Minimality of p-adic rational maps with good reduction

A rational map with good reduction in the field $\mathbb{Q}\_p$ of $p$-adic numbers defines a $1$-Lipschitz dynamical system on the projective line $\mathbb{P}^1(\mathbb{Q}\_p)$ over $\mathbb{Q}\_p$. The dynamical structure of such a system is completely described by a minimal decomposition. That is to say, $\mathbb{P}^1(\mathbb{Q}\_p)$ 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 prime $p$, a criterion of minimality for rational maps with good reduction is obtained. When $p=2$, 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 degree $2$, $3$ and $4$ can never be minimal on the whole space $\mathbb{P}^1(\mathbb{Q}\_2)$.


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, let Q p be the field of p-adic numbers and Z p be the ring of integers in Q p .Ergodic theory on the ring Z p is extremely important for applications to automata theory, computer science and cryptology, especially in connection with pseudorandom numbers and uniform distribution of sequences.The minimality, or equivalently the ergodicity with respect to the Haar measure, of non-expanding dynamical systems on Z p are 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 space C p of p-adic complex numbers [5,23,24,28] and in the adelic space [10].The Fatou and Julia theory of the rational maps on C p , and on the Berkovich space over C p , are also developed [7,6,20,26,27].However, the global dynamical structure of rational maps on Q p remains unclear, though the rational maps of degree one are totally characterized in [15] .
Let φ ∈ Q p (z) be a rational map of degree d ≥ 2. Then φ induces a dynamical system on the projective line P 1 (Q p ) over Q p , denoted by (P 1 (Q p ), φ).Let E ⊂ P 1 (Q p ) be a subset such that φ(E) ⊂ E. Then restricted to E, φ defines a subsystem (E, φ| E ).The subsystem (E, φ| E ) is called minimal if for any point x ∈ E, the orbit of x under φ is dense in E. In this article, we suppose that φ has good reduction (see the definition below).The minimality of (P 1 (Q p ), φ) and its subsystems will be fully investigated.As we will see, any rational map φ having good reduction is 1-Lipschitz continuous on P 1 (Q p ) which is equipped with its spherical metric.This suggests that (P 1 (Q p ), φ) shares many properties with the polynomial dynamics on Z p , which are 1-Lipschtz with respect to the metric induced by the p-adic absolute value.
Let denote the reduction modulo p from Z p to Z p /pZ p such that a → a with a ≡ a (mod p).For a polynomial f (z) = d i=0 a i z i ∈ Z p [z], the reduction of f modulo p is defined as Note that φ can be written as a quotient of polynomials f, g ∈ Z p [z] having no common factors, such that at least one coefficient of f or g has absolute value 1.Let F p = Z p /pZ p be the finite field of p elements.The degree of a rational map φ, denoted by deg φ, is the maximum of the degrees of its denominator and numerator without common factors.The reduction of φ modulo p is the rational function (of degree at most deg φ) Assume that φ has good reduction.Let ρ(•, •) be the spherical metric on P 1 (Q p ) (see the definition in Section 2).Then the map φ is 1-Lipschitz continuous (everywhere nonexpanding [27, p.59]): ρ(φ(P 1 ), φ(P 2 )) ≤ ρ(P 1 , P 2 ) for all P 1 , P 2 ∈ P 1 (Q p ).
The 1-Lipschitz continuity of φ leads us to investigate the minimality and minimal decomposition of the dynamical system (P 1 (Q p ), φ), as in the case of polynomial dynamical systems on Z p studied in [14].
Let P 1 (F p ) be the projective line over F p .The reduction φ induces a transformation from P 1 (F p ) into itself.We denote by φ k the k-th iteration of φ.A criterion of the minimality of the system (P 1 (Q p ), φ) is given in the following theorem.
Theorem 1.1.Let φ ∈ Q p (z) be a rational map of deg φ ≥ 2 with good reduction.Then the dynamical system (P 1 (Q p ), φ) is minimal if and only if the following conditions are satisfied: (1) The reduction φ is transitive on P 1 (F p ). ( Under the spherical metric ρ(•, •), P 1 (Q p ) 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 called Gauss point.Other vertices correspond to balls with radius strictly less than one (with respect to the spherical metric).The points in P 1 (Q p ) are the boundary points of the tree.A vertex is said to be at level n (n ≥ 1), if there are n edges between the vertex and the Gauss point.Then for n ≥ 1, there are (p + 1)p n−1 vertices at level n.Since a rational map with good reduction is 1-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 consider φ as action on the tree, the condition (1) in Theorem 1.1 means that φ is transitive on the set of vertices at level 1, the conditions (1) and ( 2) imply that φ is transitive on the set of vertices at level 2, while the condition (3) implies that φ 1 FIGURE 1. Tree structure of P 1 (Q 2 ).The points of P 1 (Q 2 ) are considered as the boundary points of the infinite tree. is transitive on the set of vertices at level 3 if conditions (1) and ( 2) are also satisfied.We will see in Theorem 4.1 that the transitivity of φ on these finite levels is sufficient for the minimality of φ on the whole space.The conclusion of Theorem 1.1 can be interpreted as follows.
The dynamical system (P 1 (Q p ), φ) is minimal if and only φ is transitive on the set of vertices at level 2 for p ≥ 5, and at level 3 for p = 2 or 3.
When the system (P 1 (Q p ), φ) is not minimal, we describe the dynamical structure of the system by showing all its minimal subsystems.Theorem 1.2.Let φ ∈ Q p (z) be a rational map of deg φ ≥ 2 with good reduction.We have the following decomposition where P is the finite set consisting of all periodic points of φ, M = i M i is the union of all (at most countably many) clopen invariant sets such that each M i is a finite union of balls and each subsystem φ : M i → M i is minimal, and points in B lie 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: The decomposition in Theorem 1.2 will be referred to as the minimal decomposition of the system φ : We remark that the possible lengths of periodic orbits were investigated in [29], see also [27, pp.62-64].
A finite periodic orbit of φ is by definition a minimal set.But for the convenience of the presentation of the paper, only the sets M i 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 any 1-Lipschitz continuous map, a minimal component must be a Legendre set and that any Legendre set is a minimal component of some 1-Lipschitz continuous map.Recall that E ⊂ P 1 (Q p ) is a Legendre set (see [8, p. 778]) if for every integer n ≥ 0, every non-empty intersection of E with a ball of radius p −n contains the same number of balls of radius p −(n+1) .
We will show that for a rational map with good reduction, the minimal components M i are Legendre sets of special forms.Further, the dynamics of the minimal subsystems are adding machines.Let (p s ) s≥1 be a sequence of positive integers such that p s |p s+1 for every s ≥ 1.We denote by Z (ps) the inverse limit of Z/(p s Z), called an odometer.The sequence (p s ) s≥1 is called the structure sequence of the odometer.The map z → z + 1 on Z (ps) is the adding machine on Z (ps) .The structures of minimal components are determined as follows.
Theorem 1.3.Let φ ∈ Q p (z) be a rational map of deg φ ≥ 2 with good reduction.
Assume that E ⊂ P 1 (Q p ) is a minimal component of φ.Then φ : E → E is conjugate to the adding machine on an odometer Z (ps) , where with integers k and satisfying that 1 ≤ k ≤ p + 1 and | (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 for p = 2.It seems to be a much more harder task for p ≥ 3.
The following theorem provides, in some sense, a principle for constructing minimal rational maps on P 1 (Q 2 ).For a degree d ≥ 2 rational map of form Theorem 1.4.Let φ be defined as (1.1).If φ has good reduction and is minimal on Conversely, the condition ( 1.2) implies that φ is 1-Lipschitz continuous and minimal on Corollary 1.5.Let φ ∈ Q 2 (z) be a rational map of degree 2, 3 or 4 with good reduction.Then the dynamical system (P 1 (Q 2 ), φ) is not minimal.
A complete characterization of minimal polynomial maps on Z p in terms of their coefficients are given in [22] for p = 2 and in [12] for p = 3.However, there is still no complete description for p ≥ 5. On the other hand, the characterizations of minimal Mahler series 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 in Q 2 (z) of degree 4 which is 1-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 of P 1 (Q p ), including the spherical metric and the tree structure of P 1 (Q p ). 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 case p = 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 when p = 3.

PROJECTIVE LINE
Any point in the projective line P 1 (Q p ) may be given in homogeneous coordinates by a pair [x 1 : x 2 ] of points in Q p which are not both zero.Two such pairs are equal if they differ by an overall (nonzero) factor λ ∈ Q * p : The field Q p may be identified with the subset of P 1 (Q p ) given by This subset contains all points in P 1 (Q p ) except one: the point of infinity, which may be given as . The points of P 1 (Q 3 ) are considered as the boundary points of the infinite tree.
The spherical metric defined on P 1 (Q p ) is analogous to the standard spherical metric on the Riemann sphere.
Remark that the restriction of the spherical metric on the ring Z p := {x ∈ Q p , |x| ≤ 1} of p-adic integers is same as the metric induced by the absolute value | • | p .
A rational map φ ∈ Q p (z) induces a transformation on P 1 (Q p ). Rational maps are always Lipschitz continuous on P 1 (Q p ) with respect to the spherical metric (see [27, Theorem 2.14.]).In particular, rational maps with good reduction are 1-Lipschitz continuous.
Lemma 2.1 ([27], p.59).Let φ ∈ Q p (z) be a rational map with good reduction.Then the map φ : For a ∈ P 1 (Q p ) and an integer n ≥ 1, denote by For each integer n ≥ 1, every ball of radius p −n consists of p disjoint balls of radius As mentioned in Section 1, the projective line P 1 (Q p ) over Q p could be considered as an infinite (p + 1)-Caylay tree: the branch index of each vertex is p + 1, i.e. each vertex is an endpoint of p + 1 edges.There is a centered vertex which is called Gauss point.Excluding Gauss point, each vertex corresponds to a ball in P 1 (Q p ) of radius strictly less then one (with respect to the spherical metric).The Gauss point can be considered as the ball of radius 1 centered at the origin which is actually the whole space P 1 (Q p ).The set of edges of the tree consists of the pairs of balls (B, B ) of radius ≤ 1 such that where r(B) and r(B ) are the radii of balls B and B respectively.
The points in P 1 (Q p ) are the boundary points of the tree.Remind that a vertex is said to be at level n(n ≥ 1) if there are n edges between Gauss point and the vertex.A vertex at level n corresponds a ball of radius p −n .The projective line P 1 (Q p ) consists of (p + 1)p n−1 disjoint balls of radius p −n .There are (p + 1)p n−1 vertices at level n.For example, see Figures 1 and 2 for the tree structures of P 1 (Q 2 ) and P 1 (Q 3 ).

INDUCED DYNAMICS
For each positive integer n, P 1 (Q p ) consists of (p + 1)p n−1 disjoint balls of radius p −n .Denote by B n the set of the (p + 1)p n−1 disjoint balls of radius p −n .In general, any 1-Lipschitz continuous map φ : As P 1 (Q p ) is considered as an infinite tree, for each positive integer n, there is a one-toone correspondence between B n and the vertices of the tree at level n.The 1-Lipschitz continuous map φ induces a transformation on the tree under which the set of vertices of each level is invariant.
Many properties of the dynamics φ on P 1 (Q p ) are linked to those of φ n on B n .
Proposition 3.1.Let φ : P 1 (Q p ) → P 1 (Q p ) be a 1-Lipschitz continuous map.Then the system (P 1 (Q p ), φ) is minimal if and only if the finite system (B n , φ n ) is minimal for all integers n ≥ 1.
The equivalence in Proposition 3.1 is similar to that for 1-Lipschitz continuous maps on the ring Z p or 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.Suppose that (P 1 (Q p ), φ) is not minimal.Then there exist an open set S and a point z ∈ P 1 (Q p ) such that φ k (z) / ∈ S for all integers k ≥ 1.Since the set S is open, there exists a ball B n0 (z 0 ) ⊂ S for some z 0 ∈ P 1 (Q p ) and some integer n 0 large enough.Recall that (B n , φ n ) is minimal for all integers n > 0. So there exists an integer k 0 such that φ k0 n0 (B n0 (z)) = B n0 (z 0 ).This implies ρ(φ k0 (z), z 0 ) ≤ p −n0 , which contradicts the fact that φ k0 (z) / ∈ S.
Recall that a rational map with good reduction is 1-Lipschitz continuous.For a rational map φ ∈ Q p (z) with good reduction, to study the minimality of the system (P 1 (Q p ), φ), it suffices to study the induced dynamics (B n , φ n ) for all integers n ≥ 1.Now let us first study the dynamics (B 1 , φ 1 ) at level 1.Note that each point in P 1 (F p ) corresponds to a ball in B 1 .The induced map φ 1 of φ can be considered as the reduction φ acting on P 1 (F p ).The map φ : P 1 (F p ) → P 1 (F p ) admits some periodic orbits.A periodic orbit is call a cycle of φ.The points outside any cycle will be mapped into some cycle after several iterations. Let To simplify notation, we consider a transformation f ∈ PGL 2 (Z p ) with f (x 1 ) = 0, where x 1 is a point in the ball corresponding to x 1 .Replacing x 1 and φ with 0 and f • φ • f −1 respectively, we may assume that x 1 = 0.Under this assumption, we will see in the following that φ k acting on pZ p is conjugate to a power series acting on Z p .
For two rational maps θ, ω ∈ Q p (z) with good reduction, the composition θ•ω has good reduction, and θ • ω = θ • ω (see [27, p.59, Theorem 2.18]).So φ k has good reduction and 0 is a fixed point of φ k .We write φ k in the form The fact that 0 is a fixed point of φ k implies that φ k (0) = a 0 /b 0 ≡ 0 (mod p).Since φ k has good reduction, we have a 0 ∈ pZ p and b 0 ∈ Z p \ pZ p .Otherwise, the numerator and denominator of the reduction have the common factor z, which implies deg Multiplying numerator and denominator by b −1 0 , we may thus write φ k in the form Then by the following Lemma 3.2, φ k is 1-Lipschitz from pZ p to itself and Lemma 3.2.Let φ ∈ Q p (z) be a rational map of the form Then φ is 1-Lipschitz on pZ p .Furthermore, with λ i ∈ Z p for all i ≥ 2.
Proof.The Taylor expansion of φ around z = 0, which in this case can be obtained by simple long division, gives with A(z), B(z) ∈ Z p [z] and λ = φ (0) ∈ Z p .Observe that on pZ p , we have Hence we can write φ(z) as the form (3.1).Obviously, φ induces a 1-Lipschitz map on pZ p .Now we study the dynamical system (pZ p , φ k ).For simplicity, we write ϕ instead of φ k .Note that every power series converges on pZ p .If λ 0 ∈ pZ p , then ϕ(pZ p ) ⊂ pZ p and ϕ induces a 1-Lipschitz map from pZ p to itself.
In the remainder of this section we assume ] with λ 0 ∈ pZ p .The dynamical system (pZ p , ϕ) is conjugate to a system (Z p , χ) by the transformation f (z) = z/p, i.e. where ] converges on Z p and induces a map from Z p to itself.
The dynamical system of χ on Z p and its induced dynamics on Z p /p n Z p was studied in [16].Actually, the convergent series with coefficient in the integral ring O K of a finite extension K of Q p were studied as dynamical systems on O K in [16].The system (Z p , θ) is a special case when K = Q p .
In the following we shall translate results about θ in the language of results about ϕ.For more details on this subject, the reader may consult [14,16].
For each n ≥ 1, denote by ϕ n the induced map of ϕ on pZ p /p n Z p , i.e., for all x ∈ pZ p .In the sprit of Proposition 3.1, the minimality and minimal decomposition of the system (pZ p , ϕ) can be deduced form its induced dynamics (pZ p /p n Z p , ϕ n ).So we need to study the finite systems (pZ p /p n Z p , ϕ n ).To this end, the main idea, which comes from [11,29], is to study the behaviour of systems (pZ p /p n Z p , ϕ n ) 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 in In the following we are concerned with the behavior of the finite dynamics ϕ n+1 on the set X σ and determine all cycles in X σ of ϕ n+1 , which will be called lifts of σ.Remark that the length of any lift σ of σ is a multiple of k.Let X i = x i + p n Z p be the closed disk of radius p −n corresponding to x i ∈ σ and be the clopen set corresponding to the cycle σ.
Let ψ := ϕ k be the k-th iteration of ϕ.Then, any point in σ is fixed by ψ n , the n-th induced map of ψ.For any point x ∈ X σ , we have (ψ(x) − x)/p n ∈ Z p .Let One can check that α n (x) (mod p) is always constant on X σ .We should remark that under the condition that α n ≡ 1 (mod p), β n (mod p) is also constant on X σ [14,16] .We distinguish the following four behaviours of ϕ n+1 on X σ : (a) If α n ≡ 1 (mod p) and β n ≡ 0 (mod p), then ϕ n+1 restricted to X σ preserves a single cycle of length pk.In this case we say σ grows.
(b) If α n ≡ 1 (mod p) and β n ≡ 0 (mod p), then ϕ n+1 restricted to X σ preserves p cycles of length k.In this case we say σ splits.
(c) If α n ≡ 0 (mod p), then ϕ n+1 restricted to X σ preserves one cycle of length k and the remaining points of X σ are mapped into this cycle.In this case we say σ grows tails.
(d) If α n ≡ 0, 1 (mod p), then ϕ n+1 restricted to X σ preserves one cycle of length k and p−1 cycles of length k , where is the order of α n in the multiplicative group F p \{0}.In this case we say σ partially splits.
For n ≥ 1, let σ = (x 1 , . . ., x k ) ⊂ pZ p /p n Z p be a k-cycle and let σ be a lift of σ which is a of ϕ n+1 contained in X σ .To illustrate the change of nature for a cycle to its lifts, we shall show the relationship between (α n , β n ) and (α n+1 , β n+1 ).For the calculation of (α n+1 mod p, β n+1 mod p) from (α n mod p, β n mod p), the interested reader may consult to [14, Lemma 2] and [16, p.1636].The following propositions predict the behaviours of the lifts of a cycle σ.(1) If σ grows or splits, then any lift σ grows or splits.
(3) If σ partially splits, then the lift σ of the same length as σ partially splits, and the other lifts of length k grow or split.Proposition 3.4 ( [14,16]).Let σ be a growing cycle of ϕ n and σ be the unique lift of σ.
For more details of Proposition 3.4, we refer the reader to [14, Proposition 2] for the first two assertion, and to [14, 4]).Then by Proposition 3.4, a criterion of the minimality of the system ϕ on pZ p can be obtained.
As we investigate the induced dynamical systems level by level, the possible periods of ϕ on pZ p can also be obtained, as a consequence of Proposition 3.4.
(1) If p > 3, the period of a periodic orbit must be a factor of p − 1.
(2) If p = 3, the period of a periodic orbit must be 1, 2 or 3.
(3) If p = 2, the period of a periodic orbit must be 1 or 2.
Furthermore, the dynamical structure of ϕ on pZ p is fully illustrated by the following minimal decomposition.
Suppose ϕ n = id for all n ≥ 1.We have the following decomposition where P is the finite set consisting of all periodic points of ϕ, M = i M i is the union of all (at most countably many) clopen invariant sets such that each M i is a finite union of balls and each subsystem ϕ : M i → M i is minimal, and the points in B lie in the attracting basin of P M.
The sets M i in the above decomposition are called minimal components.To completely characterize the dynamical system (pZ p , ϕ), each minimal component is described as the adding machine on an odometer by giving the structure sequence of the odometer.
is a minimal clopen invariant set of ϕ, then ϕ : E → E is conjugate to the adding machine on the odometer Z (ps) , where where ≥ 1 is an integer dividing p − 1.

PROOF OF THEOREMS 1.1-1.3
Proof of Theorem 1.1.We begin with proving the " if " part of the theorem.The reduction φ is minimal on P 1 (F p ) which implies that φ 1 is minimal on B 1 .Let ψ = φ p+1 be the p+1-th iteration of φ.Then the ball B 1 (0) = pZ p is an invariant set of ψ.By the argument of Section 3, ψ can be written as a power series with coefficients in Z p which converges on pZ p , i.e.
for all i ≥ 1 and λ 0 ∈ pZ p .Noting that φ is 1-Lipschitz, ψ is also 1-Lipschitz.The minimality of the dynamical system (pZ p , ψ) leads to ψ is isometric on pZ p , see [3,8].So φ is isometric on P 1 (Q p ) with respect to the spherical metric.Hence, ψ is minimal on each ball of radius 1/p which implies the minimality of (P 1 (Q p ), φ).It suffices to show that the dynamical system (pZ p , ψ) is minimal.
For the cases p > 3, it follows from Corollary 3.5 that the dynamical system (pZ p , ψ) is minimal.
Proof of Theorem 1.2.The space P 1 (Q p ) is a union of p + 1 balls with radius p −1 .Each ball can be identified with a point in P 1 (F p ).The reduction map φ on P 1 (F p ) admits some cycles.By iteration of φ, the points outside the cycles are attracted by the cycles.The ball corresponding to such a point will be put into the third part B in the minimal decomposition.
Let (x 1 , • • • , x k ) ⊂ P 1 (F p ) be a cycle of φ.Without loss of generality, we assume x 1 = 0. Let ψ = φ k be the k-th iteration of φ.Then the ball B 1 (0) = pZ p is an invariant set of ψ.Noting that φ is a rational map of deg φ ≥ 2, φ n is a rational map of degree (deg φ) n .So φ n = id for all n ≥ 1.By Theorem 3.7, the dynamical system (pZ p , ψ) has a minimal decomposition which then gives a minimal decomposition of the system Applying the same argument to all the cycles of φ, we obtain the minimal decomposition of the whole system (P 1 (Q p ), φ).
Moreover, by Corollary 3.6, the possible lengths of periodic orbits are obtained.
Proof of Theorem 1.3.Let k denote the length of the induced periodic orbit of φ on the minimal set E at the first level.Each point at this first level corresponds to a point in Without loss of generality, we assume that 0 ∈ E. The set B 1 (0) ∩ E is invariant by φ k .Consider the dynamical system (B 1 (0) ∩ E, φ k ).Noting φ have good reduction, by (3.1) and Theorem 3.8, we deduce that φ k : B 1 (0) ∩ E → B 1 (0) ∩ E is conjugate to the adding machine on the odometer Z (ps) , where , which implies φ : E → E is conjugate to the adding machine on the odometer Z (ps) , where

MINIMAL RATIONAL MAP FOR THE
Since φ(0) = ∞, for the convenience of calculation, we shall select a suitable coordinate.Let Then we have (5.3) Conversely, the condition (5.3) implies that φ is 1-Lipschitz continuous and (B 1 , φ 1 ) is minimal.Moreover, the Taylor expansion of φ 3 at 0 is of the form Proof.Assume that φ has good reduction, the coefficients a i and b j are in Z 2 .Otherwise, the degree of the reduction map of φ is strictly less than d, which implies that φ has bad reduction.Let the reduction of φ modulo p.The condition that φ has good reduction implies that the polynomials f , g have no common zero.As we have already indicated above, we can assume φ(0) = ∞ and φ(∞) = 1.So we have a 0 ≡ 1 (mod 2).The minimality of the system (B 1 , φ 1 ) means that φ(1) = 0.So we have f (1) = 0 which implies A φ ≡ 0 (mod 2).Since the polynomials f , g have no common zero, we have g(1) = 0 which means that B φ ≡ 1 (mod 2). Since we have A φ,1 ≡ 1 (mod 2).
By the last condition φ(1) ≡ 2 (mod 4) in (5.5), we immediately get For the proof of Theorem 1.4, we need calculate the second derivative of φ 3 at 0. For simplicity, denote A φ := i≥2 i(i − 1)a i and B φ := i≥2 i(i − 1)b i .Before the proof let us first calculate the second derivatives ψ (0), ϕ (0) and φ (1): Now we are ready to prove Theorem 1.4.
Secondly, if the first condition of (5.6) is satisfied then by (5.5), 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 condition φ 6 (0) ≡ 4 (mod 8) of (5.6) which will lead to the last condition of (1.2).
For the other part of Theorem 1.4, note that the first four conditions of (1.2) imply the 1-Lipschitz continuity of φ.Further, φ 3 can be written as a form of power series with coefficients in Z p .Thus from the proof of Theorem 1.1, we deduce that the minimality of φ is 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 of φ imply that φ is isometric on P 1 (Q 2 ).
For a rational map φ of degree 2 which has good reduction and is minimal on P 1 (Q 2 ), each point has either 0 or 2 pre-images since φ can not have critical point in P 1 (Q 2 ).This contradicts to the fact that φ is isometric.
For a rational map φ of degree 3 which is defined as (1.1), the good reduction property and minimality give the reduction φ(z) = (z−1)f (z) zg(z) , where f, g ∈ F 2 [z] are two quadratic irreducible polynomials.However, 1+z+z 2 is the unique quadratic irreducible polynomial over F 2 .Hence, f = g, which contradicts that φ has good reduction.
For a rational map φ of degree 4 which is defined as (1.1), the good reduction property and minimality give the reduction φ(z) = (z−1)f (z) zg(z) , with f, g ∈ F 2 [z] being two different cubic irreducible polynomials.It is known that 1 + z + z 3 and 1 + z 2 + z 3 are the only two cubic irreducible polynomials over F 2 .Hence, φ(z) can only have two cases: By simple calculations, in both cases, a 3 − b 3 ≡ 0 (mod 2), which contradicts the seventh equation in (1.2).

5.2.
Minimal rational maps of degree 4.This section is devoted to investigate the minimal rational maps of degree 4 which are 1-Lipschitz continuous and their induced orbits on B 3 .For a rational map Actually, the two balls B 1 (0) and B 1 (0) have no difference.To avoid confusion, the elements in B 1 (0) are denoted by z, such as 1, 2.
Then we can define a map φ from P 1 (Q 2 ) to itself as follows: Instead of investigating the dynamical system (P 1 (Q 2 ), φ), we study the dynamical system ( P 1 (Q 2 ), φ), so that we can do modulo calculation by computer.We obtain all rational maps of degree 4 with coefficients in {0, 1, 2, 3} which have good reduction and are minimal on P 1 (Q p ).We associate each of the 12 vertices at level 3 a label i, where i ∈ {0, 1, 2 • • • 7}, if the vertex corresponds to the ball B n (i), i, where i ∈ {0, 2, 4, 6}, if the vertex corresponds to the ball B n (1/i).
The rational maps and their induced orbits at level 3 are showed in the following table.
obtained by canceling common factors in the reductions f (z), g(z).If deg φ = deg φ, we say φ has good reduction.If deg φ < deg φ, we say φ has bad reduction.
φ(z) = a 0 + a 1 z + • • • + a d−1 z d−1 + a d z d b 1 z + • • • + b d−1 z d−1 + b d z d of degree d with a i , b j ∈ Q 2 and a d = b d = 1.Assume that φ 1-Lipschitz continuous and minimal on P 1 (Q 2 ).The orbit of φ 1 on B 1 is 0 → ∞ → 1 → 0.For the convenience of investigating the induced orbit on B n with n ≥ 2, we choose the local coordinate around ∞ by f (z) = 1/z.Denote by B 1 (0) the image of B 1 (∞) under f .we have the following communicating graph.
[8,,llary 1]for the third assertion.Remark that the results in Proposition 3.4 are presented only for polynomials in Z p [z].Actually, these results also hold for convergent series in Z p [[z]] (see[16, Propositions 4.7, 4.8, 4.10, 4.11]for a more general setting).It is now well known that the subsystem (E, ϕ) is minimal if and only if the induced map ϕ n : E/p n Z p → E/p n Z p is minimal for any n ≥ 1 (see[1, p. 111], [2, Theorem 1.2] and[8, Corollary Let E ⊂ pZ p be a ϕ-invariant compact open set.Let E/p n Z p := {x ∈ Z p /p n Z p : ∃ y ∈ E such that x ≡ y (mod p n )}.