佈於非阿基米德體的動態系統的朱利葉集和週期點

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(1)國立臺灣師範大學數學系碩士班碩士論文. 指導教授：. 紀文鎮. 博士. The Julia Sets and The Periodic Points in Dynamical Systems over Nonarchimedean Fields. 研究生：王怡文. 中華民國一百年一月.

(2) 中文摘要: 在本碩士論文中，我們會先介紹一些 nonarchimedean metrics 和一些和其相關的性質，接著在 dynamical system 中我們去計算一些函數的 periodic points 和一些函數的 Julia sets，最後介紹已知的定理”Julia set 會包含在 periodic point 的 closure 內”，我們實際討論一些例子的 Julia sets 和其 repelling periodic points 的 closure，並說明其相等。.

(3) THE JULIA SETS AND THE PERIODIC POINTS IN DYNAMICAL SYSTEMS OVER NONARCHIMEDEAN FIELDS WANG, YI-WUN. Contents 1. Preliminaries. 2. 1.1. Nonarchimedean Metrics. 2. 2. Dynamical Systems. 7. 2.1. Families of Dynamical Systems. 7. 2.2. Symbolic Dynamics. 10. 2.3. The Closure of the Periodic Points over a Nonarchimedean Field. 13. 3. Examples. 13. 3.1. The Dynamics of φ(z) =. z 2 −z p. 13. 3.2. The Dynamics of φ(z) =. z p −z p. 19. References. 22. 1.

(4) 2. WANG, YI-WUN. 1. Preliminaries 1.1. Nonarchimedean Metrics We begin by recalling the definition and basic properties of nonarchimedean absolute values. Definition 1.1.1. An absolute value on a field K is a map | · | : K → R with the following properties: (1) |α| ≥ 0, and |α| = 0 if and only if α = 0. (2) |αβ| = |α||β| for all α, β ∈ K. (3) |α + β| ≤ |α| + |β| for all α, β ∈ K. (triangle inequality) We say that the absolute value is nonarchimedean, if the property (3) is replaced by (30 ) |α + β| ≤ max{|α|, |β|} for all α, β ∈ K. Example 1.1.2. The field Q has the usual real absolute value |α|∞ = max{α, −α}. For each prime p it also has p-adic absolute value defined as follows. Every nonzero Q rational number α has a unique factorization of the form α = ± p prime pep (α) with ep (α) ∈ Z. Then |α|p = p−ep (α) . The p-adic absolute values are nonarchimedean. Proposition 1.1.3. Let K be a field with a nonarchimedean absolute value | · |v and let α, β ∈ K. Then |α|v 6= |β|v implies that |α + β|v = max{|α|v , |β|v }. Proof. Without loss of generality, we can assume that |α|v > |β|v . Then |α|v = |α + β − β|v ≤ max{|α + β|v , |β|v }. Since |β|v < |α|v , this inequality implies that |α + β|v ≥ |α|v . On the other hand, the inequality |α + β|v ≤ max{|α|v , |β|v } = |α|v implies that |α + β|v = |α|v = max{|α|v , |β|v }.. . Let K be a field. The projective line P1 (K) is given by homogeneous coordinates P1 (K) ∼ = (K 2 \{(0, 0)})/ ∼,.

(5) THE JULIA SETS AND THE PERIODIC POINTS IN DYNAMICAL SYSTEMS OVER NONARCHIMEDEAN FIELDS 3. where [X1 , Y1 ] ∼ [X2 , Y2 ] if there exists λ ∈ K ∗ such that (X1 , Y1 ) = (λX2 , λY2 ). We may also view that P1 (K) ∼ =K. [. {∞}.. In terms of homogeneous coordinates, a rational map φ : P1 → P1 is given by a pair of homogeneous polynomials φ(X, Y ) = [F ∗ (X, Y ), G∗ (X, Y )] of degree d. These are related to the description φ(z) =. F (z) G(z). by the relations. F ∗ (X, Y ) = X d F (X/Y ) and G∗ (X, Y ) = Y d G(X/Y ). Definition 1.1.4. Let K be a field with a nonarchimedean absolute value | · |v and let P1 = [X1 , Y1 ] and P2 = [X2 , Y2 ] be points in P1 (K). The v-adic chordal metric on P1 (K) is ρv (P1 , P2 ) :=. |X1 Y2 − X2 Y1 |v . max{|X1 |v , |Y1 |v } max{|X2 |v , |Y2 |v }. The following proposition shows that the v-adic chordal metric is a nonarchimedean metric defined on P1 (K). Proposition 1.1.5. Let K be a nonarchimedean field and Pi points in P1 (K). Then the v-adic chordal metric ρv has the following properties. (1) 0 ≤ ρv (P1 , P2 ) ≤ 1. (2) ρv (P1 , P2 ) = 0 if and only if P1 = P2 . (3) ρv (P1 , P2 ) = ρv (P2 , P1 ). (4) ρv (P1 , P3 ) ≤ max{ρv (P1 , P2 ), ρv (P2 , P3 )}. Proof. The lower bound in (1) and the part (3) is obtained directly by the definition. Let Pi = [Xi , Yi ] be points in P1 (K). Since the absolute value is nonarchimedean, we get |X1 Y2 − X2 Y1 |v ≤ max{|X1 Y2 |v , |X2 Y1 |v } ≤ max{|X1 |v , |Y1 |v } max{|X2 |v , |Y2 |v }. Therefore, the upper bound in (1) is 1. The value ρv (P1 , P2 ) is 0 if and only if |X1 Y2 − X2 Y1 |v is 0, that is equivalent to X1 Y2 = X2 Y1 . Thus the definition of the projective line P1 (K) implies that P1 = P2 . The proof of (4) requires the.

(6) 4. WANG, YI-WUN. consideration of several cases. The Lemma 1.1.6 makes the proof more transparent by allowing some freedom to change coordinates. We will prove it after Lemma 1.1.6.. . Let K be a field. A linear fractional transformation (or M¨obius transformation) is a map from P1 (K) into P1 (K) of the form [X, Y ] 7→ [aX + bY, cX + dY ] We also write it as the form z 7→. where a, b, c,d ∈ K with ad − bc 6= 0. az+b cz+d. when it is in the affine coordinates. It. defines an automorphism of P1 (K), and composition of transformations corresponds ! to multiplication of the corresponding matrices. a b. . Two matrices give the. c d. same transformation if and only if they are scalar multiples of one another. It is well-known that these are the only automorphisms of P1 (C). So we have Aut(P1 (C)) = PGL2 (C) = GL2 (C)/C∗ . Lemma 1.1.6. Let K be a nonarchimedean field with the absolute value | · |v . Let R = {α ∈ K | |α|v ≤ 1} be the ring of integers of K and let f : P1 (K) → P1 (K) be a linear fractional transformation of the form f ([X, Y ]) = [aX+bY, cX+dY ],. where a, b, c,d ∈ K with ad − bc 6= 0,i.e., f ∈ PGL2 (K).. Then there exists a positive constant C such that ρv (f (P1 ), f (P2 )) ≤ Cρv (P1 , P2 ). for all points P1 , P2 ∈ P1 (K).. Furthermore, if a, b, c and d are integers with ad − bc is an unit,i.e., f ∈ PGL2 (R), then ρv (f (P1 ), f (P2 )) = ρv (P1 , P 2) for all points P1 , P2 ∈ P1 (K). Proof. Since the points Pi are in the projective line P1 (K), we can write each points as Pi = [Xi , Yi ] where Xi , Yi ∈ R and one of Xi or Yi is an unit. Then max{|Xi |v , |Yi |v } = 1 and ρv (P1 , P2 ) = |X1 Y2 − X2 Y1 |v ..

(7) THE JULIA SETS AND THE PERIODIC POINTS IN DYNAMICAL SYSTEMS OVER NONARCHIMEDEAN FIELDS 5. Write δ = ad − bc, and consider the identities d(aXi + bYi ) − b(cXi + dYi ) =δXi c(aXi + bYi ) − a(cXi + dYi ) = − δYi . Thus we have the inequality |δ|v |Xi |v = |d(aXi + bYi ) − b(cXi + dYi )|v ≤ max{|d(aXi + bYi )|v , |b(cXi + dYi )|v } ≤ max{|d|v , |b|v } max{|aXi + bYi |v , |cXi + dYi |v }. Similarly, we get |δ|v |Yi |v ≤ max{|c|v , |a|v } max{|aXi + bYi |v , |cXi + dYi |v }. Therefore |δ|v = |δ|v max{|Xi |v , |Yi |v } ≤ λ max{|aXi + bYi |v , |cXi + dYi |v },. (1). where λ = max{|a|v , |b|v , |c|v , |d|v }. A direct computation leads to |(aX1 +bY1 )(cX2 +dY2 )−(aX2 +bY2 )(cX1 +dY1 )|v = |δ(X1 Y2 −X2 Y1 )|v = |δ|v ρv (P1 , P2 ). Thus ρv (f (P1 ), f (P2 )) = ≤ Take C =. λ2 , |δ|v. |(aX1 + bY1 )(cX2 + dY2 ) − (aX2 + bY2 )(cX1 + dY1 )|v max{|aX1 + bY1 |v , |cX1 + dY1 |v } max{|aX2 + bY2 |v , |cX2 + dY2 |v } |δ|v ρv (P1 , P2 ) λ2 = ρv (P1 , P2 ). (|δ|v /λ)2 |δ|v. and then the lemma follows.. If f ∈ PGL2 (R), then |δ|v = 1. The inequality 1 = |ad − bc|v ≤ max{|ad|v , |bc|v } ≤ λ2 ≤ 1 implies that λ = 1. Since the numbers are integers, we have max{|aXi + bYi |v , |cXi + dYi |v } ≤ 1. Since |δ|v = λ = 1 the inequalities (1) and (2) imply max{|aXi + bYi |v , |cXi + dYi |v } = 1.. (2).

(8) 6. WANG, YI-WUN. Therefore ρv (f (P1 ), f (P2 )) = |(aX1 + bY1 )(cX2 + dY2 ) − (aX2 + bY2 )(cX1 + dY1 )|v = ρv (P1 , P2 ). Now we return to give the proof of Proposition 1.1.5 (4). Write each point as Pi = [Xi , Yi ] with Xi , Yi ∈ R and at least one of Xi and Yi is in R∗ . Then max{|Xi |v , |Yi |v } = 1, and ρv (Pi , Pj ) = |Xi Yj − Xj Yi |v . If |X2 |v ≤ |Y2 |v = 1, we apply the map f (X, Y !) = [Y2 X − X2 Y, Y ]. Then the absolute value of the determinant of. Y2 −X2 0. is 1. The Lemma 1.1.6 implies. 1. that ρv (P1 , P3 ) = ρv (f (P1 ), f (P3 )) =. |Y3 (Y2 X1 − X2 Y1 ) − Y1 (Y2 X3 − X2 Y3 )|v max{|Y2 X1 − X2 Y1 |v , |Y1 |v } max{|Y2 X3 − X2 Y3 |v , |Y3 |v }. = |Y3 (Y2 X1 − X2 Y1 ) − Y1 (Y2 X3 − X2 Y3 )|v = max{|Y3 (Y2 X1 − X2 Y1 )|v , |Y1 (Y2 X3 − X2 Y3 )|v } ≤ max{|(Y2 X1 − X2 Y1 )|v , |(Y2 X3 − X2 Y3 )|v } = max{ρv (P1 , P2 ), ρv (P2 , P3 )}. If |Y2 |v ≤ |X2 |v = 1, we apply the map f (X, Y ) = [Y2 X − X2 Y, X]. Then f is in PGL2 (R). Similarly, the Lemma 1.1.6 implies that ρv (P1 , P3 ) = ρv (f (P1 ), f (P3 )) ≤ max{ρv (P1 , P2 ), ρv (P2 , P3 )}. The proof of the Proposition 1.1.5 is complete.. .

(9) THE JULIA SETS AND THE PERIODIC POINTS IN DYNAMICAL SYSTEMS OVER NONARCHIMEDEAN FIELDS 7. 2. Dynamical Systems 2.1. Families of Dynamical Systems A dynamical system is a pair (S, φ) consisting of a set S and a self-map φ : S → S. We write φn = φ ◦ φ ◦ · · · φ which is called the n-th iterate of φ. By convention, | {z } n times. we let φ0 be the identity map on S. The orbit of an element x in S is the subset Oφ(x) of points obtained by applying the iterates of φ to x, i.e., Oφ (x) = {φn (x) ∈ S | for all integer n ≥ 0}. The goal of dynamical systems is to study the behavior of points in S as φ is applied repeatedly. Definition 2.1.1. Let (S, φ) be a dynamical system. The point α is periodic if φn (α) = α for some integer n ≥ 1. This point α is also called a periodic point. The point α is preperiodic if some iterate φm (α) is periodic. That is φm+n (α) = φm (α) for some integer n ≥ 1. The set of periodic and preperiodic points of φ in S are denoted respectively by Per(S, φ) : = {α ∈ S | φn (α) = α for some n ≥ 1}, PrePer(S, φ) : = {α ∈ S | φm+n (α) = φm (α) for some m ≥ 0 and n ≥ 1} = {α ∈ S | Oφ (α) is finite}. The points in PrePer(S, φ) are called preperiodic points, and the others are called wandering points. Therefore if α is an wondering point, then Oφ (α) is an infinite set. If α is a periodic point, then the smallest positive integer n such that φn (α) = α is called the exact period of α. Let α be a point of exact period n for φ. We define the multiplier of φ at α to be λα (α) := (φn )0 (α). The chain rule implies that (φn )0 (φ) = φ0 (α)φ0 (φ(α))φ0 (φ2 (α)) · · · φ0 (φn−1 (α))..

(10) 8. WANG, YI-WUN. In other words, λα (φ) is the product of the values of φ0 at each of the n distinct points in the orbit of α. Thus if |λα (φ)| < 1, then a small neighborhood of α will shrink each time it returns to α. If |λα (φ)| > 1, then it will expand. This observation prompts the following definitions. Definition 2.1.2. Let α be a periodic point for a rational function φ, and let λα (φ) be the corresponding multiplier. Then α is called superattracting if λα (φ) = 0, attracting if |λα (φ)| < 1, neutral if |λα (φ)| = 1, repelling if |λα (φ)| > 1. Neutral periodic points are also sometimes called indifferent. Example 2.1.3. We study the iteration of the polynomial map φ(z) = z 2 − 1 on the finite field F19 .. Figure 1. Action of φ(z) = z 2 − 1 on the field F19 . The point 5, 15 are fixed points. The points 6, 8 and 16 are perodic points with exact period 3, and the points 0 and 18 are perodic points with exact period 2. All other points are preperiodic, but not periodic. Since F19 is a finite set, there obviously are no wandering points..

(11) THE JULIA SETS AND THE PERIODIC POINTS IN DYNAMICAL SYSTEMS OVER NONARCHIMEDEAN FIELDS 9. Example 2.1.4. Now, we also use the same polynomial φ(z) = z 2 − 1, but consider its action on some infinite set. (1) First of all, we consider its action on Z. By direct calculation, we get 1 → 0 −1. Thus 0 and −1 are periodic with exact period 2, and 1 is preperiodic but not periodic. If |z| ≥ 2, then |z 2 − 1| = |z 2 | − 1 > |z|. Thus limn→∞ φn (z) = ∞, and this implies that every other element of Z is wandering. (2) More generally, the only φ-preperiodic point in Q are {1, 0, −1}. Let z be a rational number. Since the integers are considered above, we may assume that z is not an integer. Let p be a prime divisor of the denominator of z. Then z is of the form. t ps. where t and s are integers and gcd(t, ps) = 1.. Consider the p-adic absolute value as φ acting on z, we have |φ(z)|p = |. t2 − p2 s2 |p ≥ p2 . p2 s2. For each n ≥ 0, we get the inequality |φn (z)|p ≥ p2n and then lim |φn (z)|p = ∞.. n→∞. If z is a preperiodic point, the orbit Oφ (z) is a finite set and then |φn (z)|p will be bounded. Thus z is not a preperiodic point. (3) If we consider its action on the complex numbers C, then there are infinitely many complex preperiodic points. Note that φn (x) is a polynomial of degree 2n for each n ∈ N. Since C is algebraic closed, the solutions of φm+n (z) = φm (z) are preperiodic points in C for any m ≥ 0 and n ≥ 1. Definition 2.1.5. Let (S1 , ρ1 ) and (S2 , ρ2 ) be metric spaces, and let Φ be a collection of maps from S1 into S2 . The collection Φ is said to be equicontinuous at the point α ∈ S1 if for every ε > 0, there exists δ > 0 such that ρ1 (α, β) < δ implies that ρ2 (φ(α), φ(β)) < ε for every φ ∈ Φ. For an individual map φ from S into itself, we say that φ is equicontinuous at the point α ∈ S if the collection of iterates {φn | n ∈ N} is equicontinuous at α..

(12) 10. WANG, YI-WUN. Definition 2.1.6. Let S be a metric space and φ a map from S into itself. The Fatou set F(φ) of φ is the maximal open set on which φ is equicontinuous. The Julia set J (φ) of φ is the complement of the Fatou set. Informally, we may say that points in the Julia set J (φ) tend to wander away from one another as φ is iterated, and the nearby points in the Fatou set F(φ) tend to stay together. Example 2.1.7. If α is an attracting periodic point of φ, then the points close to α are all attracted to the points in the orbit of α. Thus the attracting periodic point α is in the Fatou set F(φ). Similarly, if α is a repelling periodic point of φ, since a small neighborhood of it will expand, the point α is in the Julia set J (φ). Example 2.1.8. Let φ be a map from P1 (C) into P1 (C) given by φ(z) = z d . Since n. the n-th iterate of φ is φn (z) = z d , we get   0 if |z| < 1, n lim φ (z) = n→∞  ∞ if |z| > 1. Therefore a point α with absolute value |α| = 6 1 is in the Fatou set F(φ). The Julia set J (φ) is the unit circle in C (i.e., J (φ) = {z ∈ C | |z| = 1}). If α is a point with |α| = 1, the n-th iterate of φ at α has absolute value |φn (α)| = 1 for all n ∈ N. For any open neighborhood N of α, there exists a point β with |β| = 6 1. Thus N contains a point β such that limn→∞ φn (β) = 0 or limn→∞ φn (β) = ∞. Therefore φ is not equicontinuous at α; that is the unit circle is not in the Fatou set. Since the points of C except the unit circle are in the Fatou set, the Julia set is the unit circle. 2.2. Symbolic Dynamics Symbolic dynamics is a tool for modeling seemingly more complicated dynamical systems. Let S = {σ1 , σ2 , . . . , σs } be a finite set of symbols and let S N = {[β0 β1 β2 · · · ] | βn ∈ S}.. (3).

(13) THE JULIA SETS AND THE PERIODIC POINTS IN DYNAMICAL SYSTEMS OVER NONARCHIMEDEAN FIELDS 11. We want to define a metric on S N . Fix a number p > 1 and let α = [α0 α1 · · · ] and β = [β0 β1 · · · ] be elements in S N . Then we set ρ(α, β) = p− inf{n|αn 6=βn } .. (4). By direct calculation, ρ is a nonarchimedean metric whith the following properties • 1 ≥ ρ(α, β) ≥ 0. • ρ(α, β) = 0 if and only if α = β. • ρ(α, γ) ≤ max{ρ(α, β), ρ(β, γ)} where α, β and γ are elements in S N . Example 2.2.1. Let S be a set of s symbols. An important map on S N is the left shift map L : S N −→ S N. [β0 β1 β2 β3 · · · ] 7−→ [β1 β2 β3 β4 · · · ].. More formally, the sequence L(β) is defined by L(β)n = βn+1 . The set S N cantains exactly sn points satisfying Ln (α) = α. To see this, note that a sequence α ∈ S N satisfies Ln (α) = α if and only if its first n terms repeat, i.e., it has the form α = [α0 α1 α2 · · · αn−1 α0 α1 α2 · · · αn−1 α0 α1 α2 · · · αn−1 · · · ]. {z }| {z }| {z } | initial n terms. same n terms. same n terms. There are s choices for each αi , i = 0, 1, . . . n − 1, so there exist sn elements. By the same point of view, the preperiodic point β ∈ S N satisfing Lm+n (β) = Lm (β) has the form 0 β = [β00 β10 β20 · · · βm−1 β0 β1 β2 · · · βn−1 β0 β1 β2 · · · βn−1 β0 β1 β2 · · · βn−1 · · · ]. {z }| {z }| {z } {z }| | the first m terms. initial n terms. same n terms. same n terms. Proposition 2.2.2. Let S be a set of s symbols and S N the space of S-sequences as above with associated metric ρ. Let L : S N → S N be the left shift map. (1) If ρ(α, β) < 1, then ρ(L(α), L(β)) = p · ρ(α, β). (2) L is continuous (indeed Lipschitz), and it is uniformly expanding on each of the disks {α ∈ S N | α0 = σi },. i = 1, 2, . . . , s.. (3) The set S N contains exactly sn points satisfying Ln (α) = α..

(14) 12. WANG, YI-WUN. (4) The periodic points of L are dense in S N . (5) There exists a point γ ∈ S N whose orbit OL (γ) = {Ln (γ) | n ≥ 0} is dense in S N . Proof. Let α = [α0 α1 α2 α3 · · · ] and β = [β0 β1 β2 β3 · · · ] be elements in S N . If α = β, then (1) is ture because the values of both sides are 0. Suppose that α 6= β and n = min{s | αs 6= βs }. Then we get ρ(α, β) = p−n . That ρ(α, β) < 1 implies that n > 0. Since L(α)n = αn+1 and n > 0, min{s | L(α)s 6= L(β)s } = n − 1 is well-defined. Thus ρ(L(α), L(β)) = p−n+1 and (1) is ture. Since ρ(α, β) ≤ 1 for all α, β ∈ S N , the item (1) implies that ρ(L(α), L(β)) ≤ p · ρ(α, β) for all α and β. Hence L is Lipschitz. Since ρ(α, β) < 1 if and only if they are in the same disk, when it is the case we get. ρ(L(α), L(β)) = pρ(α, β) > ρ(α, β).. Hence the map L is expanding on each of those disks. The example above says (3) is ture. To prove (4), for an element α ∈ S N we take a sequence {υn }n≥0 where the υn is the periodic point with the first initial n terms coinciding with the first n terms of α for each n ≥ 0. Thus we have. ρ(α, υn ) ≤ p−n , for all n ∈ N.. Therefore we have that limn→∞ υn = α, and thus the periodic points of L are dense in S N . Finally to prove (5). Let γ be an element in S N such that the first s terms are all possible blocks of length 1, then the next 2s2 terms are all possible blocks of length 2 and so on . Now, let α be an element in S N . For each n ≤ 0 there exists m ∈ N such that the first n terms of Lm (γ) coincide with the first n terms of α. Similarly, every element in S N is a limit point of a sequence in OL (γ). Hence the orbit OL (γ) is dense.. .

(15) THE JULIA SETS AND THE PERIODIC POINTS IN DYNAMICAL SYSTEMS OVER NONARCHIMEDEAN FIELDS 13. 2.3. The Closure of the Periodic Points over a Nonarchimedean Field In this subsection, we introduce a theorem about the closure of the periodic points over a nonarchimedean field from the paper of L.C. Hsia. It tells us that the relationship between the Julia set and the cloure of period points. We will see some examples in section 3. Theorem 2.3.1. Let K be a nonarchimedean field and φ : P1 (K) → P1 (K) be a rational map defined over K with deg(φ) ≥ 2. Then the Julia set J (φ) is contained in the closure of the set of periodic points Per(φ) of φ. Proof. See [Hsia00], Theorem 3.1.. 3. Examples. 3.1. The Dynamics of φ(z) =. z 2 −z p. Let p be an odd prime number, Cp the completion of the algebraic closure of Qp . Let | · | be the nonarchimedean absolute value defined on Cp with |p| = p1 . In this subsection we want to study the dynamics of the map φ(z) =. z2 − z . p. Let α ∈ Cp with |α| > 1. Then |α| = |α − 1| and we have |φ(α)| = |. α2 − α | = p|α||α − 1| = p|α|2 . p n −1. Thus for each n ∈ N, the absolute value of φn (α) is p2. n. |α|2 . Therefore if |α| > 1,. then α is attracted to ∞. Thus the Julia set J (φ) is a subset of Λ where Λ := {α ∈ Cp | |φn (α)| ≤ 1, ∀n ∈ N}. Denote the sets 1 I0 :={α ∈ Cp | |α| ≤ }, p. (5). 1 I1 :={α ∈ Cp | |α − 1| ≤ }. p. (6). Since the absolute value is nonarchimedean, these two disks are disjoint. We want to show that the set Λ is a subset of the union I0 ∪ I1 ..

(16) 14. WANG, YI-WUN. Proposition 3.1.1. Let φ be the map defined on P1 (Cp ) such that φ(z) = any z ∈ Cp with |z| >. 1 p. z 2 −z . p. For. and |z − 1| > p1 , we have limn→∞ |φn (z)| = ∞,i.e., a point. z which is not in the union of I0 and I1 must be attracted to ∞. In particular, the Julia set is a subset of Λ ⊆ I0 ∪ I1 . Proof. Let z be an element in Cp but not in I0 ∪ I1 . By the above discussion, we may assume that |z| ≤ 1. Thus max{|z|, |z − 1|} = 1, and then |φ(z)| = |. z(z − 1) | = p|z||z − 1| > 1. p. Hence φ(z) is attracted to ∞, and so is z. Since a point ouside of the union I0 ∪ I1 is attracted to the supperattracting fixed point at ∞, so such a point is in the Fatou set. Thus Λ is a subset of I0 ∪ I1 .. . The set Λ is not equal to the union of I0 and I1 . Note that for any α ∈ Cp with |α| ≤ 1 we get φ(pα) = pα2 − α ≡ −α. (mod p).. (7). Let α = 1 + px for some x with |x| ≤ 1. Then pα is in the set I0 since the absolute value of pα is p1 . The equation (7) implies that φ(pα) ≡ −1 (mod p). Thus we have |φ(pα)| = |φ(pα) − 1| = 1. Therefore φ(pα) is not in the union of I0 and I1 , then Proposition 3.1.1 implies that φ(pα) is attracted to ∞. Since the orbit Oφ (z) for z ∈ Λ is contained in the set Λ which is a subset of I0 ∪ I1 , for a point z ∈ Λ, we have φn (z) ∈ Iβn where βn ∈ {0, 1} for all non-negative integers. Thus we get the itinerary map β which is defined as β : Λ −→. {0, 1}N. z 7−→ [β0 β1 β2 · · · ]. Lemma 3.1.2. Let F be a complete discrete valuation field with ring of integer O and the maximal ideal M. Let f (x) be a monic polynomial with coefficients in O. Let f (α0 ) ∈ M2s+1 , f 0 (α0 ) 6∈ Ms+1 for some α0 ∈ O and integer s ≥ 0. Then there exists α ∈ O such that α − α0 ∈ Ms+1 and f (α) = 0. This lemma is given in [FVo93], II.1.3, Corollary 3..

(17) THE JULIA SETS AND THE PERIODIC POINTS IN DYNAMICAL SYSTEMS OVER NONARCHIMEDEAN FIELDS 15. Lemma 3.1.3. Let w be an element in I0 ∪ I1 . Then φ−1 (w) consists of two points z0 , z1 such that zu ∈ Iu for u = 0, 1, and further, if w ∈ Qp , then z0 and z1 are in Qp . Proof. Consider the monic polynomial f (x) = x2 − x − pw. The set of roots of f (x) is the set φ−1 (w). We need to show that f (x) has two roots such that one is in I0 and the other is in I1 . Let M be the maximal ideal of Cp and k the integer such that p ∈ Mk \Mk+1 . Suppose that w is in Iu where u = 0 or 1. Since |w − u| ≤ p−1 , the element (w − u) is in Mk . Let s = k − 1. Then f (−pu) = p2 u2 + pu − pw = p2 u2 − p(w − u) ∈ M2k ⊂ M2s+1 , f (1 + pu) = (1 + pu)2 − (1 + pu) − pw = p2 u2 − p(w − u) ∈ M2k ⊂ M2s+1 , and f 0 (−pu) = −2pu − 1 6∈ M ⊃ Ms+1 , f 0 (1 + pu) = 2pu + 1 6∈ M ⊃ Ms+1 . Thus the Lemma 3.1.2 implies that f (x) has two roots z0 and z1 such that (z0 +pu) ∈ Ms+1 = Mk and (z1 − 1 − pu) ∈ Mk . Since pu is in Mk , we have that z0 and (z1 − 1) are in Mk . In particular, the absolute values of z0 and (z1 − 1) are less than |p| = p−1 . Therefore z0 and z1 are in the set I0 and I1 respectively. If w is in Qp , then the Hensel Lemma (see [FVo93] II.1.2 Corollary 1. or [Neu99] II.4.6) implies that f (x) splits in Qp . Thus z1 and z0 are in Qp .. . Proposition 3.1.4. Let β be the itinerary map from Λ into {0, 1}N . Then it has the following properties. (1) The itinerary map β is injective. (2) The restriction of β on Λ ∩ Qp is surjective, i.e., β(Λ ∩ Qp ) = {0, 1}N . (3) For all z and w ∈ Λ, we have |z − w| = ρ(β(z), β(w))..

(18) 16. WANG, YI-WUN. (4) Let L be the left shift map on {0, 1}N . Then the following diagram is commutative: Λ. φ. −→. ↓β. Λ ↓β. (8). L. {0, 1}N −→ {0, 1}N . Proof. (1) Let u be 0 or 1. Then we want to show that if z, w ∈ Iu , then |φ(z) − φ(w)| = p|z − w|.. (9). Since |φ(z)−φ(w)| = p|z−w||z+w−1|, it would be successful in showing |z+w−1| = 1. The absolute values of (z − u) and (w − u) are less than p1 . Since u is 0 or 1, the value (2u − 1) is 1 or −1. Thus |2u − 1| = 1 and strictly greater than |w − u| and |z − u|. Hence we have |z + w − 1| = |(z − u) + (w − u) + (2u − 1)| = |2u − 1| = 1. Let z and w be elements in Λ such that β(z) = β(w). Then φn (z) and φn (w) are in the same set for all n ≥ 0. Thus the equation (9) implies |φn (z) − φn (w)| = p|φn−1 (z) − φn−1 (w)| = pn |z − w|. Since z and w are in Λ, the absolute value of (φn (z) − φn (w)) is bounded. Let n tend to ∞. Then we have |z − w| = 0. (2) For any binary sequence of α0 α1 α2 · · · αn with n ≥ 1, define a set Jα0 α1 α2 ···αn := {z ∈ Qp | φi (z) ∈ Iαi , ∀0 ≤ i ≤ n},. (10). where φ0 is the identity map as before. If z ∈ Jα0 α1 α2 ···αn ∩ Λ, then β(z)i = αi for all 0 ≤ i ≤ n. Since the sets satisfy Jα0 α1 α2 ···αn = Qp ∩ Jα0 ∩ φ−1 (Jα1 ) ∩ · · · ∩ φ−n (Jαn ), they are nested in the sense Jα0 α1 α2 ···αn = (Jα0 α1 α2 ···αn−1 ∩ φ−n (Jαn )) ⊂ Jα0 α1 α2 ···αn−1 . Since φ is continuous and J0 = I0 ∩ Qp , J1 = I1 ∩ Qp are closed, the sets φ−i (Jαi )’s are closed and so is the intersection Jα0 α1 α2 ···αn . We claim that it is nonempty. The.

(19) THE JULIA SETS AND THE PERIODIC POINTS IN DYNAMICAL SYSTEMS OVER NONARCHIMEDEAN FIELDS 17. set J0 and J1 are nonempty, since they are disks in Zp with radius p1 . To prove this by induction, suppose that Jβ0 β1 ···βn−1 is nonempty for any binary sequence of length n. Since Jα1 α2 ···αn is nonempty and Lemma 3.1.3 says that #(φ−1 (z) ∩ I0 ) = #(φ−1 (z) ∩ I1 ) = 1 for any z ∈ I0 ∪ I1 , we have φ−1 (Jα1 α2 ···αn ) ∩ Ji is nonempty for i = 0, 1. Hence the set Jα0 α1 α2 ···αn = Jα0 ∩ φ−1 (Jα1 α2 α3 ···αn ) is nonempty. Let α = [α0 α1 α2 · · · ] be an element in {0, 1}N and let Jα := ∩n≥0 Jα0 α1 ···αn . Since Jα is the intersection of nested sequence of nonempty closed bounde subsets of Qp , by compactness, Jα is nonempty. Thus for any point z ∈ Jα , we have φn (z) ∈ Iαn and |φn (z)| is bounded for all n ≥ 0 . Hence we have β(Λ ∩ Qp ) = {0, 1}N . (3) If z = w, then both sides are zero. Assume that z 6= w and let k = min{n ≥ 0 | β(z)n 6= β(w)n }. Then ρ(β(z), β(w)) = p−k . Since β(z)n = β(w)n for all 0 ≤ n < k and they are in the same disk for all 0 ≤ n < k. The equation (9) implies that |φk (z) − φk (w)| = pk |z − w|. On the oher hand, since φk (z) and φk (w) are not in the same disk, the absolute value of (φk (z) − φk (w)) is 1. Therefore we have |z − w| = p−k = ρ(β(z), β(w)). (4) Let z be an element in Λ and α = [α0 α1 α2 · · · ] be the image of β in {0, 1}N . Then β ◦ φ(z) = [α1 α2 α3 · · · ] = L([α0 α1 α2 · · · ]) = L ◦ β(z). The proof is complete.. . The proposition above allows us to identify the dynamics of the polynomial map φ(z) =. z 2 −z p. with the dynamics of the shift map on the space {0, 1}N . It becomes. an easy matter to read the dynamics of φ(z) from the elementary properties of the shift map..

(20) 18. WANG, YI-WUN. Theorem 3.1.5. Let p be an odd prime, and let φ(z) :=. z2 − z , p. Λ :={z ∈ Cp | |φn (z)| is bounded for all non-negative integer n}. Then we have the following properties. (1) The Julia set J (φ) is equal to Λ and Λ is a subset of Qp . (2) Every periodic point of φ other than ∞ is repelling. (3) The repelling periodic points are dense in the Julia set J (φ). Proof. (1) Let β be the itinerary map from Λ into {0, 1}N . Proposition 3.1.4 (1) says that β is injective, and it is surjective on Λ ∩ Qp . It follows that Λ is a subset of Qp . To prove that J (φ) = Λ, we note that Proposition 3.1.4 implies that β is bijective and respects the metrics. Since β ◦ φ = L ◦ β where L is the shift map on {0, 1}N , the dynamical properties of φ acting on Λ is identical to the dynamical properties of L acting on {0, 1}N . So, J (φ) is a subset of Λ implies β(J (φ)) = J (L). However Proposition 2.2.2 says that L is uniformly expanding on each of the disks {α ∈ {0, 1}N | α0 = i}, where i = 0 or 1. A uniformly expanding map is nowhere equicontinuous, so J (L) = {0, 1}N . Thus we deduce that J (φ) = Λ. (2) Let α be a periodic point with exact period n for φ. Since the point outside Λ is attracted to ∞, so α is an element in Λ. Proposition 3.1.4 (3)-(4) implies that φn (z) − φn (α) ρ(β ◦ φn (z), β ◦ φn (α)) ρ(Ln ◦ β(z), Ln ◦ β(α)) | |= = . z−α ρ(β(z), β(α)) ρ(β(z), β(α)) Since L is the left shift map , if ρ(s, w) = p−k for some integer k, then ρ(Ln (s), Ln (w)) = p−k+n for any n ≤ k. If z and α are close enough, then |. φn (z) − φn (α) ρ(Ln ◦ β(z), Ln ◦ β(α)) |= = pn . z−α ρ(β(z), β(α)). This implies that |(φn )0 (α)| = pn . Therefore every periodic point of φ other than ∞ is repelling. (3) Proposition 2.2.2 (4) says that the periodic points of L are dense in {0, 1}N . Via the itinerary map β, the periodic points of φ is dense in Λ. Since every periodic.

(21) THE JULIA SETS AND THE PERIODIC POINTS IN DYNAMICAL SYSTEMS OVER NONARCHIMEDEAN FIELDS 19. point other than ∞ is repelling, the repelling periodic points are dense in the Julia set J (φ). The proof is complete.. 3.2. The Dynamics of φ(z) =. . z p −z p. The notations is the same as the subsection 3.1. We consider the map φ(z) =. zp − z . p. Let S be the subset of Cp consisting of the (p − 1)-th root of unity and 0. Then S is a set of representative of the residue class field Fp of Qp in Cp . Denote that Λ :={α ∈ Cp | |φn (α)| ≤ 1, ∀n ≥ 0}, 1 Iu :={α ∈ Cp | |α − u| ≤ }, p [ and IS := Iu .. for each u ∈ S,. u∈S. Let α ∈ Cp with |α| > 1. Then |α|p > |α| and we get |φ(α)| = |. αp − α | = p|αp − α| = p|α|p . p. Hence |φn (α)| diverges to infinity. So α is in the Fatou set F(φ) of φ since α is attracted to the superattracting fixed point at infinity. Therefore the Julia set is a subset of Λ. Let α ∈ Cp \IS such that |α| ≤ 1. Since S is a set of representative of Fp , there is at most one element y ∈ S such that. 1 p. < |α − y| < 1 and the other. u ∈ S satifies |α − u| = 1. Thus |φ(α)| = p|αp − α| = p. Y. |α − u| > 1.. u∈S. Since φ(α) is attracted to infinity, the point α is not in the Julia set and then J (φ) ⊆ Λ ⊆ IS . Therefore the orbit Oφ (z) of z ∈ Λ is contained in IS . For a point z ∈ Λ, let βn ∈ S such that φn (z) ∈ Iβn for all nonnegative integers n. Then the itinerary map β is defined as β : Λ −→. SN. z 7−→ [β0 β1 β2 · · · ]..

(22) 20. WANG, YI-WUN. T Lemma 3.2.1. Let w ∈ IS . Then φ−1 (w) Iu is nonempty for all u ∈ S. In T T T particular, if w ∈ IS Qp , then φ−1 (w) Iu Qp is nonempty for all u ∈ S. Proof. Consider the monic polynomial f (x) = xp − x − pw. The set φ−1 (w) is the set of roots of f (x). Thus we want to show that for each v ∈ S, the polynomial f (x) has a root in the set Iv . Moreover, if w is in Qp then the root is in Qp . Let M be the maximal ideal of Cp and k the integer such that p ∈ Mk \Mk+1 , and set w be in Iu for some u ∈ S. Since |w − u| ≤ p−1 , the element (w − u) is in Mk . For any v ∈ S, we have v p = v and then f (v + pu) = (v + pu)p − (v + pu) − pw ≡ v p + pv p−1 pu − v + pu − pw ≡ p(u − w) ≡ 0. (mod M2k ). (mod M2k ),. and f 0 (v + pu) = p(v + pu)p−1 − 1 6∈ M. Let s = k − 1. Then f (v + pu) is in M2s+1 and f 0 (v + pu) is not in Ms+1 . Thus Lemma 3.1.2 implies that there exists α such that (α−(v+pu)) ∈ Mk and f (α) = 0. Since the element pu is in Mk , the element (α − v) is in Mk ,i.e., |α − v| ≤ p−1 . In particular, for any v ∈ S, there exists α ∈ Iv such that f (α) = 0. If w is in Qp , then f (x) is a polynomial over the Zp and it splits into p distinct linear polynomials over the residue class field of Qp . Thus Hensel Lemma says that f (x) splits completely over Qp and so the roots are in Qp .. . Let u ∈ S and x, y ∈ Iu . Let M be the maximal ideal of Cp . Since |φ(x) − φ(y)| = p|x − y||. p−1 Y. xi y p−1−i − 1|. i=0. and p−1 Y i=0. i p−1−i. xy. −1≡. p−1 Y. ui up−1−i − 1 ≡ −2. mod M,. i=0. we have that |φ(x) − φ(y)| = p|x − y| if x and y are in the same disk.. (11).

(23) THE JULIA SETS AND THE PERIODIC POINTS IN DYNAMICAL SYSTEMS OVER NONARCHIMEDEAN FIELDS 21. Thus the itinerary map β is injective by the same proof as Proposition 3.1.4 (1). By Lemma 3.2.1 and the same proof as Proposition 3.1.4 (2), the itinerary map β is T surjective when it restricts to Λ Qp . The equation (11) also implies that for any x, y ∈ Λ, we have |x − y| = ρ(β(x), β(y)).. (12). Hence we have the following proposition. Proposition 3.2.2. Let β be the itinerary map from Λ into {0, 1}N . Then it has the following properties. (1) The itinerary map β is injective. (2) The restriction of β on Λ ∩ Qp is surjective, i.e., β(Λ ∩ Qp ) = {0, 1}N . (3) For all x and y ∈ Λ, we have |x − y| = ρ(β(x), β(y)). (4) Let L be the left shift map on S N . Then the following diagram is commutative: Λ. φ. −→. Λ. ↓β. ↓β. SN. −→ S N .. (13). L. Theorem 3.2.3. Let p be an odd prime number, and let φ(z) :=. zp − z , p. Λ :={z ∈ Cp | |φn (z)| is bounded for all non-negative integer n}. Then we have the following properties. (1) The Julia set J (φ) is equal to Λ and Λ = Zp . (2) Every periodic point of φ other than ∞ is repelling. (3) The repelling periodic points are dense in the Julia set J (φ). Proof. The proof of this theorem is the same as the proof of the Theorem 3.1.5. So we shall only prove that Λ = Zp . Since Λ is a subset of Qp , IS respectively, so Λ T is a subset of Qp IS = Zp . If α ∈ Zp , then α ¯p = α ¯ where α ¯ is the element in the residue class field corresponding to α. Thus αp ≡ α (mod p) and then φ(α) ∈ Zp ..

(24) 22. WANG, YI-WUN. Hence if α ∈ Zp , then φn (α) ∈ Zp . Moreover, |φn (α)| ≤ 1 for all n ≥ 0. Therefore Zp is a subset of Λ. The proof is complete.. . Remark. Let p be an odd prime number and φn (z) =. z n −z p. defined on Cp where n. is a positive integer such that p - (n − 1). Let K be a subfield of Cp such that the (n − 1)-th roots of unity is contained in it. Then using the same method, we will get that the Julia set is contained in the ring of integers of K, and the Julia set is equal to the closure of the repelling periodic points. References [Ben10] R. L. Benedetto, “Non-archimedean Dynamics in Dimension One”, Arizona Winter School, 2010. [FVo93] I. B. Fesenko, S. V. Vostokov, “Local Fields and Their Extensions”, Translations of Math. Monographs, Vol.121, AMS, 1993. [Hsia00] L. C. Hsia, “Closure of periodic points over a non-Archimedean field”, J. London Math. Soc(2), 62(3):685-700, 2000. [Neu99] J.. Neukirch,. “Algebraic. Number. Theory”,. Grundlehren. der. mathematischen. Wissenchaften, Vol.322, Springer, 1999. [Sil07] J. H. Silverman, “The Arithmetric of Dynamical Systems”, Springer, New York, 2007. [Sil10] J. H. Silverman, “Lecture Note on Arithmetric Dynamics”, Arizona Winter School March 13-17, 2010..

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