HAL Id: hal-01227597
https://hal.archives-ouvertes.fr/hal-01227597v3
Submitted on 5 Dec 2016
HAL
is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire
HAL, estdestinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
MINIMALITY OF p-ADIC RATIONAL MAPS WITH GOOD REDUCTION
Ai-Hua Fan, Shilei Fan, Lingmin Liao, Yuefei Wang
To cite this version:
Ai-Hua Fan, Shilei Fan, Lingmin Liao, Yuefei Wang. MINIMALITY OF p-ADIC RATIONAL MAPS
WITH GOOD REDUCTION. Discrete and Continuous Dynamical Systems - Series A, American
Institute of Mathematical Sciences, 2017. �hal-01227597v3�
MINIMALITY OFp-ADIC RATIONAL MAPS WITH GOOD REDUCTION
AIHUA FAN, SHILEI FAN, LINGMIN LIAO, AND YUEFEI WANG
ABSTRACT. A rational map with good reduction in the fieldQpofp-adic numbers defines a1-Lipschitz dynamical system on the projective lineP1(Qp)overQp. The dynamical structure of such a system is completely described by a minimal decomposition. That is to say,P1(Qp)is decomposed into three parts: finitely many periodic orbits; finite or countably many minimal subsystems each consisting of a finite union of balls; and the attracting basins of periodic orbits and minimal subsystems. For any primep, a criterion of minimality for rational maps with good reduction is obtained. Whenp= 2, a condition in terms of the coefficients of the rational map is proved to be necessary for the map being minimal and having good reduction, and sufficient for the map being minimal and 1-Lipschitz. It is also proved that a rational map having good reduction of degree2,3and 4can never be minimal on the whole spaceP1(Q2).
1. INTRODUCTION
The present study contributes to the theory of p-adic dynamical systems which has recently been intensively and widely developed. For this development, one can consult the books [3,6,27] and their bibliographies therein.
For a prime number p, letQp be the field ofp-adic numbers andZp be the ring of integers inQp. Ergodic theory on the ringZp is extremely important for applications to automata theory, computer science and cryptology, especially in connection with pseudo- random numbers and uniform distribution of sequences. The minimality, or equivalently the ergodicity with respect to the Haar measure, of non-expanding dynamical systems on Zpare extensively studied in [2,8,9,12,13,14,17,18,19,21,22,25], and so on.
The dynamical properties of the fixed points of the rational maps have been studied in the spaceCpofp-adic complex numbers [5,23,24,28] and in the adelic space [10]. The Fatou and Julia theory of the rational maps onCp, and on the Berkovich space overCp, are also developed [7,6,20,26,27]. However, the global dynamical structure of rational maps onQpremains unclear, though the rational maps of degree one are totally characterized in [15] .
Letφ∈Qp(z)be a rational map of degreed≥2. Thenφinduces a dynamical system on the projective lineP1(Qp)overQp, denoted by(P1(Qp), φ). LetE ⊂P1(Qp)be a subset such thatφ(E)⊂ E. Then restricted toE,φdefines a subsystem(E, φ|E). The subsystem(E, φ|E)is calledminimalif for any pointx∈E, the orbit ofxunderφis dense inE. In this article, we suppose thatφhas good reduction (see the definition below). The minimality of(P1(Qp), φ)and its subsystems will be fully investigated. As we will see, any rational mapφhaving good reduction is1-Lipschitz continuous onP1(Qp)which is
2010Mathematics Subject Classification. Primary 37P05; Secondary 11S82, 37B05.
Key words and phrases. p-adic dynamical system, minimal decomposition, projective line, good reduction, rational map .
A. H. FAN was supported by NSF of China (Grant No. 11471132); S. L. FAN was supported by NSF of China (Grant No.s 11401236 and 11611530542); L. M. LIAO was supported by 12R03191A-MUTADIS; Y. F.
WANG was supported by NSF of China (Grant No. 11231009).
1
equipped with its spherical metric. This suggests that(P1(Qp), φ)shares many properties with the polynomial dynamics onZp, which are 1-Lipschtz with respect to the metric induced by thep-adic absolute value.
Let denote thereduction modulopfromZp toZp/pZp such thata 7→ awitha≡ a(modp). For a polynomialf(z) =Pd
i=0aizi ∈Zp[z], thereduction off modulopis defined as
f(z) =
d
X
i=0
aizi.
Note thatφcan be written as a quotient of polynomialsf, g∈Zp[z]having no common factors, such that at least one coefficient offorghas absolute value1. LetFp=Zp/pZp
be the finite field ofpelements. Thedegreeof a rational mapφ, denoted bydegφ, is the maximum of the degrees of its denominator and numerator without common factors. The reduction ofφmodulopis the rational function (of degree at mostdegφ)
φ(z) =f(z)
g(z) ∈Fp(z),
obtained by canceling common factors in the reductionsf(z), g(z). Ifdegφ= degφ, we sayφhasgood reduction. Ifdegφ <degφ, we sayφhasbad reduction.
Assume thatφhas good reduction. Letρ(·,·)be thespherical metriconP1(Qp)(see the definition in Section 2). Then the mapφis 1-Lipschitz continuous (everywhere non- expanding [27, p.59]):
ρ(φ(P1), φ(P2))≤ρ(P1, P2) for allP1, P2∈P1(Qp).
The 1-Lipschitz continuity ofφ leads us to investigate the minimality and minimal de- composition of the dynamical system(P1(Qp), φ), as in the case of polynomial dynamical systems onZpstudied in [14].
LetP1(Fp)be the projective line overFp. The reductionφinduces a transformation fromP1(Fp)into itself. We denote byφk thek-th iteration ofφ. A criterion of the mini- mality of the system(P1(Qp), φ)is given in the following theorem.
Theorem 1.1. Letφ∈Qp(z)be a rational map ofdegφ≥2with good reduction. Then the dynamical system(P1(Qp), φ)is minimal if and only if the following conditions are satisfied:
(1) The reductionφis transitive onP1(Fp).
(2) (φ(p+1))0(0)≡1 (mod p)and|φ(p+1)(0)|p = 1/p.
(3) For the casep= 2or3, additionally,|φ(p+1)p(0)|p= 1/p2.
Under the spherical metricρ(·,·),P1(Qp)can be considered as an infinite symmetric tree (see Figures 1 and 2). This tree is in fact an infinite(p+ 1)-Cayley tree. The centered vertex which is the root of the tree, is calledGauss point. Other vertices correspond to balls with radius strictly less than one (with respect to the spherical metric). The points inP1(Qp)are the boundary points of the tree. A vertex is said to beat leveln(n≥ 1), if there arenedges between the vertex and the Gauss point. Then forn ≥ 1, there are (p+ 1)pn−1vertices at leveln. Since a rational map with good reduction is1-Lipschitz continuous with respect to the spherical metric, it will induce an action on the tree under which the Gauss point is fixed and the sets of vertices at the same level are invariant.
Remark that if we considerφas action on the tree, the condition (1) in Theorem 1.1 means thatφis transitive on the set of vertices at level 1, the conditions (1) and (2) imply thatφis transitive on the set of vertices at level2, while the condition (3) implies thatφ
1
FIGURE1. Tree structure ofP1(Q2). The points ofP1(Q2)are consid- ered as the boundary points of the infinite tree.
is transitive on the set of vertices at level3if conditions (1) and (2) are also satisfied. We will see in Theorem 4.1 that the transitivity ofφon these finite levels is sufficient for the minimality ofφon the whole space. The conclusion of Theorem 1.1 can be interpreted as follows.
The dynamical system(P1(Qp), φ)is minimal if and onlyφis transitive on the set of vertices at level2forp≥5, and at level3forp= 2or3.
When the system(P1(Qp), φ)is not minimal, we describe the dynamical structure of the system by showing all its minimal subsystems.
Theorem 1.2. Letφ∈ Qp(z)be a rational map ofdegφ ≥2with good reduction. We have the following decomposition
P1(Qp) =PG MG
B
wherePis the finite set consisting of all periodic points ofφ,M=F
iMiis the union of all (at most countably many) clopen invariant sets such that eachMiis a finite union of balls and each subsystemφ:Mi → Miis minimal, and points inBlie in the attracting basin of a periodic orbit or of a minimal subsystem. Moreover, the length of a periodic orbit has one of the following forms:
kork`, ifp≥5, kork`orkp, ifp= 2or3, where1≤k≤p+ 1and`|(p−1).
The decomposition in Theorem 1.2 will be referred to as theminimal decompositionof the systemφ:P1(Qp)→P1(Qp).
We remark that the possible lengths of periodic orbits were investigated in [29], see also [27, pp.62-64].
A finite periodic orbit ofφ is by definition a minimal set. But for the convenience of the presentation of the paper, only the setsMi in the above decomposition are called minimal components. What kind of set can be a minimal component of a good reduction rational map? In a recent work [8], the authors showed that for any1-Lipschitz continuous map, a minimal component must be a Legendre set and that any Legendre set is a minimal
component of some1-Lipschitz continuous map. Recall thatE ⊂P1(Qp)is aLegendre set(see [8, p. 778]) if for every integern ≥0, every non-empty intersection ofEwith a ball of radiusp−ncontains the same number of balls of radiusp−(n+1).
We will show that for a rational map with good reduction, the minimal componentsMi
are Legendre sets of special forms. Further, the dynamics of the minimal subsystems are adding machines. Let(ps)s≥1be a sequence of positive integers such thatps|ps+1for every s≥1. We denote byZ(ps)the inverse limit ofZ/(psZ), calledan odometer. The sequence (ps)s≥1is called thestructure sequenceof the odometer. The mapz → z+ 1onZ(ps)
is theadding machineonZ(ps). The structures of minimal components are determined as follows.
Theorem 1.3. Let φ ∈ Qp(z) be a rational map of degφ ≥ 2 with good reduction.
Assume thatE⊂P1(Qp)is a minimal component ofφ. Thenφ:E →Eis conjugate to the adding machine on an odometerZ(ps), where
(ps) = (k, k`, k`p, k`p2,· · ·) with integerskand`satisfying that1≤k≤p+ 1and`|(p−1).
In applications, one usually would like to construct or determine a minimal rational (polynomial) map by its coefficients. However, this is far from easy, although a criterion of the minimality of rational maps with good reduction is given in Theorem 1.1. For rational maps of degree one, a complete description is given in [15]. Here we give the description of rational maps of degree at least two with good reduction, but only forp= 2. It seems to be a much more harder task forp≥3.
For a rational mapφ ∈Qp(z)of degree at least2, the number of periodic points of a fixed period must be finite. Hence, there exists az0 ∈Qpsuch thatz0, φ(z0), φ2(z0)are distinct. Consider the linear fraction
g(z) =(z−z0)(φ2(z0)−φ(z0)) (z−φ(z0))(φ2(z0)−z0).
Theng(z0) = 0, g(φ(z0)) =∞andg(φ2(z0)) = 1. Letψ=g◦φ◦g−1be the conjugation ofφbyg. Then we haveψ(0) = ∞andψ(∞) = 1and the rational functionψcan be written as
ψ(z) = a0+a1z+· · ·+ad−1zd−1+zd b1z+· · ·+bd−1zd−1+zd
whereai, bj∈Qp(0≤i < d,1≤j < d)andd≥2is the degree ofφ. Therefore, without loss of generality, we can always assume thatφ(0) =∞andφ(∞) = 1.
The following theorem provides, in some sense, a principle for constructing minimal rational maps onP1(Q2). For a degreed≥2rational map of form
φ(z) = a0+a1z+· · ·+ad−1zd−1+adzd
b1z+· · ·+bd−1zd−1+bdzd (1.1) withai, bj ∈ Q2 andad = bd = 1, we setAφ := P
i≥0ai,Bφ := P
j≥1bj,Aφ,1 :=
P
i≥0a2i+1,Aφ,2:=P
i≥0a4i+1andAφ,3:=P
i≥0a4i+3.
Theorem 1.4. Letφbe defined as (1.1). Ifφhas good reduction and is minimal onP1(Q2), then
ai, bj∈Z2, for0≤i≤d−1and1≤j≤d−1, a0≡1 (mod 2),
Bφ≡1 (mod 2), Aφ≡2 (mod 4), Aφ,1≡1 (mod 2), b1≡1 (mod 2),
ad−1−bd−1≡1 (mod 2),
a0b1(ad−1−bd−1)(Aφ,2−Aφ,3)Bφ+
2(b2−a1+ad−2−bd−2+bd−1+Aφ,3)≡1 (mod 4).
(1.2)
Conversely, the condition( 1.2)implies thatφis1-Lipschitz continuous and minimal on P1(Q2).
Corollary 1.5. Letφ∈Q2(z)be a rational map of degree2,3or4with good reduction.
Then the dynamical system(P1(Q2), φ)is not minimal.
A complete characterization of minimal polynomial maps onZpin terms of their coeffi- cients are given in [22] forp= 2and in [12] forp= 3. However, there is still no complete description forp ≥ 5. On the other hand, the characterizations of minimal Mahler se- ries and van der Put series by their coefficients are investigated in [1,3] and in [4,21]
respectively.
The above condition (1.2) could be quickly checked by computer, since there are only arithmetic operations ‘+,−,×’ in the equations of (1.2). By condition (1.2) and a quick computer computation, we obtain a series of minimal rational maps inQ2(z)of degree4 which is1-Lipschitz but do not have good reduction. See the table in Section 5.2.
The paper is organized as follows. In Section 2, we give some preliminaries ofP1(Qp), including the spherical metric and the tree structure ofP1(Qp). Section 3 is devoted to the induced dynamical systems on the vertices. The proofs of Theorems 1.1–1.3 are given in Section 4. In Section 5, we characterize the minimal rational maps with good reduction in terms of their coefficients for the casep = 2. Theorem 1.4 and Corollary 1.5 will be proved in this section. To illustrate our result, in the last section, we present two rational maps with good reduction whose exact minimal decompositions are described whenp= 3.
2. PROJECTIVE LINE
Any point in the projective lineP1(Qp)may be given in homogeneous coordinates by a pair[x1 :x2]of points inQpwhich are not both zero. Two such pairs are equal if they differ by an overall (nonzero) factorλ∈Q∗p:
[x1:x2] = [λx1:λx2].
The fieldQpmay be identified with the subset ofP1(Qp)given by [x: 1]∈P1(Qp)|x∈Qp .
This subset contains all points inP1(Qp)except one: the point of infinity, which may be given as∞= [1 : 0].
1
FIGURE2. Tree structure ofP1(Q3). The points ofP1(Q3)are consid- ered as the boundary points of the infinite tree.
The spherical metric defined onP1(Qp)is analogous to the standard spherical metric on the Riemann sphere. IfP = [x1, y1]andQ = [x2, y2]are two points inP1(Qp), we define
ρ(P, Q) = |x1y2−x2y1|p
max{|x1|p,|y1|p}max{|x2|p,|y2|p} or, viewingP1(Qp)asQp∪ {∞}, forz1, z2∈Qp∪ {∞}we define
ρ(z1, z2) = |z1−z2|p
max{|z1|p,1}max{|z2|p,1} ifz1, z2∈Qp, and
ρ(z,∞) =
1, if|z|p≤1, 1/|z|p, if|z|p>1.
Remark that the restriction of the spherical metric on the ringZp:={x∈Qp,|x| ≤1}
ofp-adic integers is same as the metric induced by the absolute value| · |p.
A rational mapφ ∈ Qp(z)induces a transformation onP1(Qp). Rational maps are always Lipschitz continuous onP1(Qp)with respect to the spherical metric (see [27, The- orem 2.14.]). In particular, rational maps with good reduction are1-Lipschitz continuous.
Lemma 2.1([27], p.59). Letφ∈Qp(z)be a rational map with good reduction. Then the mapφ:P1(Qp)7→P1(Qp)is1-Lipschitz continuous:
ρ(φ(P), φ(Q))≤ρ(P, Q), ∀P, Q∈P1(Qp).
Fora∈P1(Qp)and an integern≥1, denote by
Bn(a) :={x∈P1(Qp) :ρ(x, a)≤p−n}
theballof radiusp−ncentered ata. The projective lineP1(Qp)consists ofp+ 1disjoint balls of radiusp−1,
P1(Qp) =B1(∞)G G
0≤i≤p−1
B1(i) .
For each integer n ≥ 1, every ball of radius p−n consists of pdisjoint balls of radius p−n−1. For example,B1(∞)consists ofB2(1/p), B2(2/p),· · ·, B2((p−1)/p), B2(∞).
As mentioned in Section 1, the projective lineP1(Qp)overQpcould be considered as an infinite(p+ 1)-Caylay tree: the branch index of each vertex isp+ 1, i.e. each vertex is an endpoint ofp+ 1edges. There is a centered vertex which is called Gauss point.
Excluding Gauss point, each vertex corresponds to a ball inP1(Qp)of radius strictly less then one (with respect to the spherical metric). The Gauss point can be considered as the ball of radius1centered at the origin which is actually the whole spaceP1(Qp). The set of edges of the tree consists of the pairs of balls(B, B0)of radius≤1such that
B0⊂B, r(B) =p·r(B0) wherer(B)andr(B0)are the radii of ballsBandB0respectively.
The points in P1(Qp) are the boundary points of the tree. Remind that a vertex is said to be at leveln(n≥ 1)if there arenedges between Gauss point and the vertex. A vertex at levelncorresponds a ball of radiusp−n. The projective lineP1(Qp)consists of (p+ 1)pn−1disjoint balls of radiusp−n. There are(p+ 1)pn−1vertices at leveln. For example, see Figures 1 and 2 for the tree structures ofP1(Q2)andP1(Q3).
3. INDUCED DYNAMICS
For each positive integern, P1(Qp)consists of (p+ 1)pn−1 disjoint balls of radius p−n. Denote byBn the set of the(p+ 1)pn−1disjoint balls of radiusp−n. In general, any1-Lipschitz continuous mapφ:P1(Qp)→P1(Qp)induces a transformation onBn. Denote byφnthe induced map ofφonBn, i.e.
φn(Bn(x)) =Bn(φ(x)), ∀x∈P1(Qp).
AsP1(Qp)is considered as an infinite tree, for each positive integern, there is a one-to- one correspondence betweenBnand the vertices of the tree at leveln. The1-Lipschitz continuous mapφinduces a transformation on the tree under which the set of vertices of each level is invariant.
Many properties of the dynamicsφonP1(Qp)are linked to those ofφnonBn. Proposition 3.1. Letφ: P1(Qp)→ P1(Qp)be a1-Lipschitz continuous map. Then the system(P1(Qp), φ)is minimal if and only if the finite system(Bn, φn)is minimal for all integersn≥1.
The equivalence in Proposition 3.1 is similar to that for1-Lipschitz continuous maps on the ringZpor on a discrete valuation domain (see [1, p. 111], [2, Theorem 1.2] and [8, Corollary 4]).
Proof. The “ only if ” part of the statement is obvious, we prove only the “ if ” part. Sup- pose that (P1(Qp), φ)is not minimal. Then there exist an open set S and a pointz ∈ P1(Qp)such thatφk(z)∈/ Sfor all integersk ≥1. Since the setS is open, there exists a ballBn0(z0) ⊂ S for some z0 ∈ P1(Qp)and some integern0large enough. Recall that(Bn, φn)is minimal for all integersn > 0. So there exists an integerk0such that φkn00(Bn0(z)) = Bn0(z0). This impliesρ(φk0(z), z0)≤p−n0, which contradicts the fact
thatφk0(z)∈/S.
Recall that a rational map with good reduction is1-Lipschitz continuous. For a rational mapφ∈Qp(z)with good reduction, to study the minimality of the system(P1(Qp), φ), it suffices to study the induced dynamics(Bn, φn)for all integersn≥1.
Now let us first study the dynamics(B1, φ1)at level1. Note that each point inP1(Fp) corresponds to a ball inB1. The induced mapφ1ofφcan be considered as the reductionφ acting onP1(Fp). The mapφ:P1(Fp)→P1(Fp)admits some periodic orbits. A periodic orbit is call acycleofφ. The points outside any cycle will be mapped into some cycle after several iterations.
Let(x1,· · ·, xk)⊂P1(Fp)be a cycle ofφ. To simplify notation, we consider a trans- formationf ∈PGL2(Zp)withf(x1) = 0, wherex1is a point in the ball corresponding to x1. Replacingx1andφwith0andf◦φ◦f−1respectively, we may assume thatx1= 0.
Under this assumption, we will see in the following thatφkacting onpZpis conjugate to a power series acting onZp.
For two rational mapsθ, ω∈Qp(z)with good reduction, the compositionθ◦ωhas good reduction, andθ◦ω=θ◦ω(see [27, p.59, Theorem 2.18]). Soφkhas good reduction and 0is a fixed point ofφk. We writeφkin the form
φk(z) =a0+a1z+· · ·anzn b0+b1z+· · ·bnzn
with coefficientsa0,· · ·ad, b0,· · ·bd ∈ Zpand at least one coefficient inZp\pZp. The fact that0is a fixed point ofφk implies thatφk(0) =a0/b0 ≡0 (modp). Sinceφk has good reduction, we havea0 ∈ pZp andb0 ∈ Zp\pZp. Otherwise, the numerator and denominator of the reduction have the common factorz, which impliesdegφk <degφk. Multiplying numerator and denominator byb−10 , we may thus writeφkin the form
φk(z) =a0+a1z+· · ·anzn 1 +b1z+· · ·bnzn .
Then by the following Lemma 3.2,φkis1-Lipschitz frompZpto itself and φk(z) =a0+λz+λ2z2+λ3z3+· · · ,
withλi∈Zpfor alli≥2.
Lemma 3.2. Letφ∈Qp(z)be a rational map of the form a0+a1z+· · ·anzn
1 +b1z+· · ·bnzn , withaj, bj∈Zp. Thenφis1-Lipschitz onpZp. Furthermore,
φ(z) =a0+λz+λ2z2+λ3z3+· · · (3.1) withλi∈Zpfor alli≥2.
Proof. The Taylor expansion of φaround z = 0, which in this case can be obtained by simple long division, gives
φ(z) =a0+λz+ A(z) 1 +zB(z)z2
withA(z), B(z)∈Zp[z]andλ=φ0(0)∈Zp. Observe that onpZp, we have 1
1 +zB(z)= 1−zB(z) +z2(B(z))2−z3(B(z))3+· · · .
Hence we can writeφ(z)as the form (3.1). Obviously, φinduces a1-Lipschitz map on
pZp.
Now we study the dynamical system(pZp, φk). For simplicity, we writeϕinstead of φk. Note that every power series
ϕ(z) =
∞
X
i=0
λizi∈Zp[[z]]
converges onpZp. Ifλ0 ∈ pZp, thenϕ(pZp) ⊂ pZp andϕinduces a1-Lipschitz map frompZpto itself.
In the remainder of this section we assumeϕ(z) =P∞
i=0λizi∈Zp[[z]]withλ0∈pZp. The dynamical system(pZp, ϕ)is conjugate to a system (Zp, χ)by the transformation f(z) =z/p, i.e.
pZp ϕ //
z/p
pZp z/p
Zp χ //Zp, whereχ(z) =P∞
i=0pi−1λizi ∈ Zp[[z]]converges onZp and induces a map fromZpto itself.
The dynamical system ofχonZpand its induced dynamics onZp/pnZpwas studied in [16]. Actually, the convergent series with coefficient in the integral ringOK of a finite extensionKofQpwere studied as dynamical systems onOKin [16]. The system(Zp, θ) is a special case whenK=Qp.
In the following we shall translate results aboutθ in the language of results aboutϕ.
For more details on this subject, the reader may consult [14,16].
For eachn≥1, denote byϕnthe induced map ofϕonpZp/pnZp, i.e., ϕ(xmodpn)≡ϕ(x) (modpn),
for allx∈pZp. In the sprit of Proposition 3.1, the minimality and minimal decomposition of the system(pZp, ϕ)can be deduced form its induced dynamics(pZp/pnZp, ϕn). So we need to study the finite systems(pZp/pnZp, ϕn). To this end, the main idea, which comes from [11, 29], is to study the behaviour of systems (pZp/pnZp, ϕn)by induction. We refer the reader to [14,16] for the application of this idea to give a minimal decomposition theorem for any convergent power series inZp[[z]].
A collectionσ= (x1,· · ·, xk)ofkdistinct points inpZp/pnZpis called acycleofϕn
of lengthkor ak-cycleat leveln, if
ϕn(x1) =x2,· · ·, ϕn(xi) =xi+1,· · ·, ϕn(xk) =x1. Set
Xσ :=
k
G
i=1
Xi where Xi:={xi+pnt+pn+1Zp; t= 0,1,· · · , p−1} ⊂pZp/pn+1Zp. Then
ϕn+1(Xi)⊂Xi+1(1≤i≤k−1) and ϕn+1(Xk)⊂X1.
In the following we are concerned with the behavior of the finite dynamicsϕn+1on the setXσand determine all cycles inXσ ofϕn+1, which will be calledliftsof σ. Remark that the length of any liftσ˜ofσis a multiple ofk.
LetXi =xi+pnZpbe the closed disk of radiusp−ncorresponding toxi∈σand Xσ:=
k
G
i=1
Xi
be the clopen set corresponding to the cycleσ.
Letψ := ϕk be thek-th iteration ofϕ. Then, any point inσis fixed byψn, then-th induced map ofψ. For any pointx∈Xσ, we have(ψ(x)−x)/pn∈Zp. Let
αn(x) :=ψ0(x) =
k−1
Y
j=0
ϕ0(ϕj(x)) (3.2)
βn(x) := ψ(x)−x
pn = ϕk(x)−x
pn . (3.3)
One can check that αn(x) (modp)is always constant on Xσ. We should remark that under the condition thatαn≡1 (modp),βn(modp)is also constant onXσ[14,16] . We distinguish the following four behaviours ofϕn+1onXσ:
(a) Ifαn ≡1 (modp)andβn 6≡ 0 (modp), thenϕn+1restricted toXσ preserves a single cycle of lengthpk. In this case we sayσgrows.
(b) Ifαn ≡ 1 (modp)andβn ≡0 (modp), thenϕn+1 restricted toXσ preservesp cycles of lengthk. In this case we sayσsplits.
(c) Ifαn≡0 (modp), thenϕn+1restricted toXσpreserves one cycle of lengthkand the remaining points ofXσare mapped into this cycle. In this case we sayσgrows tails.
(d) Ifαn 6≡0,1 (modp), thenϕn+1restricted toXσ preserves one cycle of lengthk andp−1` cycles of lengthk`, where`is the order ofαnin the multiplicative groupFp\{0}.
In this case we sayσpartially splits.
Forn ≥ 1, letσ = (x1, . . . , xk) ⊂ pZp/pnZp be ak-cycle and letσ˜ be a lift ofσ which is a ofϕn+1 contained inXσ. To illustrate the change of nature for a cycle to its lifts, we shall show the relationship between(αn, βn)and(αn+1, βn+1). For the calcu- lation of(αn+1modp, βn+1modp)from (αnmodp, βnmodp), the interested reader may consult to [14, Lemma 2] and [16, p.1636]. The following propositions predict the behaviours of the lifts of a cycleσ.
Proposition 3.3([14] Proposition 1, and [16] Proposition 4.4). Letσbe a cycle ofϕn. (1)Ifσgrows or splits, then any lift˜σgrows or splits.
(2)Ifσgrows tails, then the single liftσ˜also grows tails.
(3)Ifσpartially splits, then the lift˜σof the same length asσpartially splits, and the other lifts of lengthk`grow or split.
Proposition 3.4([14,16]). Letσbe a growing cycle ofϕnandσ˜be the unique lift ofσ.
(1)Ifp >3andn≥1, thenσ˜also grows.
(2)Ifp= 3andn≥2, thenσ˜also grows.
(3)Ifp= 2andσ˜grows, then the lift ofσ˜grows.
For more details of Proposition 3.4, we refer the reader to [14, Proposition 2] for the first two assertion, and to [14, Corollary 1] for the third assertion. Remark that the results in Proposition 3.4 are presented only for polynomials inZp[z]. Actually, these results also hold for convergent series inZp[[z]](see [16, Propositions 4.7, 4.8, 4.10, 4.11] for a more general setting).
LetE⊂pZpbe aϕ-invariant compact open set. Let
E/pnZp:={x∈Zp/pnZp:∃y∈Esuch thatx≡y(mod pn)}.
It is now well known that the subsystem(E, ϕ)is minimal if and only if the induced map ϕn:E/pnZp →E/pnZpis minimal for anyn≥1(see [1, p. 111], [2, Theorem 1.2] and [8, Corollary 4]). Then by Proposition 3.4, a criterion of the minimality of the systemϕ onpZpcan be obtained.
Corollary 3.5. The dynamical system(pZp, ϕ)is minimal if and only if (1)the finite system(pZp/p3Zp, ϕ3)is minimal forp= 2or3;
(2)the finite system(pZp/p2Zp, ϕ2)is minimal forp≥5.
As we investigate the induced dynamical systems level by level, the possible periods of ϕonpZpcan also be obtained, as a consequence of Proposition 3.4.
Corollary 3.6. Letϕ(z) =P∞
i=0λizi∈Zp[[z]]withλ0∈pZpand consider the dynami- cal system(pZp, ϕ).
(1)Ifp >3, the period of a periodic orbit must be a factor ofp−1.
(2)Ifp= 3, the period of a periodic orbit must be1,2or3.
(3)Ifp= 2, the period of a periodic orbit must be1or2.
Furthermore, the dynamical structure ofϕonpZp is fully illustrated by the following minimal decomposition.
Theorem 3.7([16], Theorem 1.1). Let ϕ(z) = P∞
i=0λizi ∈ Zp[[z]]with λ0 ∈ pZp. Supposeϕn 6=id for alln≥1. We have the following decomposition
pZp=PG MG
B,
whereP is the finite set consisting of all periodic points ofϕ,M =F
iMiis the union of all (at most countably many) clopen invariant sets such that eachMiis a finite union of balls and each subsystem ϕ : Mi → Mi is minimal, and the points inB lie in the attracting basin ofPF
M.
The setsMiin the above decomposition are calledminimal components. To completely characterize the dynamical system(pZp, ϕ), each minimal component is described as the adding machine on an odometer by giving the structure sequence of the odometer.
Theorem 3.8([16], Theorem 1.1). Letϕ(z) = P∞
i=0λizi ∈Zp[[z]]withλ0 ∈pZp. If E ⊂ pZp is a minimal clopen invariant set ofϕ, thenϕ : E → E is conjugate to the adding machine on the odometerZ(ps), where
(ps) = (`, `p, `p2,· · ·) where`≥1is an integer dividingp−1.
4. PROOF OFTHEOREMS1.1–1.3
Proof of Theorem 1.1. We begin with proving the “ if ” part of the theorem. The reduction φis minimal onP1(Fp)which implies thatφ1 is minimal onB1. Letψ =φp+1 be the p+1-th iteration ofφ. Then the ballB1(0) =pZpis an invariant set ofψ. By the argument of Section 3,ψcan be written as a power series with coefficients inZpwhich converges onpZp, i.e.
ψ(z) =λ0+λ1z+λ2z2+λ3z3+· · ·
withλi ∈ Zp for alli ≥ 1 andλ0 ∈ pZp. Noting thatφis 1-Lipschitz, ψ is also 1- Lipschitz. The minimality of the dynamical system(pZp, ψ)leads toψis isometric on pZp, see [3,8]. Soφis isometric onP1(Qp)with respect to the spherical metric. Hence,
ψis minimal on each ball of radius1/pwhich implies the minimality of(P1(Qp), φ). It suffices to show that the dynamical system(pZp, ψ)is minimal.
Forn = 1, the cycle (0)is the unique cycle of ψ1. The condition (φ(p+1))0(0) ≡ 1 (mod p)implies that the cycle(0) grows or splits. By the condition|φ(p+1)(0)|p = 1/p, the cycle(0)grows which then means that the dynamical system(pZp/p2Zp, ψ2)is minimal.
For the casesp >3, it follows from Corollary 3.5 that the dynamical system(pZp, ψ) is minimal.
Note that(φp(p+1))0(0)≡ (φ(p+1))0(0) ≡1 (mod p). For the casesp = 2or3, the additional condition implies that the unique lift of(0)at level2grows. Thus the system (pZp/p3Zp, ψn)is minimal. By Corollary 3.5, the dynamical system(pZp, ψ)is minimal.
Now we prove the “ only if ” part. Letφbe a minimal rational map with good reduc- tion. By Proposition 3.1, the system (Bn, φn)is minimal for all integers n ≥ 1. The minimality of system(B1, φ1)implies that the reductionφis minimal onP1(Fp), while the minimality of system(B2, φ2)implies that the cycle(0)ofψ1 =φp+11 grows. Hence we obtain(φ(p+1))0(0)≡1 (mod p)and|φ(p+1)(0)−0|p = 1/p. Furthermore, by the minimality of system(B3, φ3), the unique lift of(0)ofφ(p+1)at level2grows. So we
have|φp(p+1)(0)|p= 1/p2.
In the above proof of Theorem 1.1, we have indeed established the following result.
Theorem 4.1. Letφ : P1(Qp) 7→ P1(Qp) be a rational map ofdegφ ≥ 2 with good reduction. Then the dynamical system(P1(Qp), φ)is minimal if and only if the following condition is satisfied:
(1) the system(B3, φ3)is minimal forp= 2or3, (2) the system(B2, φ2)is minimal forp≥5.
Proof of Theorem 1.2. The spaceP1(Qp)is a union ofp+ 1balls with radiusp−1. Each ball can be identified with a point inP1(Fp). The reduction mapφon P1(Fp)admits some cycles. By iteration ofφ, the points outside the cycles are attracted by the cycles.
The ball corresponding to such a point will be put into the third partB in the minimal decomposition.
Let(x1,· · ·, xk) ⊂ P1(Fp)be a cycle ofφ. Without loss of generality, we assume x1= 0. Letψ=φkbe thek-th iteration ofφ. Then the ballB1(0) =pZpis an invariant set ofψ. Noting that φis a rational map ofdegφ ≥ 2,φn is a rational map of degree (degφ)n. Soφn 6=idfor alln ≥1. By Theorem 3.7, the dynamical system(pZp, ψ) has a minimal decomposition which then gives a minimal decomposition of the system (Fk
i=1B1(xi), φ).
Applying the same argument to all the cycles ofφ, we obtain the minimal decomposition of the whole system(P1(Qp), φ).
Moreover, by Corollary 3.6, the possible lengths of periodic orbits are obtained.
Proof of Theorem 1.3. Letkdenote the length of the induced periodic orbit ofφon the minimal setE at the first level. Each point at this first level corresponds to a point in P1(Fp). Sok≤p+ 1.
Without loss of generality, we assume that0 ∈ E. The setB1(0)∩E is invariant by φk. Consider the dynamical system(B1(0)∩E, φk). Notingφhave good reduction, by (3.1) and Theorem 3.8, we deduce thatφk :B1(0)∩E →B1(0)∩Eis conjugate to the
adding machine on the odometerZ(ps), where
(ps) = (`, `p, `p2,· · ·)
with` | (p−1), which impliesφ : E → E is conjugate to the adding machine on the odometerZ(ps), where
(ps) = (k, k`, k`p, k`p2,· · ·).
5. MINIMAL RATIONAL MAP FOR THE CASEp= 2 5.1. Minimal Conditions. Let
φ(z) = a0+a1z+· · ·+ad−1zd−1+adzd b1z+· · ·+bd−1zd−1+bdzd
be a rational map of degreed ≥ 2 with ai, bj ∈ Q2 and ad = bd = 1. Recall that Aφ =P
i≥0ai,Bφ =P
j≥1bj,Aφ,1 = P
i≥0a2i+1,Aφ,2 =P
i≥0a4i+1andAφ,3 = P
i≥0a4i+3.
Sinceφ(0) =∞, for the convenience of calculation, we shall select a suitable coordi- nate. Let
ψ(z) := 1
φ(z) = b1z+· · ·+bd−1zd−1+bdzd
a0+a1z+· · ·+ad−1zd−1+adzd (5.1) and
ϕ(z) :=φ(1
z) = ad+ad−1z+· · ·+a1zd−1+a0zd
bd+bd−1z+· · ·+b1zd−1 . (5.2) Then we haveφ3=φ◦ϕ◦ψ.
Lemma 5.1. Ifφhas good reduction and(B1, φ1)is minimal, then
ai, bj∈Z2, for0≤i≤d−1and1≤j≤d−1, a0≡1 (mod 2),
Aφ≡0 (mod 2), Bφ≡1 (mod 2).
(5.3)
Conversely, the condition(5.3)implies thatφis1-Lipschitz continuous and(B1, φ1)is minimal. Moreover, the Taylor expansion ofφ3at0is of the form
φ3(z) =φ(1) +λz+λ2z2+λ3z3+· · ·, whereλ, λ2, λ3· · · ∈Z2.
Proof. Assume thatφhas good reduction, the coefficientsaiandbjare inZ2. Otherwise, the degree of the reduction map ofφis strictly less thand, which implies thatφhas bad reduction.
Let
φ(z) =f(z)
g(z) = a0+a1z+· · ·+ad−1zd−1+zd b1z+· · ·+bd−1zd−1+zd
be the reduction ofφ modulop. The condition that φhas good reduction implies that the polynomialsf , ghave no common zero. As we have already indicated above, we can assumeφ(0) = ∞andφ(∞) = 1. So we have a0 ≡ 1 (mod 2). The minimality of the system(B1, φ1)means that φ(1) = 0. So we havef(1) = 0which impliesAφ ≡ 0 (mod 2). Since the polynomialsf , ghave no common zero, we haveg(1) 6= 0which means thatBφ≡1 (mod 2).
Now we assume that the coefficients ofφsatisfy the condition (5.3). By (5.3), we can check directly that(B1, φ1)is minimal.
By Lemma 3.2, the conditionsAφ ≡ 0 (mod 2), andBφ ≡ 1 (mod 2)imply that φ(z+ 1)is1-Lipschitz frompZp topZp and can be written as the form of (3.1). Recall thatφ3 = φ◦ϕ◦ψand note that ψ(0) = 0andϕ(0) = 1. It suffices to prove thatψ andϕ−1are both1-Lipschitz frompZpto itself and have the form (3.1). These can be obtained by direct calculations and by applying Lemma 3.2.
In the following lemma, we will calculate the derivative ofφ3at0. Using the chain rule, we obtain
(φ3)0(0) =ψ0(0)·ϕ0(0)·φ0(1). (5.4) For simplicity, we denoteA0φ := P
i≥1iai andB0φ := P
i≥1ibi. By calculation, we have
η1:=ψ0(0) =b1/a0, η2:=ϕ0(0) =ad−1−bd−1, and
η:=φ0(1) =A0φBφ−Bφ0Aφ
B2φ .
Lemma 5.2. Assume that the rational mapφhas good reduction. Then the system(B2, φ2) is minimal if and only if
ai, bj ∈Z2, for0≤i≤d−1and1≤j ≤d−1, a0≡1 (mod 2),
Bφ≡1 (mod 2), Aφ≡2 (mod 4) Aφ,1≡1 (mod 2), b1≡1 (mod 2),
ad−1−bd−1≡1 (mod 2).
Proof. The rational mapφhas good reduction and(B2, φ2)is minimal implies that(B1, φ1) is also minimal. SopZp isφ3-invariant. By the classification of the lifts of the cycles, (B2, φ2)is minimal if and only if
(B1, φ1)is minimal, (φ3)0(0)≡1 (mod 2), andφ3(0)φ(1)≡2 (mod 4) (5.5) under the condition thatφhas good reduction.
By Lemma 5.1,(B1, φ1)is minimal if and only ifa0 ≡1 (mod 2),Bφ ≡1 (mod 2) andAφ≡0 (mod 2).
Recall that
(φ3)0(0) =ψ0(0)·ϕ0(0)·φ0(1) = b1
a0(ad−1−bd−1)A0φBφ−Bφ0Aφ
Bφ2 . The second condition(φ3)0(0)≡1 (mod 2)of (5.5) implies that
b1≡1 (mod 2), (ad−1−bd−1)≡1 (mod 2), and
A0φ≡1 (mod 2).
Since
A0φ=X
i≥0
(2i+ 1)a2i+1+X
i≥1
2ia2i≡X
i≥0
a2i+1(mod 2), we haveAφ,1≡1 (mod 2).
By the last conditionφ(1)≡2 (mod 4)in (5.5), we immediately get Aφ≡2 (mod 4).
For the proof of Theorem 1.4, we need calculate the second derivative ofφ3at0. For simplicity, denoteA00φ :=P
i≥2i(i−1)aiandB00φ :=P
i≥2i(i−1)bi. Before the proof let us first calculate the second derivativesψ00(0),ϕ00(0)andφ00(1):
ξ1:=ψ00(0) =2b2a20−2a1b1a0
a30 ,
ξ2:=ϕ00(0) = 2(ad−2−bd−2) + 2(b2d−1−ad−1bd−1), ξ:=φ00(1) = A00φBφ2−Bφ00AφBφ+ 2(Aφ(B0φ)2−A0φBφ0Bφ)
Bφ3 .
Now we are ready to prove Theorem 1.4.
Proof of Theorem 1.4. Under the condition thatφhas good reduction, Theorem 4.1 implies that(P1(Qp), φ)is minimal if and only if(B3, φ3)is minimal which is equivalent to the following three conditions
(B2, φ2)is minimal, (φ6)0(0)≡1 (mod 2), and φ6(0)≡4 (mod 8). (5.6) In the following, we will characterize the three conditions of (5.6).
Firstly, the minimality of the system(B2, φ2)has already been characterized by coeffi- cients in Lemma 5.2. Then we obtain all the conditions in (1.2) except the last one.
Secondly, if the first condition of (5.6) is satisfied then by (5.5), (φ6)0(0)≡((φ3)0(0))2≡1 (mod 2).
Thus the second condition of (5.6) is in fact included in the first condition of (5.6).
Finally, we are going to investigate the relation among the coefficients by using the last conditionφ6(0)≡4 (mod 8)of (5.6) which will lead to the last condition of (1.2).
Note that all the Taylor’s coefficients ofφ3expanded atz= 0belong toZ2. Hence for z∈2Z2, we have
φ3(z)≡φ(1) + (φ3)0(0)z+(φ3)00(0)
2 z2(mod 8).
Thus
φ6(0)≡φ(1) + (φ3)0(0)φ(1) +(φ3)00(0)
2 φ(1)2(mod 8). (5.7) Notice thatφ(1) ≡ 2 (mod 4)and(φ3)0(0) ≡ 1 (mod 2). Then dividing both sides of (5.7) byφ(1), we deduce that the conditionφ6(0)≡4 (mod 8)is equivalent to
1 + (φ3)0(0) +(φ3)00(0)
2 φ(1)≡2 (mod 4), which in turn, by noting that(φ3)00(0)/2∈Z2, is equivalent to
(φ3)0(0) + (φ3)00(0)≡1 (mod 4). (5.8)
To continue, let us calculate(φ3)0(0) (mod 4),(φ3)00(0) (mod 4) and simplify their expressions. Usinga0 ≡1 (mod 2)andBφ ≡1 (mod 2), we have1/a0≡a0(mod 4) andB2φ≡1 (mod 4). Then
(φ3)0(0)≡a0b1(ad−1−bd−1)(A0φBφ−AφBφ0) (mod 4).
SinceAφ≡2 (mod 4)andα1≡α2≡1 (mod 2), we have
(φ3)0(0)≡a0b1(bd−1−bd−1)A0φBφ−AφB0φ(mod 4).
For the second derivative ofφ3at0, by the chain rule, we have
(φ3)00(0) =ηη2ξ1+ηη21ξ2+ξη12η22. (5.9) Sinceη≡η1≡η2≡1 (mod 2), we haveη2≡η21≡η22≡1 (mod 4). Consequently, we obtain
(φ3)00(0)≡ηη2ξ1+ηξ2+ξ(mod 4). (5.10) Apply the expressions ofξ, ξ1andξ2to Equation (5.10). Using the facts thata0 ≡b1 ≡ ad−1−bd−1≡1 (mod 2)andBφ≡1 (mod 2), we deduce
(φ3)00(0)≡2ηη2(a0b2−a1b1) + 2η(ad−2−bd−2+b2d−1−ad−1bd−1)+
+A00φBφ−2A0φB0φ−AφBφ00(mod 4)
≡2(b2−a1+ad−2−bd−2+bd−1) +A00φBφ−2A0φBφ0 −AφB00φ(mod 4).
Observing thatBφ00≡0 (mod 2)andAφ≡0 (mod 2), we have (φ3)0(0) + (φ3)00(0)
≡a0b1(ad−1−bd−1)A0φBφ−AφB0φ+ 2(b2−a1+ad−2−bd−2+bd−1) +A00φBφ−2A0φBφ0 (mod 4).
SinceAφ≡2 (mod 4)andA0φ≡Aφ,1≡1 (mod 2), we get (φ3)0(0) + (φ3)00(0)
≡a0b1(ad−1−bd−1)A0φBφ+ 2(b2−a1+ad−2−bd−2+bd−1) +A00φBφ(mod 4).
Note that
A0φ≡X
i≥0
a4i+1+ 2X
i≥0
a4i+2−X
i≥0
a4i+3(mod 4) and
A00φ≡2X
i≥0
a4i+2+ 2X
i≥0
a4i+3(mod 4).
Then
(φ3)0(0) + (φ3)00(0)
≡a0b1(ad−1−bd−1)(Aφ,2−Aφ,3)Bφ+ 2(b2−a1+ad−2−bd−2+bd−1) + 2X
i≥0
a4i+2+ 2(X
i≥0
a4i+2+Aφ,3) (mod 4)
≡a0b1(ad−1−bd−1)(Aφ,2−Aφ,3)Bφ
+ 2(b2−a1+ad−2−bd−2+bd−1+Aφ,3) (mod 4).
By (5.8), we obtain the last condition of (1.2). Hence, we obtain the first part of Theorem 1.4.
For the other part of Theorem 1.4, note that the first four conditions of (1.2) imply the 1-Lipschitz continuity ofφ. Further, φ3 can be written as a form of power series with coefficients inZp. Thus from the proof of Theorem 1.1, we deduce that the minimality of φis equivalent to the conditions (5.6). Hence the above proof of the first part leads to the
conclusion of the second part of the theorem.
Proof of Corollary 1.5. Remark that the good reduction property and minimality ofφim- ply thatφis isometric onP1(Q2).
For a rational mapφof degree 2 which has good reduction and is minimal onP1(Q2), each point has either0or2pre-images sinceφcan not have critical point inP1(Q2). This contradicts to the fact thatφis isometric.
For a rational mapφof degree3which is defined as (1.1), the good reduction property and minimality give the reductionφ(z) = (z−1)f(z)zg(z) , wheref, g∈F2[z]are two quadratic irreducible polynomials. However,1+z+z2is the unique quadratic irreducible polynomial overF2. Hence,f =g, which contradicts thatφhas good reduction.
For a rational mapφof degree4which is defined as (1.1), the good reduction property and minimality give the reductionφ(z) =(z−1)f(z)zg(z) , withf, g∈F2[z]being two different cubic irreducible polynomials. It is known that1 +z+z3and1 +z2+z3are the only two cubic irreducible polynomials overF2. Hence,φ(z)can only have two cases:
φ(z) =(z−1)(1 +z+z3)
z(1 +z2+z3) or φ(z) = (z−1)(1 +z2+z3) z(1 +z+z3) .
By simple calculations, in both cases,a3−b3≡0 (mod 2), which contradicts the seventh
equation in (1.2).
5.2. Minimal rational maps of degree4. This section is devoted to investigate the min- imal rational maps of degree4which are1-Lipschitz continuous and their induced orbits onB3.
For a rational map
φ(z) = a0+a1z+· · ·+ad−1zd−1+adzd b1z+· · ·+bd−1zd−1+bdzd
of degreedwithai, bj∈Q2andad=bd= 1. Assume thatφ1-Lipschitz continuous and minimal onP1(Q2). The orbit ofφ1onB1is0→ ∞ →1→0. For the convenience of investigating the induced orbit onBnwithn≥2, we choose the local coordinate around
∞byf(z) = 1/z. Denote byBe1(0)the image ofB1(∞)underf. we have the following communicating graph.
@
@
@
@
@
@ R
Be1(0) B1(0)
ψ
φ PP
PPPq φ
B1(∞)
?
φ 1 f
B1(1)
ϕ
Actually, the two balls B1(0) and Be1(0) have no difference. To avoid confusion, the elements inBe1(0)are denoted byez, such ase1,e2.
