn
to be tame, if there are positive real numbers u,v R such that
P >0
∈
(2.1.2) u γ /γ v
n 1 n
≤ | − | ≤
for sufficiently large integers n (i.e. for n N for some N Z ).
P P 0
≥ ∈ ≥
This implies that there are positive constants c ,c with c c so that
1 2 1 2
≤
(2.1.3) c v n γ c u n
1 − n 2 −
≤ | | ≤
for sufficiently large integer n Z (actually, put c = γ vNP and
c = γ uNP for n N ). Let∈us ≥co0nsider two limit1valu|eNs:P|
2 | NP| ≥ P
(2.1.4) u r :=1/lim γ 1/n R :=1/lim γ 1/n v.
P n P n
≤ n | | ≤ n | | ≤
→∞ →∞
Cauchy-Hadamard Theorem says that P is convergent of radius r .
P
Example 1. Let Γ be a group or a monoid with a finite generator
system G. Then the length l(g) of an element g Γ is the shortest
∈
length of words expressing g in the letter G. Set Γ := g Γ
n
{ ∈ |
l(g) n and γ := #(Γ ). Then the growth function (Poincar´e
n n
≤ }
series) for Γ with respect to G is defined by PΓ,G(t) := ∞n=0γntn. The
Tsehqeureenfocere{,γbny}nch∈Zo≥o0siinsginuc=re1a/siγngaannddvs=em1i,-mthueltgirpolwictahtivPseerγiems+ins≤taγmmeγ.n.
1
2. Ramsey’s theorem says that, for any n Z , there exists a
>0
∈
positive integer N such that if the edges of the complete graph on
N vertices are colored either red or blue, then there exists n vertices
such that all edges joining them have the same colour. The least such
integer N is denoted by R(n), and is called the nth diagonal Ramsey
number, e.g. R(1)=1,R(2)=2,R(3)=6,R(4)=18 (c.f. [SR]). Then,
the following estimates are known due to Erd¨os [E] and Szekeres:
2n/2 R(n) 22n.
≤ ≤
So, R(t) := ∞ R(n)tn (where put R(0)=1) form a tame series.
n=0
P
OPPOSITE POWER SERIES 5
2.2. The space Ω(P) of opposite series.
Let P be a tame power series. Then, there is a positive integer N
P
such that γ is invertible for all n N . Therefore, for n Z , we
n ≥ P ∈ ≥NP
define the opposite polynomial of degree n by
(2.2.1) X (P) := n γn k sk.
n k=0 γ−n
Regarding X (P) as a seqPuence in the space C[[s]] of formal
power series{, wnhere}nC≥[N[sP]] is equipped with the classical topology, i.e.
the product topology of coefficient-wise convergence in classical topol-
ogy, we define the space of opposite series by
the set of accumulation points of the sequence
(2.2.2) Ω(P) :=
(2.2.1) with respect to the classical topology.
That is, an element of Ω(P) can be viewed as an equivalence class of
infinite convergent subsequences X (P) of opposite polynomials.
{ nm }m
The first statement on Ω(P) is the following.
Assertion 1. Let P be a tame series. Then Ω(P) is a non-empty
compact closed subset of C[[s]].
Proof. For each k Z , the kth coefficient γn k of the polynomial
X (P)forsufficient∈ly la≥r0gen Z withrespect tγo−n P andk (i.e. forn
n 0
N +k 1)hastheapproximat∈ion≥uk γn k = γn 1 γn 2 γn k ≥
P − ≤ | γ−n | | γ−n ||γn−1|···|γn −k+1| ≤
vk, i.e. it lies in the compact annulus − −
D¯(0,uk,vk) := a C uk a vk .
{ ∈ | ≤| |≤ }
Thus, for each fixed m Z , the image of the sequence (2.2.1) under
0
tahcecutmruunlactaetsiotnomanapnoπn≤-em∈m:p≥Cty[[cso]]m→paCctms+u1b,set o∞k=f0amkskD7→¯(0(,au0k,,·v·k·),,asmay)
P k=0
Ω . Then, we have:
m
≤ Q
Ω(P) = ∩∞m=0 (π≤m)−1Ω≤m ∩ ∞k=0D¯(0,uk,vk) ,
where the RHS, as an in(cid:0)tersection of decQreasing sequence(cid:1)of compact
(cid:3)
sets, is non-empty and compact.
An element a(s) = Σ a sk of Ω(P) is called an opposite series.
∞k=0 k
Its kth coefficients a , i.e. an oscillation value of period k, belongs to
k
D¯(0,uk,vk). Given an opposite series a(s), the constant term a is
0
equal to 1. The coefficient a , i.e. oscillation value of period 1, is called
1
the initial of the opposite series a, and denoted by ι(a).
For later use, let us introduce an auxiliary space of the initials:
γ
n 1
(2.2.3) Ω1(P) := the accumulation set of the sequence − ,
γ
n n 0
n o ≫
6 KYOJISAITO
which is a compact subset in D¯(0,u,v). The projection map Ω(P)
→
Ω (P), a ι(a) is surjective but may not be injective (see 3.5 Ex.).
1
7→ §
2.3. The τ -action on Ω(P).
Ω
We introduce a continuous map τ form Ω(P) to itself.
Ω
AItwhfhestoshseseeerqstlueiimqoeunnietcne2dc.ee{paX{e.nXnLdmnes−mt1o({(nPnPlm)y)}}}ommmn∈∈∈ZZaZ≥≥≥0a00ncbaodelnsiavosesdrcugoebennssovetetqoerudgeaenbnscyeotτopopf(aoaZns)i≥.to0eTptspeheonersdniiei,tnsewgase,teohrthia∞eevsne.,
Ω
(2.3.1) τ (a) = (a 1)/ι(a)s.
Ω
−
b. Let CΩ(P) be the C-linear subspace of C[[s]] spanned by Ω(P).
Then the map τ : Ω(P) CΩ(P), a ι(a)τ (a) naturally extends
Ω
to an endomorphism of−C→Ω(P). 7→
(2.3.2) τ EndC(CΩ(P))
∈
Proof. a. By definition, for any k Z , the sequence γnm k converges
∈ ≥0 γnm−
to a constant ak ∈ D¯(uk,vk). Then, γ(nmγ−n1m)−(1k−1) = γγnnmm−k/γγnnmm−1 con-
vaenrgoepsptoositaek/saer1i.esT,hwahtoisse, t(hke s1e)qtuhencoceeffi{cXiennmt−−i1s(Peq)u}mal∈Zto≥0aco/nav.erges to
k 1
−
b. This is trivial, since a ι(a)τ (a) is a restriction on Ω(P) of an
Ω
affine linear endomorphism 7→(a 1)/s on C[[s]]. (cid:3)
−
2.4. Examples of τ -actions.
Ω
At present, except for the trivial cases when #Ω(P) = 1 so that
τ =id, there are only few examples where the action (Ω(P ),τ ) is
Ω Γ,G Ω
explicitly known: namely, the groups of the form Γ=(Z/p Z)
1
Z/p Z for some p , ,p Z (n 2) with the generator system∗ ·G··=∗
n 1 n >1
a , ,a where·a··is th∈e standa≥rd generator of Z/p Z for 1 i n,
1 n i i
{ ··· } ≤ ≤
which include Mach`ı’s example (see 3.3-4).
§
For the tame series R(t) in 2.1 Example 2, we know nothing about
§
(Ω(R),τ ). It is already a question whether #Ω(R) is equal to 1, finite
Ω
many (>1), or infinite? The author would like to expect #Ω(R)=1.
2.5. Stability of Ω(P).
In the present subsection, we are (mainly) concerned with follow-
ing type of questions, which we will call stability questions concerning
Ω(P): for a given tame series P, under which assumptions on another
power series Q, is P + Q again tame and Ω(P) =Ω(P + Q)? Or, if
Ω(P +Q) changes from Ω(P), how does it change?
We discuss some miscellaneous results related to stability questions,
but we do not pursue full generalities. Except that Assertion 3 is used
in the proof of Assertion 13, results in the present paragraph are not
OPPOSITE POWER SERIES 7
used in the present article. Therefore, the reader may choose to skip
the part of this subsection after Assertion 3 without substantial loss.
Assertion 3. Let Q= ∞n=0qntn converge in the disc of radius rQ such
that r > R . Then P +Q is tame and Ω(P) = Ω(P +Q).
Q P
P
Proof. Let c be a real number satisfying r > c > R . Then, one
Q P
has limq cn=0 and cn 1/ γ for sufficiently large n. This implies
n n
n ≥ | |
lim γ→n+∞qn=1+ lim qn =1. The required properties follow. (cid:3)
n γn n γn
→∞ →∞
Assertion 4. Let r be a positive real number with r<R . If Ω (P)
P 1
z C : z =r = . Then there exists a power series Q(t) of radius o∩f
{ ∈ | | } ∅
convergence r =r such that P+Q is tame and Ω(P+Q) Ω(P).
Q
6⊂
Proof. We define the coefficients of Q(t) = ∞n=0qntn by the following
conditions: q = r n and arg(q ) = arg(γ ). Then, for tameness of
n − n n
| | P
P +Q, we have to show some positive bounds 0<U A V for A =
n n
|γn−γn1++qqnn−1|. Since |γn+qn|=|γn|+r−n, we have An=≤|γn−11≤+/γ1n/|(+γrn/(r|nγn)|rn).
Then, evaluating term-by-term in the numerator, one gets A | |v+r=:
n
≤
V. On the other hand, according as 1 1/( γ rn) or not, we have
n
≥ | |
A u/2 or A r/2. Therefore, we may set U:=min u/2,r/2 .
n n
≥ ≥ { }
Let us find a particular element d Ω(P +Q) such that d Ω(P).
∈ 6∈
For a small positive real number ε satisfying the inequality (1 ε)/r>
−
1/R , there exists an increasing infinite sequence of integers n (m
P m
Z ) such that ((1 ε)/r)nm > γ for m Z . By choosing ∈a
≥0 − | nm| ∈ ≥0
suitable sub-sequence (denoted by the same n ), we may assume that
m
X (P +Q) converges to an element, say d, in Ω(P + Q). Its kth
nm
coefficient d is equal to the limit of the sequence (γ +q )/(γ +
k nm k nm k nm
− −
q ) for n . For each fixed n , dividing the numerator and the
nm m→∞ m
denominator by q , we get an expression (X+rkY)/(Z+1) where
nm
X = γ /γ γ rnm vk (1 ε)nm (for n >> k), Y S1,
a|nd| Z| =nm−γk rnnmm| ·<| n(m1 ε)|nm≤. Th·us,−taking the limit n ∈, we
have|X| |0,nYm |eiθk fo−r some θ R and Z 0 so thatmd→=∞rkeiθk.
k k
→ → ∈ →
On the other hand, we see that d Ω(P), since ι(d)=reiθ1 Ω (P) by
1
6∈ ∈6 (cid:3)
assumption.
WedonotusefollowingAssertioninthepresentpaper,sinceweknow
more precise information for the cases #Ω(P)< . However, it may
∞
have a significance when we study the general case with #Ω(P)= .
∞
Assertion 5. An opposite series converges with radius 1/sup a : a
{| | ∈
Ω (P) 1/R .
1 P
} ≤
Proof. Let a(s) = mlim Xnm(P) for an increasing sequence {nm}m∈Z≥0
be an opposite series→.∞By the Cauchy-Hadmard theorem, the radius of
8 KYOJISAITO
convergence of a is given by
r = 1/ lim a 1/k = 1/ lim lim γ /γ 1/k,
a k | k| k |m nm−k nm|
→∞ →∞ →∞
where the RHS is lower bounded by 1/sup a : a Ω (P) from below.
1
{| | ∈ } (cid:3)
It seems natural to ask when we can replace sup a : a Ω (P) by
1
{| | ∈ }
R ? Finally, we state a result, which is not related to the stability.
P
Assertion 6. For any positive integer m, we have the equality
(2.5.1) Ω(P) = Ω dmP
dtm
which is equivariant with the action of(cid:0)τ (cid:1)
Ω
Proof. It is sufficient to show the case m = 1. We show a slightly
sscteorrnFoivoenesrgreagarne(snsi)cnteaicftroeeafmantshedinneogts:neslqeytquhiueefne{ncsXuecebnγsmn{emqn(cid:0)umdkdeP}tntm(cid:1)co}e∈mcZ≥{∈is0XZ≥ean0qnmuda(ilPvsfoao)rl}ecmanon∈tnyZvt≥oefi0rxtghceeodesnctkvooe∈nravgZ(ees≥rs)g0.,etonthcaee
−
γnm
of the sequence (nmn−mkγ)γnnmm−k =(1−k/nm)γnγmnm−k to the same c. (cid:3)
3. Finite rational accumulation
We show that, if Ω(P) is a finite set, then it has a strong structure,
which we call the finite rational accumulation ( 3.2 Theorem and its
§
Corollary). The whole sequel of the present paper focuses on its study.
3.1. Finite rational accumulation.
We introduce the concept of finite rational accumulation. To this
end, we start with a preliminary concept: a rational subset of Z .
0
≥
The following fact is easy and well known, so we omit its proof.
Fact. The following conditions for a subset U Z are equivalent.
0
i) Put U(t) := tn. Then, U(t) is a rat⊂iona≥l function in t.
n U
ii)There exists h ∈Z and a polynomial V(t) such that U(t) = V(t).
P∈ >0 1 th
iii) There exists h Z such that n+h U iff n U for n >>−0.
>0
iv) There exists h ∈Z , a subset u Z/h∈Z and a fi∈nite set D Z
>0 0
such that U D= ∈ U[e] D, where,⊂for a class [e] Z/hZ of e,⊂pu≥t
[e] u
\ ∪ ∈ \ ∈
(3.1.1) U[e]:= n Z n e mod h .
0
{ ∈ ≥ | ≡ }
Further more, ii), iii) and iv) are equivalent for a pair (U,h). The
least such h for a fixed U will be called the period of U.
OPPOSITE POWER SERIES 9
Definition. 1. A subset U of Z is called a rational subset if it satisfies
0
≥
one of the above four equivalent conditions.
2. A finite rational partition of Z is a finite collection U of
0 a a Ω
rational subsets U Z indexed by≥a finite set Ω such that{th}ere∈is a
a 0
finite subset D of Z⊂ s≥o that one has the disjoint decomposition
0
≥
Z D = (U D).
0 a Ω a
≥ \ ∐ ∈ \
In particular, for h Z , the partition := U[e] of Z is
>0 h [e] Z/hZ 0
∈ U { } ∈ ≥
called the standard partition of period h.
3. Fora finite rational partition U of Z , the period ofa stan-
a a Ω 0
{ } ∈ ≥
dard partition, which subdivide U , is called a period of U .
a a Ω a a Ω
{ } ∈ { } ∈
The smallest period (=lcm period of U a Ω ) of a finite rational
a
{ | ∈ }
partition U is called the period of U .
a a Ω a a Ω
{ } ∈ { } ∈
We, now, arrived at the key concept of the present paper.
fiDneifitenirtaiotino.naAllyseaqcucuenmcuela{tXinng}inf∈tZh≥e0seoqfupenoicnetascicnumauHlaatuessdtoorafffisnpitaecesetis,
say Ω, such that for a system of pairwise-disjoint open neighborhoods
for a Ω, the system U for U := n Z X is a
a a a Ω a 0 n a
fiVnite rati∈onal partition of{Z }. ∈The (resp. a){per∈iod≥of|the p∈arVtit}ion is
0
≥
called the (resp. a) period of the finite rational accumulation set Ω.
3.2. τ -periodic point in Ω(P).
Ω
Generallyspeaking, finitenessoftheaccumulationsetΩofasequence
does not imply that it is finite rationally accumulating (see 3.5 Ex-
§
ample a). Therefore, the following theorem describes a distinguished
property of the accumulation set Ω(P). This justifies the introduction
of the concept of “finite rational accumulation”.
Theorem. Let P(t) be a tame power series in t. Suppose there exists
an isolated point of Ω(P), say a, which is periodic with respect to the
τ -action on Ω(P). Then Ω(P) is a finite rational accumulation set,
Ω
whose period h is equal to #Ω(P). Furthermore, we have a natural
P
bijection that identifies Ω(P) with the τ -orbit of a:
Ω
Z/h Z Ω(P)
P
(3.2.1) ≃
e mod h a[e] := limX (P),
P 7→ n e+hP·n
→∞
where the standard subdivision of the partition of Z is the exact
UhP ≥0
partition for the space Ω(P) of the opposite series of P. The shift
action [e] [e 1] in the LHS is equivariant to the τ action in the
Ω
7→ −
RHS.
Proof. The assumption on a means:
i) There exists a positive integer h Z such that
>0
∈
10 KYOJISAITO
(τΩ)ha=a=(τΩ)h′a for 0<h′<h.
ii) There exists an open neigh6 bourhood of a in C[[s]] such that
a
V
Ω(P) = a .
a
∩V { }
In particular, Ω(P) a is a closed set.
\{ }
Since Ω(P) is a compact Hausdorff space, it is a regular space, so
we may assume further that Ω(P) = a .Then, by setting U :=
a a
n Z X (P) , the seque∩ncVe X{ (}P) converges to the
{ ∈ ≥0 | n ∈ Va} { n }n∈Ua
unique limit element a. By the definition of τ in 2, the relation
Ω
§
(τ )ha = a implies that the sequence X (P) converges to a.
Ω { n−h }n∈Ua
That is, there exists a positive number N such that for any n U with
a
∈
n>N, X (P) , and hence n h belongs to U .
n h a a
Conside−r the s∈etVA:= [e] Z/hZ−there are infinitely many elements
{ ∈ |
ofU whicharecongruentto[e]moduloh . Bythedefiningpropertyof
a
}
N,if[e] A, thenU containsU[e] Z (Proof. Foranym Z with
a N N
∈ ∩ ≥ ∈ ≥
m mod h [e], there exists an integer m U such that m > m and
′ a ′
≡ ∈
m mod h = [e] by the definition of the set A. Then, by the definition
′
of N, m h U . Obviously, either m h = m or m h > m occurs.
′ a ′ ′
− ∈ − −
If m h > m then we repeat the same argument to m :=m h so
′ ′′ ′
− −
that m h = m 2h U . Repeating, similar steps, after finite
′′ ′ a
− − ∈
k-steps, we show that m kh = m U ).
′ a
− ∈
Thus, U is, up to a finite number of elements, equal to the rational
a
subset U[e]. This implies A = . Consider the rational subset
[e]A
∪ ∈ 6 ∅
U := n i n U for i = 0,1, ,h 1. Due to 2.3
(τΩ)ia { − | ∈ a} ··· − §
Assertion 2, X (P) converges to (τ )ia, so U is, up to
{ n }n∈U(τΩ)ia Ω (τΩ)ia
a finite number of elements, equal to the rational subset U[e i].
[e] A −
By the assumption a = τia for 0 i < h, any pair of ratio∪nal∈subsets
6 Ω ≤
U (0 i < h) have at most finite intersection, so A is a singleton of
(τΩ)ia ≤
the form A={[e0]} for some e0 ∈ Z and U(τΩ)ia=U[e0−i] up to a finite
number of elements. On the other hand, since the union h 1U
alreadycovers Z upto finiteelements andsinceeach X (P∪i)=−0 (τΩ)ia
≥0 { n }n∈U(τΩ)ia
converges only to (τ )ia, the opposite sequence (2.2.1) can have no
Ω
other accumulating point than the set a,τ a, ,(τ )h 1a . That
Ω Ω −
{ ··· }
is, Ω(P) is a finite rational accumulation set with the transitive h -
P
periodic action of τ . (cid:3)
Ω
Corollary. If the set of isolated points of Ω(P) is finite, then Ω(P) is
a finite rational accumulation set with the presentation (3.2.1).
Proof. Since the τ action preserves the set of isolated points of Ω(P),
Ω
(cid:3)
there should exists a periodic point.