Abstract

Berger asked the question "To what extent the preperiodic points of a stable p-adic power series determines a stable p-adic dynamical system ?" In this work we have applied the preperiodic points of a stable p-adic power series in order to determine the corresponding stable p-adic dynamical system.
ISSN 2070-0466, p-Adi c Numbers, Ultrametr ic Analysis and Applications, 2022, Vol. 14, No. 2, pp. 157–163. c
Pleiades Publishi ng, Ltd., 2022.
RESEARCH ARTICLES
Rigidity and Unlikely Intersection for Stable p-Adic Dynamical
Systems
Mabud Ali Sarkar1* and Absos Ali Shaikh1**
1Department of Mathematics, The University of Burdwan, Burdwan-713101, India
Received October 31, 2021; in nal form, January 22, 2022; accepted January 26, 2022
AbstractBerger asked the question To what extent the preperiodic points of a stable p-adic
power series determines a stable p-adic dynamical system ?In this work we have applied the
preperiodic points of a stable p-adic power series in order to determine the corresponding stable
p-adic dynamical system.
DOI: 10.1134/S2070046622020066
Key words: nonarchimedean dynamical system, power series, preperiodic points, formal
group, isogeny, rigidity, unlikely intersections.
1. INTRODUCTION AND MOTIVATION
Let Kbe the nite extension of the p-adic eld Qpwith ring of integers OK, and the unique maximal
ideal mK. We denote the units in OKby O
K.Let ¯
Kbe the algebraic closure of Kand ¯
mKbe the integral
closure of mKin ¯
K.LetCpbe the p-adic completion of ¯
Kand denote mCp={zCp||z|p<1}.
In [6], Berger studied to what extent the torsion points Tors(F)of a formal group Fover OK
determines the formal group. He proved that if Tors(F1)Tors(F2)is innite then F1=F2.Hefurther
asked the question, if Disastablep-adic dynamical system, then:
To what extent the preperiodic points Preper(D)determines D?
In this work, we have answered this question by proving our main Theorem 3.8 in Section 3. We
have also provided an alternate proof of it following some examples in Section 4. The proofs relies on the
following tools:
(a) The rst proof uses the correspondence between Tors(F)and Preper(D).
(b) The alternate proof uses the following two facts:
(i) Galois correspondence of a stable p-adic dynamical system D. Indeed, we proved that given
any stable p-adic dynamical system Dover OK, there exists a σGal(¯
K/K)and a stable
series w(x)Dsuch that σ(x)=w(x),xPreper(D).
(ii) Rigidity of power series on open unit disk mCp. We say that a subset ZmCpis Zariski
dense in mCpif every power series h(x)OK[[x]] that vanishes on Zis necessarily equal to
zero. A subset ZmCpis Zariski dense in mCpif and only if it is innite.
*E-mail: mabudji@gmail.com
**E-mail: aashaikh@math.buruniv.ac.in
157
158 SARKAR, SHAIKH
2. p-ADIC DYNAMICAL SYSTEM AND SOME RESULTS
In this section, we recall some preliminaries, and prove some helpful results:
Denition 2.1. [5] A (discrete) dynamical system consists of a set Γand a function γΓ.
Its dynamics is indeed the study of the behavior of the points in Γby repeatedly applying γon
the points of Γ, i.e., we study the iterates of γ.Ifweconsiderthenth iterate
γn(x)=γγ◦···γ(x)

n iterates
,
then the orbit of xΓis dened by Oγ(x)=x, γ(x)
2(x)
3(x),···.
(i) The point xis called periodic of period nif γn(x)=xfor some n1.
(ii) If γ(x)=x,thenxis a xed point.
(iii) A point xis preperiodic if some iterate γi(x)is periodic i.e., there exists m, n such that
γm(x)=γn(x).Inotherwords,xis preperiodic if its orbit Oγ(x)is nite.
Denition 2.2. [6] A stable p-adic dynamical system Dover OKis a collection of p-adic power
series in OK[[x]] without constant term such that the power series commutes with each other
under formal composition. A power series fin Dis called stable if f(0) is neither 0nor a root of
1.WesaythatDx·OK[[x]] is a stable p-adic dynamical system of nite height if the elements
of Dcommute with each other under composition, and if Dcontains a stable series fsuch that
f(0) mKand f(x)≡ 0mod mK(i.e., fis of nite height) as well as a stable series usuch that
u(0) O×
K. The collection Dcan be made as large as possible in the sense that whenever a stable
power series commutes with any member of D, it belongs to D. Such a collection Dis the main
object in p-adic dynamical systems [4].
Example 2.3. If Fis a formal group law of nite height over OK, then the endomorphism ring
EndOK(F)of Fis a stable p-adic dynamical system.
Proposition 2.4. For an invertible power series, preperiodic points are exactly the periodic points,
i.e., xed points of iterates of u.
Proof. Let u(x)x·OK[[x]] be invertible. For any preperiodic point αof u(x), there exists natural
numbers m, n with m>n such that um(α)=un(α).Sinceu(x)is invertible, u(n)(x)exists in
x·OK[[x]] and hence u(mn)(α)=α.
If uis an invertible series over OK, then the preperiodic points of uareexactlytheperiodicpointsby
Proposition 2.4. Now we remember that: the only periodic points of uare roots of upm(x)xfor some
mN. The full ring Zpacts on the invertible members of the dynamical system D. For, the series upm
converge to the identity in the appropriate topology, and thus the map ZDby n→ unis continuous
when Zhas the p-adic topology, so extends to ZpD. It follows from this that if mZand m=prn
with pn, then the xed points of umare the xed points of upr. To be more precise, we consider the
following two lemmas:
Lemma 2.5. Let ube an invertible series in OK[[x]]. Then for every natural number n0,forany
λ¯
Kwith v(λ)>0,ifλis a xed point of u,thenλis also a xed point of upn.
Proof. Note that u2(λ)=u(u(λ)) = u(λ)=λ. Thus by induction on n, the result follows.
Lemma 2.6. Let ube an invertible series in OK[[x]]. Then for every natural number n0,forany
λ¯
Kwith v(λ)>0,ifλis a xed point of u,thenλis also a xed point of upn. More generally,
for zZp,λis also a xed point of uz.
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RIGIDITY AND UNLIKELY INTERSECTION 159
Proof. We recall that the map ZOK[[x]] by n→ unis continuous when Zhas the p-adic topology
and OK[[x]] has (mK,x)-adic topology. This latter topology also has the property that if {Ui},Uare
invertible series in OK[[x]] with limit limiUi=U,then limiUi(λ)=U(λ). That is, evaluation at λis a
continuous map from OK[[x]] to {ξ¯
K:v(ξ)>0}.
Now suppose that λis a xed point of u,andzZp. There is a sequence of positive integers {zi}with
limit z,andsoλis a xed point of each uzi,sothatuz(λ)=ulimizi(λ) = limi(uzi(λ)) = limiλ=λ.
We dene the following two sets:
Preper(u)=
n{xOK|upn(x)=x}=all preperiodic points of an invertible series uD
T(f)=
n{x¯
mK|fn(x)=0}=all torsion points of a noninvertible series fD.(2.1)
We note the following interesting result, which says that Preper(D)is independent of choices of stable
series in D:
Proposition 2.7. [4] Let u, f Dbe invertible and noninvertible series, respectively. Then the set
of roots of iterates of fis equal to the set periodic points of u(x).Thatis,ifT(f)denotes the set
of roots of iterates of f,thenT(f)=Preper(u).
3. THE MAIN RESULTS
We start with a conjecture.
Conjecture 3.1. [7] If fand uare, respectively, two stable noninvertible and invertible power
series in a stable p-adic dynamical system D, then there exists a formal group Fwith coecients
in OK, two endomorphisms fFand uFof F, and a nonzero power series hsuch that fh=hfF
and uh=huF.Wecallhto be the isogeny from fFto f.
Remark 3.2. The conjecture 3.1 is proved in [7, Theorem. B] for K=Qp. This conjecture resembles
to that one given by Lubin in [4] while [1, 2] and [3] proved several results in the support of
Lubin’s conjecture, which says, if a noninvertible series commutes with an invertible series, there
is a formal group somehow in the background.
For the above formal group Fover OK, its endomorphism ring EndOK(F)is a stable p-adic dynamical
system. We denote by Tors(F)=nT(n, fF), the torsion points of F,whereT(n, fF)={α¯
mK:
fn
F(α)=0}. Then we have the following nice result:
Theorem 3.3. [6] If F1and F2are two formal groups over OKand if Tors(F1)Tors(F2)=innite,
then F1=F2.
Denition 3.4. Let f(x)and g(x)be two noninvertible stable power series over OKwithout
constant term. We call a power series h(x)OK[[x]] an OK-isogeny of f(x)into g(x)if hf=
gh.Ifu(x)be any invertible series in OK[[x]] then uhis also an OK-isogeny of f.
Next we prove the following lemma.
Lemma 3.5. Let f(x)and g(x)be two noninvertible stable power series over OKeach with nite
Weierstrass degree. Let hbe an isogeny of finto g,thenhmaps T(f)into T(g).Moreover,
h:T(f)T(g)is surjective.
p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS Vol. 14 No. 2 2022
160 SARKAR, SHAIKH
Proof. At rst we will show that h(0) = 0.Sinceg(h(0)) = h(f(0)) = h(0),h(0) is a xed point of g(x).
But g(x)being noninvertible can have 0as its only xedpointandhenceh(0) = 0.
Now let αTn(f)T(f),thengn(h(α)) = h(fn(α)) = h(0) = 0. This implies h(α)T(g).
This shows that hmaps T(f)to T(g).
On the other hand, take any βTm(g)T(g)for some natural number mNand let α¯
mKsuch
that h(α)=β. We need to show that αT(f).For,h(fm(α)) = gm(h(α)) = 0 implies fm(α)is a
root of h(x)which is also true for all nm.Sinceh(x)can have only nitely many roots in ¯
mK,we
must have
fnn(α)=fn(α)for some n, ˜nN.
This implies that fn(α)is a xed point of f˜n(x).Sincef˜n(x)is noninvertible, it has the only xed
point 0and hence fn(α)=0.ThusαTn(f)T(f).Thushis surjective.
Denition 3.6. We denote a stable p-adic dynamical system Dby the package (D,f,u;F, fF,u
F;h),
where Fis the background formal group of Dwith fF,u
Fnoninvertible and invertible endomor-
phisms respectively, while u, f are invertible and noninvertible power series in Drespectively,
along with an isogeny map h:fFfas in Conjecture 3.1
Now we will prove the uniqueness of the formal group Fin Conjecture 3.1.
Proposition 3.7. There exists a unique formal group Ffor each stable p-adic dynamical system
Din the Conjecture 3.1.
Proof. Let Dbe a stable p-adic dynamical system over OKconsisting of a noninvertible series fand an
invertible series u. By Conjecture 3.1, there exists a formal group Fover OKwith endomorphisms fF,
uFand an isogeny hfrom fFto f. We want to show that Fis unique. If possible let there exists another
formal group Gover OKwith endomorphisms fG,uGand an isogeny, say, hfrom fGto f. By Lemma 3.5,
we have the surjections h:T(fF)T(f)and h:T(fG)T(f). Therefore for every αPreper(D)
there exists some β1Tors(F)and some β2Tors (G)such that h(β1)=α=h(β2). This shows that
both Tors(F)and Tors(G)has innitely many points in common and thus by the Theorem 3.3, we get
F=G.
We will now prove the main result of the paper.
Theorem 3.8. If (D1,f
1,u
1;F1,f
F1,u
F1;h1)and (D2,f
2,u
2;F2,f
F2,u
F2;h2)are two dynamical
systems over OKsuch that Preper(D1)Preper(D2)is innite , then D1=D2.
Proof. By Lemma 3.5, the isogenies hidenessurjectivemapshi:T(fFi)T(fi),i=1,2.Thusfor
any βiT(fi)there exists an αiT(fFi)such that hi(αi)=βi. We note that Tors(F1)Tor s(F2)
will have innitely many points in common if Preper(D1)Preper(D2)=innite, because the isogenies
himaps T(fFi)into T(fi)by Lemma 3.5. But given that Preper(D1)Preper(D2)is innite, and
hence Tors(F1)Tors(F2)is innite. Therefore by Theorem 3.3, we conclude F1=F2.Hence by the
uniqueness property of Proposition 3.7, we must have D1=D2.
4. ALTERNATIVE PROOF OF THEOREM 3.8
In this section we give another proof of the main Theorem 3.8 which deserved to be included because
of its beauty. We are indebted to the ideas of [6]. At rst, we note the following beautiful result.
Theorem 4.1. [6] Given a formal group Fover OKwith torsion points Tors(F),thereisastable
endomorphism uFof Fand σGal(¯
K/K)such that
σ(z)=uF(z)for all zTors(F).
Now we prove a similar result for a stable p-adic dynamical system.
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RIGIDITY AND UNLIKELY INTERSECTION 161
Theorem 4.2. Let Dbe a stable p-adic dynamical system, then there exists a stable power series
w(x)Dand an σGal(¯
K/K)such that
σ(z)=w(z),for all zPreper(D).(4.1)
Proof. By the Conjecture 3.1, if fand uare two stable noninvertible and invertible power series in D,
then there exists a formal group Fwith coecients in OK, two endomorphisms fFand uFof F,anda
nonzero power series hsuch that fh=hfFand uh=huF,wherehis the isogeny from fFto
f.
By Lemma 3.5, hmaps T(fF)into T(f), and hence for every βTo rs(F),wegeth(β)Preper(D).
Moreover, by Lemma 3.5, we see h:T(fF)T(f)is also surjective. Thus for every αPreper(D)
there exists some βTors(F)such that h(β)=α. From the Theorem 3.3, we have
σ(z)=uF(z)for all zTor s(F).(4.2)
Now it remains to show that we can replace uFby an element wDin equation (3.1) such that
w/EndOK(F). Applying the isogeny hboth sides of equation (3.1) and using the relation uh=huF
from Conjecture 3.1, we get
σ(z)=uF(z)for all zTors(F),
h(σ(z)) = (huF)(z)for all zTors(F),
σ(h(z)) = u(h(z)for all zTors(F),(σ(h(z)) = h(σ(z)) (4.3)
σz)=uz),for all ˜z=h(z)Preper(D).(4.4)
The relation (4.4) follows from the relation (3.2) because hmaps T(fF)into T(f), by Lemma 3.5. Finally
denoting w(x):=u(x)Preper(D),wearedone.
The following example describes a situation when we get a relation like (4.1).
Example 4.3. Let f(x)x·OK[[x]] be a noninvertible and irreducible polynomial of degree 5 with
set of zeros Θ:={r1,r
2,r
3,r
4,r
5}such that the extension K(Θ) := K(r1,r
2,r
3,r
4,r
5)is Galois
with Galois group say, Gal(K(Θ)/K).AnyτGal(K(Θ)/K)permutes the elements of Θ.Dene
some w(x)x·OK[[x]] by w(x)=x+s(x)f(x)for some s(x)OK[[x]]. Weclaim there exist some
τGal(K(Θ)/K)so that τ(ri)=w(ri)for all riΘ.
Case I: Suppose w(x)xes one of ri,theng(x)xhas root ri,andsof(x)|(w(x)x).Inthis
case w(ri)=rifor every i=1,2,3,4,5, which implies w(x)induces the identity permutation on
the set Θ,thatis,forτ=Id Gal(K)/K)we have τ(z)=w(z)for all zΘ.
Case II: Suppose w(x)do not xanyofri,i=1,2,3,4,5. Since the splitting eld of f(x)is of
degree 5, either winduces a permutation (r1r2r3r4r5)or a permutation (r1r2r3)(r4r5).Ifwinduces
the permutation (r1r2r3)(r4r5),thenw2induces the permutation of type (r1r2r3)(r4)(r5).This
shows r4and r5are the xed points and the permutation is not identity. So by the argument
of Case I, this can not happen. Hence winduces the 5-cycle (r1r2r3r4r5). Th erefo re by re peated
composition of weach r1,r
2,r
3,r
4,r
5can be expressed as a polynomial in r1. In other words, the
splitting eld is K(Θ) = K[r1]of degree 5.Nowweclaimthatwinduces the same permutation
as a power of τ. Without loss of generality, choose the notation such that τ=(r1r2r3r4r5).Now
we have the following subcases:
(i)if w(r1)=r2then τ(w(r1)) = τ(r2)w(τ(r1)) = r3w(r2)=r3. Applying τon both sides of
w(r2)=r3,wegetw(r3)=r4. Once again, applying τon w(r3)=r4,wegetw(r4)=r5. So indeed
winduces τ.
(ii)if w(r1)=r3, similarly, winduces τ2.
(iii)if w(r1)=r4, similarly, winduces τ3.
(iv)if w(r1)=r5, similarly, winduces τ4.
Finally, since there is a continuous surjection Gal( ¯
K/K)Gal(K(Θ)/K),forthegivenτthere
exist a σGal(¯
K/K)such that σ|K(Θ) =τso that σ|K(Θ)(ri)=w(ri)for all riΘ.
Lemma 4.4. Let Dbe stable p-adic dynamical system over OKand I(x)x·OK[[x]].IfI(z)
Preper(D)for innitely many z,thenID.
p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS Vol. 14 No. 2 2022
162 SARKAR, SHAIKH
Proof. Since hmaps T(fF)into T(f)by Lemma 3.5, Theorem 4.2 implies there is a σGal(¯
K/K)
and a wDsuch that σ(z)=w(z)zPreper(D).IfzPreper(D),thenwehave
σ(I(z)) = w(I(z)).(4.5)
Since Preper(D)is stable under the action of Gal(¯
K/K),forallzPreper(D)
σ(I(z)) = I(σ(z)) = I(w(z))
σ(I(z)) = I(w(z)) (4.6)
From equations (4.5) and (3.3), we have w(I(z)) = I(w(z)) zPreper(D).ButsincePreper(D)is
innite, by Zariski dense property, we get wI=Iw. This shows ID.
Alternative proof of Theorem 3.8. By Theorem 4.2, there exists an element σGal(¯
K/K)and a
stable power series win D1such that
σ(z)=w(z)zPreper(D1).
The set Z:= Preper(D1)Preper(D2)is stable under the action of Gal(¯
K/K), and hence for all zZ,
we have σ(z)Z. Therefore w(z)Zbecause σ(z)=w(z)zPreper(D1).SinceZPreper(D2)
is innite, by the Lemma 4.4, we get wD2. This forces to conclude D1=D2.
We have produced the following two situations towards justication of the Theorem 3.8.
Example 4.5. We establish our argument rather contrapositively. We claim that there can
not be two dierentstable p-adic dynamical systems D1and D2over OKsatisfying the
statement of Theorem 3.8. For, if D1=D2satises Preper(D1)Preper(D2)=innite. Then there
exists two noninvertible series f1(x),f
2(x)respectively D1,D2such that f1f2=f2f1.But
since Preper(D1)Preper(D2)is innite, both f1f2vanishes on the innite set Preper(D1)
Preper(D2). Thus by Zariski dense property, we have f1=f2and hence f1f2=f2
1=f2f1,
which is a contradiction. Thus our claim is established.
Example 4.6. Consider the noninvertible series fF(x)=3x+x3over Z3,whereZ3is the ring
of integers of the 3-adic eld Q3. It is an endomorphism of a 1-dimensional Lubin-Tate formal
group Fover Z3. Our idea is to recoordinatize the endomorphism fFand to form its condensation
fFx1
p1p1.Letusdene a map h(x)=x2so that hfF=fh, and hence his an isogeny
from fFto f. Consider a stable p-adic dynamical system D1over Z3consisting of the noninvert-
ible series f(x):=fFx1
22=9x+6x2+x3and the invertible series u(x)=4x+x2.Itcanbe
checked that fu=uf. Here Θ1:= {0,+3,3}⊂T(fF)and Θ2:= {0,3,3}⊂T(f)
are sets of zeros of fFand f, respectively. We must note that according to construction (2.1),the
elements in T(fF)or T(f)might not belong to Q3but over some algebraic extension. Indeed, here
3/Q3.Weonlyhavetomakesurethattheisogenyhmaps the zeros of fFinto the zeros of f.
In fact, here the isogeny htakes Θ1to Θ2because h(0) = 0,h(3) = 3,h(3) = 3. Clearly
f(x)can not be an endomorphism of the formal group F(not even any formal group) because it
has repeated root. So the choice D1is nontrivial with background formal group Fand compatible
with respect to the statement of the Theorem 3.8, in other word, the dynamical systems in our
theorem exists.
The more dicult is to nd another stable p-adic dynamical system D2satisfying same criteria
as D1. We earnestlyhope that any example satisfying the statement to that of Theorem 3.3 would
lead us to nd D2. However, we can create an easier D2as follows.
For, let us consider another Lubin-Tate formal group Gover Z3with noninvertible en-
domorphism gGsatisfying gG(x)32x(mod degree 2) and gG(x)x32(mod 3Z3).Sucha
non-invertible endomorphism is gG(x)=9x+30x3+27x5+9x7+x9which commutes with an
invertible endomorphism uG(x)=5x+5x2+x5such that gG(G(x, y)) = G(gG(x),g
G(y)) and
uG(G(x, y)) = G(uG(x),u
G(y)).Wehaveformedthecondensationg(x)=81x+ 540x2+ 1386x3+
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RIGIDITY AND UNLIKELY INTERSECTION 163
1782x4+ 1287x5+ 546x6+ 135x7+18x8+x9of the endomorphism gG(x)by the isogeny h2(x)=
x2such that h2gG=gh2and h2maps the zeros of gGinto the zeros of g. Further, ghas double
roots 3and hence it can not be an endomorphism of any formal group. We have created an
invertible series ˜u(x)=25x+50x2+35x3+10x4+x5such that g˜uug. Thus we construct
D2as a stable p-adic dynamical system consisting of the invertible series ˜uand the noninvertible
series g, whose background formal group is G.
Finally, since Preper(D1)and Preper(D2)are independent of choices of stable series in D1
and D2respectively, we can take Preper(D1)=Preper(u)and Preper(D2)=T(g).But,uD1
commutes with gD2and hence Preper(D1)Preper(D2)is innite. On the other hand, fF
commutes with uGand hence Tors(F)Tors(G)is innite, which implies F=G. The uniqueness
property in Propsition 3.7 says D1=D2, and this is indeed true by our construction.
Remark 4.7. The existence of the stable p-adic dynamical systems D1and D2in the above
Example 4.6 supports the Conjecture 3.1.
Remark 4.8. The Theorem 3.3 deals with the category of stable p-adic dynamical systems which
are endomorphisms of formal groups while the main Theorem 3.8 of this paper deals with and
classies larger category of stable p-adic dynamical systems.
ACKNOWLEDGMENTS
The authors are indebted to the referee for several essential suggestions that helped us to improve the
paper. The authors would like to thank Jonathan Lubin, Laurent Berger for their insightful comments.
The rst author acknowledges The Council Of Scientic and Industrial Research, Government of
India, for the award of Senior Research Fellowship with File no.-09/025(0249)/2018-EMR-I.
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3. H.-C. Li, p-Typical dynamical systems and formal groups,Compos. Math. 130,7588 (2002).
4. J. Lubin, Nonarchimedean dynamical systems,Compos. Math. 94 (3), 321346 (1994).
5. J. H. Silverman, The Arithmetic of Dynamical Systems (Springer, New York, 2007).
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ResearchGate has not been able to resolve any citations for this publication.
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J. Lubin, "Nonarchimedean dynamical systems," Compos. Math. 94 (3), 321-346 (1994).