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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 ﬁnal form, January 22, 2022; accepted January 26, 2022

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.

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 ﬁnite extension of the p-adic ﬁeld 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={z∈Cp||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 inﬁnite 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 ﬁrst 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 σ∈Gal(¯

K/K)and a stable

series w(x)∈Dsuch that σ(x)=w(x),∀x∈Preper(D).

(ii) Rigidity of power series on open unit disk mCp. We say that a subset Z⊂mCpis Zariski

dense in mCpif every power series h(x)∈OK[[x]] that vanishes on Zis necessarily equal to

zero. A subset Z⊂mCpis Zariski dense in mCpif and only if it is inﬁnite.

*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:

Deﬁnition 2.1. [5] A (discrete) dynamical system consists of a set Γand a function γ:Γ→Γ.

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

γ◦n(x)=γ◦γ◦···◦γ(x)

n iterates

,

then the orbit of x∈Γis deﬁned by Oγ(x)=x, γ(x),γ

◦2(x),γ

◦3(x),···.

(i) The point xis called periodic of period nif γ◦n(x)=xfor some n≥1.

(ii) If γ(x)=x,thenxis a ﬁxed point.

(iii) A point xis preperiodic if some iterate γ◦i(x)is periodic i.e., there exists m, n such that

γ◦m(x)=γ◦n(x).Inotherwords,xis preperiodic if its orbit Oγ(x)is ﬁnite.

Deﬁnition 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.WesaythatD⊆x·OK[[x]] is a stable p-adic dynamical system of ﬁnite 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 ﬁnite 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 ﬁnite 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., ﬁxed 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 u◦m(α)=u◦n(α).Sinceu(x)is invertible, u◦(−n)(x)exists in

x·OK[[x]] and hence u◦(m−n)(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 u◦pm(x)−xfor some

m∈N. The full ring Zpacts on the invertible members of the dynamical system D. For, the series u◦pm

converge to the identity in the appropriate topology, and thus the map Z→Dby n→ u◦nis continuous

when Zhas the p-adic topology, so extends to Zp→D. It follows from this that if m∈Zand m=prn

with pn, then the ﬁxed points of u◦mare the ﬁxed points of u◦pr. 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 n≥0,forany

λ∈¯

Kwith v(λ)>0,ifλis a ﬁxed point of u,thenλis also a ﬁxed point of u◦pn.

Proof. Note that u◦2(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 n≥0,forany

λ∈¯

Kwith v(λ)>0,ifλis a ﬁxed point of u,thenλis also a ﬁxed point of u◦pn. More generally,

for z∈Zp,λis also a ﬁxed point of u◦z.

p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS Vol. 14 No. 2 2022

RIGIDITY AND UNLIKELY INTERSECTION 159

Proof. We recall that the map Z→OK[[x]] by n→ u◦nis 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 ﬁxed point of u,andz∈Zp. There is a sequence of positive integers {zi}with

limit z,andsoλis a ﬁxed point of each u◦zi,sothatu◦z(λ)=u◦limizi(λ) = limi(u◦zi(λ)) = limiλ=λ.

We deﬁne the following two sets:

Preper(u)=

n{x∈OK|u◦pn(x)=x}=all preperiodic points of an invertible series u∈D

T(f)=

n{x∈¯

mK|f◦n(x)=0}=all torsion points of a noninvertible series f∈D.(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 coeﬃcients

in OK, two endomorphisms fFand uFof F, and a nonzero power series hsuch that f◦h=h◦fF

and u◦h=h◦uF.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:

f◦n

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)=inﬁnite,

then F1=F2.

Deﬁnition 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 h◦f=

g◦h.Ifu(x)be any invertible series in OK[[x]] then u◦his 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 ﬁnite

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 ﬁrst we will show that h(0) = 0.Sinceg(h(0)) = h(f(0)) = h(0),h(0) is a ﬁxed point of g(x).

But g(x)being noninvertible can have 0as its only ﬁxedpointandhenceh(0) = 0.

Now let α∈Tn(f)⊂T(f),theng◦n(h(α)) = h(f◦n(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 β∈Tm(g)⊂T(g)for some natural number m∈Nand let α∈¯

mKsuch

that h(α)=β. We need to show that α∈T(f).For,h(f◦m(α)) = g◦m(h(α)) = 0 implies f◦m(α)is a

root of h(x)which is also true for all n≥m.Sinceh(x)can have only ﬁnitely many roots in ¯

mK,we

must have

f◦n+˜n(α)=f◦n(α)for some n, ˜n∈N.

This implies that f◦n(α)is a ﬁxed point of f◦˜n(x).Sincef◦˜n(x)is noninvertible, it has the only ﬁxed

point 0and hence f◦n(α)=0.Thusα∈Tn(f)⊂T(f).Thushis surjective.

Deﬁnition 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:fF→fas 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 α∈Preper(D)

there exists some β1∈Tors(F)and some β2∈Tors (G)such that h(β1)=α=h(β2). This shows that

both Tors(F)and Tors(G)has inﬁnitely 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 inﬁnite , then D1=D2.

Proof. By Lemma 3.5, the isogenies hideﬁnessurjectivemapshi:T(fFi)→T(fi),i=1,2.Thusfor

any βi∈T(fi)there exists an αi∈T(fFi)such that hi(αi)=βi. We note that Tors(F1)∩Tor s(F2)

will have inﬁnitely many points in common if Preper(D1)∩Preper(D2)=inﬁnite, because the isogenies

himaps T(fFi)into T(fi)by Lemma 3.5. But given that Preper(D1)∩Preper(D2)is inﬁnite, and

hence Tors(F1)∩Tors(F2)is inﬁnite. 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 ﬁrst, we note the following beautiful result.

Theorem 4.1. [6] Given a formal group Fover OKwith torsion points Tors(F),thereisastable

endomorphism uFof Fand σ∈Gal(¯

K/K)such that

σ(z)=uF(z)for all z∈Tors(F).

Now we prove a similar result for a stable p-adic dynamical system.

p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS Vol. 14 No. 2 2022

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 σ∈Gal(¯

K/K)such that

σ(z)=w(z),for all z∈Preper(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 coeﬃcients in OK, two endomorphisms fFand uFof F,anda

nonzero power series hsuch that f◦h=h◦fFand u◦h=h◦uF,wherehis the isogeny from fFto

f.

By Lemma 3.5, hmaps T(fF)into T(f), and hence for every β∈To rs(F),wegeth(β)∈Preper(D).

Moreover, by Lemma 3.5, we see h:T(fF)→T(f)is also surjective. Thus for every α∈Preper(D)

there exists some β∈Tors(F)such that h(β)=α. From the Theorem 3.3, we have

σ(z)=uF(z)for all z∈Tor s(F).(4.2)

Now it remains to show that we can replace uFby an element w∈Din equation (3.1) such that

w/∈EndOK(F). Applying the isogeny hboth sides of equation (3.1) and using the relation u◦h=h◦uF

from Conjecture 3.1, we get

σ(z)=uF(z)for all z∈Tors(F),

⇒h(σ(z)) = (h◦uF)(z)for all z∈Tors(F),

⇒σ(h(z)) = u(h(z)for all z∈Tors(F),(∵σ(h(z)) = h(σ(z)) (4.3)

⇒σ(˜z)=u(˜z),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).Anyτ∈Gal(K(Θ)/K)permutes the elements of Θ.Deﬁne

some w(x)∈x·OK[[x]] by w(x)=x+s(x)f(x)for some s(x)∈OK[[x]]. Weclaim there exist some

τ∈Gal(K(Θ)/K)so that τ(ri)=w(ri)for all ri∈Θ.

Case I: Suppose w(x)ﬁxes 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(Θ)/K)we have τ(z)=w(z)for all z∈Θ.

Case II: Suppose w(x)do not ﬁxanyofri,i=1,2,3,4,5. Since the splitting ﬁeld of f(x)is of

degree 5, either winduces a permutation (r1r2r3r4r5)or a permutation (r1r2r3)(r4r5).Ifwinduces

the permutation (r1r2r3)(r4r5),thenw◦2induces the permutation of type (r1r2r3)(r4)(r5).This

shows r4and r5are the ﬁxed 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 ﬁeld 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)) = r3⇒w(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 σ∈Gal(¯

K/K)such that σ|K(Θ) =τso that σ|K(Θ)(ri)=w(ri)for all ri∈Θ.

Lemma 4.4. Let Dbe stable p-adic dynamical system over OKand I(x)∈x·OK[[x]].IfI(z)∈

Preper(D)for inﬁnitely many z,thenI∈D.

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 σ∈Gal(¯

K/K)

and a w∈Dsuch that σ(z)=w(z)∀z∈Preper(D).Ifz∈Preper(D),thenwehave

σ(I(z)) = w(I(z)).(4.5)

Since Preper(D)is stable under the action of Gal(¯

K/K),forallz∈Preper(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)) ∀z∈Preper(D).ButsincePreper(D)is

inﬁnite, by Zariski dense property, we get w◦I=I◦w. This shows I∈D.

Alternative proof of Theorem 3.8. By Theorem 4.2, there exists an element σ∈Gal(¯

K/K)and a

stable power series win D1such that

σ(z)=w(z)∀z∈Preper(D1).

The set Z:= Preper(D1)∩Preper(D2)is stable under the action of Gal(¯

K/K), and hence for all z∈Z,

we have σ(z)∈Z. Therefore w(z)∈Zbecause σ(z)=w(z)∀z∈Preper(D1).SinceZ⊂Preper(D2)

is inﬁnite, by the Lemma 4.4, we get w∈D2. This forces to conclude D1=D2.

We have produced the following two situations towards justiﬁcation of the Theorem 3.8.

Example 4.5. We establish our argument rather contrapositively. We claim that there can

not be two “diﬀerent”stable p-adic dynamical systems D1and D2over OKsatisfying the

statement of Theorem 3.8. For, if D1=D2satisﬁes Preper(D1)∩Preper(D2)=inﬁnite. Then there

exists two noninvertible series f1(x),f

2(x)respectively D1,D2such that f1◦f2=f2◦f1.But

since Preper(D1)∩Preper(D2)is inﬁnite, both f1−f2vanishes on the inﬁnite set Preper(D1)∩

Preper(D2). Thus by Zariski dense property, we have f1=f2and hence f1◦f2=f◦2

1=f2◦f1,

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 ﬁeld 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

p−1p−1.Letusdeﬁne a map h(x)=x2so that h◦fF=f◦h, 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

22=9x+6x2+x3and the invertible series u(x)=4x+x2.Itcanbe

checked that f◦u=u◦f. 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 diﬃcult is to ﬁnd 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 ﬁnd 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+

p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS Vol. 14 No. 2 2022

RIGIDITY AND UNLIKELY INTERSECTION 163

1782x4+ 1287x5+ 546x6+ 135x7+18x8+x9of the endomorphism gG(x)by the isogeny h2(x)=

x2such that h2◦gG=g◦h2and 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◦˜u=˜u◦g. 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,u∈D1

commutes with g∈D2and hence Preper(D1)∩Preper(D2)is inﬁnite. On the other hand, fF

commutes with uGand hence Tors(F)∩Tors(G)is inﬁnite, 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

classiﬁes 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 ﬁrst author acknowledges The Council Of Scientiﬁc 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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p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS Vol. 14 No. 2 2022