多項式的臨界點圖樣和固定點圖樣

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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 : 林延輯博士. Critical Portraits and Fixed Point Portraits Large for Polynomial Maps. 研究生 : 楊昌宸. 中華民國九十八年七月.

(2) 致謝論文之所以能夠順利完成首先要感謝林延輯老師願意指導我完成這篇論文。當我迷惘時 , 他就像一座燈塔指引著我方向 ; 當我徬徨無助時 , 他願意聽我訴苦。對於老師諄諄教誨 , 學生沒齒難忘。感謝本系施茂祥老師和台大王藹農老師抽空擔任我的論文口試委員。感謝父母親 , 是他們讓我能無後顧之憂的全心全意地讀書。同時也要感謝親戚們的鼓勵與支持 , 尤其是二阿姨的大力支持。另外要感謝同鄉的同學彥宇、學長晉宇、物理所學弟孟昌和正宗以及英語所室友博涵為我在師大的日子增添多一點色彩。同時感謝中學同學文浩、均彥、智民、俊全和韋廷跟我談天說地。還要感謝其他人或多或少的陪伴 , 因為為數眾多 , 所以就謝天了。. 楊昌宸謹誌 2009 年 7 月.

(3) Contents 1. Introduction. 1. 2. Extenal Rays. 3. 3. Rotation Numbers. 9. 4. Critical Portraits. 14. 5. Fixed Point Portraits. 17. 6. Classification of Degree 3 Fixed Point Portraits. 20.

(4) Critical Portraits and Fixed Point Portraits for Polynomial Maps July 16, 2009 Abstract From the work of Goldberg-Milnor [GM], a candidate of fixed point portraits P under specific conditions can generate critical portraits Θ, though there are still many possible choices for Θ. Using Fisher’s Main Theorem, there is a unique monic polynomial f which is critically periodic and realizes one of the critical portrait Θ and in turn realizes the fixed point portrait P. At the end we pay our attention to the case of critical portraits of degree 3 and classify them.. Key Words. Complex dynamics, iteration for polynomial maps, rotation number, fixed point portraits.. 1. Introduction. ˆ → C ˆ The subject of complex dynamics studies iterations of rational maps f : C ˆ The Fatou set F is and discuss the behavior of forward orbits of each point z ∈ C. n ˆ on which {f } is equicontinuous and the Julia set J is the maximal open set of C the complement of F. We only discuss rational maps whose degrees are greater or equal to 2. Constant maps are trivial cases without any prerequisites. Any Möbius map which fixes only one point has the property every forward orbit converges to this unique fixed point. ˆ other than those two fixed For the other Möbius maps, any forward orbit of z ∈ C points either converges to one of the two fixed points, has a finite orbit, or is a dense subset of some circle.. 1.

(5) The Fatou set and the Julia set are invariant under f . In particular, J is the closure of a backward orbit of any point in J and it is a nonempty perfect set. Attracting cycles lie in F, while repelling ones and rationally indifferent ones lie in J , but irrationally indifferent cycles may lie in F or J . Moreover, the Julia set is the closure of the repelling periodic points. Now, we consider a more limited situation where f is a polynomial map. Note that the point of ∞ is a superattracting fixed point of f , and the basin A(∞) of ∞ is the component of F that contains ∞. The filled Julia set K is the complement of A(∞) and its boundary is J . Finally, consider the family of quadratic functions {fc (z) = z 2 + c |c ∈ C }. We mention that the Mandelbrot set M is the set consisting of c whose orbit under fc is bounded or, equivalently, never diverges to ∞. A more amazing fact is that M is also the set consisting of c with fc has the connected filled Julia set, or equivalently, fc has the connected Julia set. In this article, we devote ourselves to the question that whether given a candidate of fixed point portrait P and d, there exists a polynomial degree d ≥ 2 with connected Julia set. The organization of the article is as follow: In Section 2, we introduce basic results about external rays in [J]. Most external rays land and every rational ray must land. A fixed ray must land at a repelling or parabolic fixed point. There is at least one rational ray landing at each repelling or parabolic periodic points. In Section 3, we introduce the concepts of rotation numbers of a monotone degree one circle map and rotation sets. The main facts are Theorems 3.2 and 3.3. In Section 4, we construct d monotone degree one circle maps φj ’s from a formal critical portrait Θ with properties C1-C5. Applying the Fisher’s Main Theorem to a given critical portrait, we get a monic polynomial f which realizes it. Furthermore, we note that the fixed point portrait is the repelling periodic points of each φj . In Section 5, in a special condition that a candidate fixed point portrait P with d rational types satisfying P1-P4, we may have a polynomial f to realize P by way of those results in Section 4. In Section 6, given d = 3, we classify all possible fixed point portraits by way of critical portraits. We specifically carry out degree d pq -rotation cycles with the aid of computer.. 2.

(6) 2. Extenal Rays. ˆ →C ˆ is Standing Hypothesis 2.1. Throughout this section, we assume that f : C a monic polynomial map f (z) = z d + ad−1 z d−1 + · · · + a1 z + a0 (d ≥ 2) with the Julia set J connected, or equivalently, K is connected. Definition 2.1. With Standing Hypothesis 2.1, there is the Böttcher isomorphism ˆ \K → C ˆ \ D such that for any z in C ˆ \ K, φ (f (z)) = (φ(z))d . For any t in φ:C [0, 1), Rt = {φ−1 (re2πit ) |r > 1 } is called an external ray for K. The orthogonal trajectories of the external rays for K are called equipotential curves around K. Furthermore, φ−1 (re2πit ) converges to a point, denoted by γ(t), in J as r & 1 and we say that Rt lands at γ(t). The continuous function G : C → R defined by log |φ(z)| > 0 if z ∈ C \ K; G(z) = 0 if z ∈ K is called the Green’s function for K, and G (f (z)) = nG(z) for all complex numbers z. Definition 2.2. An external ray Rt is called rational if t ∈ Q/Z. It is called periodic if there is a positive number p such that dp t ≡ t (mod Z) and is called eventually periodic if there is nonnegative number q such that dq t (mod Z) is periodic. Remark In particular, Rt is rational if and only if Rt is eventually periodic. Rt is periodic if and only if t is rational with denominator relatively prime to d. Lemma 2.1. If the ray Rt lands at a point γ(t) ∈ J , then the ray Rdt lands at the point f (γ(t)). Furthermore, each of the d rays of the form R t+j lands at one of the d points in f −1 (γ(t)), and every point in f −1 (γ(t)) is the landing point of at least one such ray. Proof. In case that z = γ(t) is not a critical point, there exist a neighborhood N of z and a neighborhood N 0 of f (z) such that f : N → N 0 is biholomorphic by the Inverse Function Theorem. Note that d f ◦ φ−1 rei2πs = φ−1 rei2πs = φ−1 rd ei2πds , 3.

(7) then γ(ds) = f ◦ γ(s) as r & 1. Similarly, f −1 ◦ γ (ds) = γ(s). Hence R t+j lands d at w ∈ f −1 (γ(t)). In the other case, there exist a neighborhood N of z, and a neighborhood N 0 of f (z) such that f : N → N 0 is a branched covering with finite sheets. A similar argument shows that R t+j lands at w ∈ f −1 (γ(t)). d In the argument, we see that every point in f −1 (γ(t)) has such a ray landing at it. Lemma 2.2. If a periodic ray lands at z0 , then only finite rays land at z0 , and these rays are all periodic of the same period. Proof. Here we consider the rays case by case, the first for the fixed rays, the second for the periodic rays. j Assume Rt is a fixed ray, say t = d−1 , 0 ≤ j < d − 1. Clearly, z0 is a fixed point of f . Let X = {x ∈ R/Z |Rx lands at z0 }. Assume that x ∈ X, f maps a neighborhood of z0 diffeomorphically onto a neighborhood of z0 , preserving the cyclic order of the rays landing at z0 , then the d-map α 7→ dα (mod Z) is injective and preserves the cyclic order. Now, x 6≡ t (mod Z) implies for integers k, xk 6≡ t (mod Z), any nonnegative. ∞ then we can construct a sequence xk = dk x (mod Z) k=0 . Neither x1 > x0 nor x1 < x0 hold, hence x is a fixed point of d-map; otherwise, xk → xˆ which is a fixed point of d-map but there are only repelling fixed ones. Hence X is the set consisting of fixed points of d-map. More precisely, these rays are also fixed rays. Assume Rt is a periodic ray with period p > 1. Note that Rt is a fixed ray of f p , then the same argument shows the conclusion. Standing Hypothesis 2.2. From Theorem 2.1 to Theorem 2.3, we assume ψ : D → ˆ is a simply connected open set. U is a conformal isomorphism and U ⊂ C We refer the proof of Theorem 2.1 to p. 177 of [J]. Theorem 2.1 ([J], p. 177). Suppose ψ : D → U is a conformalisomorphism and. ˆ is a simply connected open set. For each eiθ ∈ ∂D, reiθ |0 ≤ r < 1 U ⊂ C maps under ψ to a curve of finite spherical length in U . In particular, the radial limit lim ψ reiθ ∈ ∂U exists for almost every θ. r%1. However, if we fix any particular point u0 ∈ ∂U , then the set θ such that this radial limit is equal to u0 has Lebesgue measure zero. 4.

(8) Corollary 2.1. If θ1 6= θ2 land at the same point u0 ∈ ∂U , then u0 disconnects ∂U . Proof. Note that θ1 , θ2 divide the unit circle into two open arcs A, B with nonzero length. It follows from Theorem 2.1 that there are a ∈ A, b ∈ B such that Ra , Rb landing at different points of ∂U . Hence u0 does disconnect ∂U . We refer the proof of Theorem 2.2 to p. 183 of [J]. Theorem 2.2 ([J], p. 183). ψ : D → U can be extended to a continuous map ¯ → U¯ if and only if ∂U is locally connected if and only if C ˆ \ U is locally ψ¯ : D connected. Theorem 2.3. If ∂U is a Jordan curve, then ψ : D → U extends to a homeomor¯ → U¯ . phism ψ¯ : D Proof. It suffices to show that ψ¯ is injective by Theorem 2.2. Otherwise, there are two distinct points θ1 , θ2 landing at the same point u0 ∈ ∂U which implies that u0 disconnects ∂U that is a Jordan curve, which is a contradiction. Theorem 2.4. Suppose X is a locally connected and compact space and f : X → Y is continuous and onto, where Y is a Haussdorff space, then Y is locally connected and compact. Proof. Since f is continuous from a compact space X onto Y , then Y is compact. It remains to show that Y is locally connected. Suppose y ∈ Y , V is an open neighborhood of y. Clearly, f −1 (V ) is open in X. For all x ∈ f −1 (y), there exists [ a connected open neighborhood Vx of x contained −1 in f (V ). It follows that f (Vx ) is connected. x∈f −1 (y). Note that f (X \ Vx ) is closed in Y for each x in f −1 (y) and [ \ \ Y \ f (Vx ) = (Y \ f (Vx )) ⊂ f (X \ Vx ) , x∈f −1 (y). then. \. x∈f −1 (y). x∈f −1 (y). f (X \ Vx ) is closed and. x∈f −1 (y). \. y∈Y \. f (X \ Vx ) ⊂. x∈f −1 (y). [ x∈f −1 (y). This completes our proof. 5. f (Vx ) ⊆ V..

(9) Theorem 2.5. For almost every t in R/Z, the ray Rt has a well-defined landing point γ(t) ∈ J . Furthermore, for any fixed z0 ∈ J , X0 = {t ∈ R/Z |γ(t) = z0 } has measure zero. ˆ \ D → D be difined as ϕ(w) = 1 ∀w ∈ C ˆ \ D. Now let Proof. Let ϕ : C w ˆ \ K as ψ = φ−1 ◦ ϕ−1 and apply Theorem 2.1. For almost t in R/Z, ψ:D→C lim φ−1 ◦ ϕ−1 (re−i2πt ) ∈ J . Furthermore, X0 is the set consisting of t in R/Z. r%1. whose radial limit equals z0 . This completes our proof. Theorem 2.6 ([J], p. 191). Given a polynomial f with connected J , the following are equivalent: (1) γ(t) is continuous. (2) J is locally connected. (3) K is locally connected. ˆ D ¯ → C\K ˆ (4) φ−1 : C\ can be extended continuously over ∂D, and the extension 2πit carries each e ∈ ∂D to γ(t) ∈ J . Proof. Consider ψ in the proof of Theorem 2.5 and Theorem 2.2, and this completes the proof the equivalence of the latter three statements. It is straightforward that the fourth statement implies the first one. It remains to prove the converse is true. γ (R/Z) is compact and locally connected by Theorem 2.4. Without loss of generality, say γ(0) ∈ J , then O− (γ(0)) ⊆ γ (R/Z). Hence J = O− (γ(0)) ⊆ γ (R/Z) ⊆ J implies γ (R/Z) = J . Corollary 2.2. J is a Jordan curve if and only if γ : R/Z → J homeomorphically. Proof. The necessary condition is a direct consequence of Theorem 2.3 and the sufficient one is the verification of what a Jordan curve is. We refer the proof of Theorem 2.3 to p. 168 of [J], which is called the Snail lemma.. 6.

(10) Lemma 2.3 ([J],p. 168). Suppose g (z) = λz + a2 z 2 + a3 z 3 + · · · . A path p : [0, ∞) in V \ {0} where V is neighborhood of 0 converges to 0 and g (p (t)) = p (t + 1) , ∀ t ≥ 0,. (2.1). then either |λ| < 1 or λ = 1. Corollary 2.3. If (2.1) is replaced with g (p (t)) = p (t − 1) , ∀ t ≥ 1,. (2.2). then either |λ| > 1 or λ = 1. Proof. Note that the orbit · · · 7→ p(2) 7→ p(1) 7→ p(0) is repelled by the origin, then λ 6= 0. We have the holomorphic inverse g −1 : W → V of f where W ⊂ f (V ) is a neighborhood of 0 with the derivative λ−1 at 0. From (2.2), we see that g −1 (p(t)) = p (t + 1) , ∀ t sufficiently large. Apply Lemma 2.3 to g −1 on W , we see either λ = 1 or |λ| > 1. Lemma 2.4. If a fixed ray Rt lands at z0 , then z0 is either a repelling or parabolic fixed point. Proof. It is straightforward that z0 is a fixed point. ˆ ˆ Since φ : C\K → C\D is a conformally isomorphism and the Green’s function + ˆ G : C \ K → R , then ˆ \ K 3 φ(z) = exp(es + 2πit), ∀s ∈ R, es > 0, exp(es ) > 1, ∃!z ∈ C (2.3) i.e., z ∈ Rt . ˆ \ K as follows: Define p : R → C p(s) 7→ z 3 φ(z) = exp(es + 2πit) ⇒ G(z) = es . Hence p is well-defined and continuous by (2.3). Also, note G (f (p(s))) = log |φ (p(s))|n = nG (p(s)) , then we see that p (s + log n) = f (p(s)) after taking the logarithm on both sides. Let p0 : [k, ∞) → V, p0 (t) = p(−t) where V contains p ((−∞, −k]) , for some k > 0. Hence,by Corollary 2.3 , either |f 0 (z0 )| > 1 or f 0 (z0 ) = 1. 7.

(11) Finally, we state Theorem 2.7 and Theorem 2.8 without proof. Their proofs are in Chapter 18 of [J]. They state that rational rays must land and repelling and parabolic points are landing points. Theorem 2.7 ([J],p. 195). Every periodic external ray lands at a either repelling or parabolic periodic point. Moreover, a rational but non-periodic ray lands at a strictly eventually periodic point. Theorem 2.8 ([J],p. 195). Every repelling or parabolic periodic point is the landing point of at least one periodic ray.. 8.

(12) 3. Rotation Numbers. Definition 3.1. ψ : R/Z → R/Z is called a monotone degree one circle map if there is a lift Ψ : R → R, monotone, continuous and Ψ(u + 1) = Ψ(u) + 1, u ∈ R. Proposition 3.1. Given a monotone degree one circle map ψ : R/Z → R/Z , there exists a unique lift Ψ : R → R up to addition of an integer constant. Theorem 3.1. Given a lift Ψ of a monotone degree one circle map ψ and a real number u ∈ R, the limit Ψn (u) − u (3.1) lim n→∞ n exists. This limit is independent of u. Proof. For any two real numbers u, v, there exists an integer k such that such that u ≤ v + k < u + 1. Apply the monotonicity of Ψ and use the induction, then Ψn (u) ≤ Ψn (v + k) < Ψn (u + 1). Substracting it with u + 1, we see that Ψn (u) − u − 1 < Ψn (v) − v < Ψn (u) − u + 1. Hence, ∀u, v ∈ R, ∀n ∈ N, Ψn (u) − u − 1 < Ψn (v) − v < Ψn (u) − u + 1.. (3.2). Uniqueness follows from (3.2). It remains to show the existence of the limit. Assume that u is a real number. For any two natural numbers p, q, set v = Ψq (u). From (3.2), we see that Ψp+q (u) < Ψq (u) + Ψp (u) − u + 1,. (3.3). and Ψ2p (u) < 2Ψp (u) − u + 1 as p = q. Inductively, we have Ψkp (u) − u < k (Ψp (u) − u) + (k − 1). (3.4). for each postive integer k. At last, for each positive integer n, there are integers k, i, k ≥ 0, 0 ≤ i < p such that n = kp + i. First, we see Ψn (u) − u < Ψkp (u) + Ψi (u) − u + 1 − u in (3.3) and deduce that n Ψ (u) − u < k (Ψp (u) − u + 1) + Ψi (u) − u via (3.4). Note that (Ψp (u) − u + 1) Ψi (u) − u Ψn (u) − u < + n p n and fix p, let n → ∞, and let p → ∞, then we see that lim sup n→∞. F n (u) − u F p (u) − u ≤ lim inf . p→∞ n p. 9. (3.5).

(13) The following definition is well-defined via Theorem 3.1. Definition 3.2. Given a lift Ψ of a monotone degree one circle map ψ, we define Ψn (u) − u , for some u ∈ R. n→∞ n The limit is called the translation number of Ψ. Trans (Ψ) = lim. (3.6). The following two propositions are easy calculation rules for the translation number. Proposition 3.2. For any lifts Ψ, Ψ + k of a monotone degree one circle map ψ where k ∈ Z, Trans (Ψ + k) = Trans (Ψ) + k. (3.7) Proposition 3.3. For any lift Ψ of a monotone degree one circle map ψ, assume m ∈ N, k ∈ Z Trans (Ψm + k) = m Trans (Ψ) + k. (3.8) Proof. Choose u ∈ R. Note that, for any positive number n, (Ψm + k)n (u) − u = (Ψm )n (u) + nk − u, then we may take the limit after dividing the identity by n. The following definition is well-defined via Proposition 3.3. Definition 3.3. Given a monotone degree one circle map ψ : R/Z → R/Z, define the rotation number of ψ to be ρ(ψ) = Trans (Ψ). mod Z , where Ψ is a lift of ψ.. (3.9). Theorem 3.2. Assume that ψ is a monotone degree one circle map. The rotation number of ψ is rational, say ρ(ψ) = pq in reduced form, if and only if ψ has a periodic point with period q. Furthermore, every orbit under ψ is either periodic or tends asymptotically to a one-sided attracting or attracting periodic orbit. Proof. It is clear that the sufficient condition holds. For the necessary condition, given ρ(ψ) = pq , let Ψ : R → R with Trans(Ψ) = pq and G(t) = Ψq (t) − p implies Trans(G) = 0. Consider the value G(0), there are three possiblities G(0) = 0, G(0) > 0, and G(0) < 0. The first condition G(0) = 0 implies that 0 is a fixed point of G. For the second possibility, we have 0 < G(0) < G2 (0) < · · · ≤ 1 which achieves our proof. The same reason for the third possibility. In fact, we can have G(t) = t, t < G(t) < G2 (t) < · · · ≤ t + 1, or t > G(t) > G2 (t) > · · · ≥ t − 1 in this fashion. This finishes our last assertion. 10.

(14) Definition 3.4. A d-fold covering map fd : S 1 → S 1 where fd (θ) = dθ mod Z is given. Suppose Θ = {0 ≤ θ0 < θ1 < · · · < θn−1 < 1} where 0 ≤ m ≤ n satisfies ∀i = 0, 1, 2, . . . , n, fd (θi ) = θ(i+m). mod n ,. then Θ is called a degree d m -rotation set. n Also, if m = kp, n = kq where k ∈ N, p, q ∈ Z , then Θ will be composed of k q-cycles. Each q-cycle is called a degree d pq -rotation cycle. And the rotation number of Θ is pq . Lemma 3.1. ∀q ≥ 2, the q-cycles under fd are in one-to-one correspondence with q-orbits consisting of numbers with base d numeral system in (0, 1) under d-shift. Proof. For each i = 0, 1, . . . , d−1, let Ai =. i i+1 , d d. . . Define γ :. d−1 [. Ai → {0, 1, . . . , d − 1}. i=0. as follows: θ ∈ Ai 7→ i. Suppose {θ0 < θ1 < · · · < θq−1 } is a q-cycle under fd , then, for any j = 0, 1, . . . , q − 1, let aj = 0.γ(θj )γ (fd (θj )) . . . γ fdq−1 (θj ) . In fact, ! γ fdq−1 (θj ) γ(θj ) γ (fd (θj )) dq aj = + + · · · + · . d d2 dq dq − 1. 1 Definition 3.5. 0 < θ1 < · · · < θn−1 } ⊆ S . ∀j = 1, 2, . . . , d −

(15) Let Θ = j {θ . Clearly, s1 ≤ s2 ≤ · · · ≤ sd−1 = n. Then 1, sj = # θi

(16) θi ∈ 0, d−1 (s1 , s2 , . . . , sd−1 ) is called the degree d deployment sequence of Θ.. Definition 3.6. Let θ ∈ S 1 . θ is advancing if fd (θ) > θ; otherwise, θ is retreating. Lemma 3.2. A degree d rotation set is completely determined by its rotation number p together with its deployment sequence s1 ≤ s2 ≤ · · · ≤ sd−1 = kq. q j−1 j Proof. For each j = 1, 2, . . . , d − 1, let Uj = d−1 . , d−1 Uj,adv = Uj,adv =. j−1 j , d−1 d. . j j , d d−1. . ⊂. 11. ⊂. j−1 j , d d. . j j+1 , d d. . (3.10). (3.11).

(17) Suppose Θ = {θ0 < θ1 < · · · < θkq−1 } ⊂ S 1 . Clearly, we see that fd advances θ0 , θ1 , . . . , θkq−kp−1 and retreats the others. For each element in Θ, follow the proof of Lemma 3.1 and (3.10) and (3.11), and we may write down the cyclic fractions with base d. Definition 3.7. Let Θ = {θ0 ≤ θ1 ≤ · · · ≤ θkq−1 } ⊂ S 1 be points which are not fixed by fd . Then the complement of Θ is composed of kq arcs joining θj and θj+1 mod kq which is denoted by Aj whose length `(Aj ) = θj+1 − θj for j = 0, 1,· · · ,

(18) kq − 2 and `(Akq−1 ) = 1 + θ0 − θkq−1 , and whose weight ω(Aj ) = i

(19) i ∈ A . In the case Θ = {θ0 < θ1 < · · · < θkq−1 }, we may have the # d−1 j d−1 length of every arc is nonzero. Lemma 3.3. Let Θ = {θ0 < θ1 < · · · < θkq−1 } ⊂ S 1 be a degree d rotation set with rotation number pq and complementary arcs A0 , A1 , . . . Akq−1 . Then d`(Ai ) = ` (Ai+kp. mod kq ). ∼ =. Also, fd : Ai → Ai+kp. mod kq. + ω(Ai ) ∀i = 0, 1, . . . , kq − 1.. (3.12). if and only if ω(Ai ) = 0.. Proof. Note that d`(Ai ) = = = =. d(θi+1 mod kq − θi ) fd (θi+1 mod kq ) − fd (θi ) + z θ(i+1+kp) mod kq − θ(i+kp) mod kq + z `(A(i+kp) mod kq ) + z. for some z ∈ Z and z = ω(Ai ). Lemma 3.4. For each i = 0, 1, 2, . . . , d − 1, (d − 1) `(Ai ) + `(Ai+k ) + · · · + `(Ai+k(q−1) ) = ω(Ai ) + ω(Ai+k ) + · · · + ω(Ai+k(q−1) ).. (3.13). Given the rotation set Θ = {θ0 < θ1 < · · · < θkq−1 }, this equation is greater than zero. Therefore there is a nonnegative number z such that ω(Ai+kz ) > 0. If ω(Ai ) = 0, then Ai ∼ = fd (Ai ); otherwise, ω(Ai ) > 0. Proof. Given this i, apply (3.12) to i, i + k, i + 2k, . . . , i + k(q − 1). This completes our proof. 12.

(20) Lemma 3.5. A sequence 0 ≤ s1 ≤ s2 ≤ · · · ≤ sd−1 = kq is realized by a degree d rotation set if and only if {[0] , [1] , . . . [k − 1]} = {[s1 ] , [s2 ] , . . . [sd−1 ]}. Proof. Because (3.13) holds, it is positive if and only if one of the length of some arc is positive which means the strict inequality holds. Corollary 3.1. k ≤ d−1. Furthermore, a degree d rotation set with rotation number p has at most (d − 1)q points. q Proof. Otherwise, the deployment sequence cannot be realized . Theorem 3.3. A degree d rotatation set is uniquely determined by its rotation number pq and its deployment sequence 0 ≤ s1 ≤ s2 ≤ · · · ≤ sd−1 = kq. Conversely, a indivisible fraction pq and candidate deployment sequence 0 ≤ s1 ≤ s2 ≤ · · · ≤ sd−1 = kq determine a rotation set if and only if {[0] , [1] , . . . [k − 1]} = {[s1 ] , [s2 ] , . . . [sd−1 ]} Proof. This is the consequence of Lemma 3.2 and Lemma 3.5. Theorem 3.4. fd has d+q−2 rotation cycles with rotation number pq . q Proof. A deployment sequence 0 ≤ s1 ≤ s2 ≤ · · · ≤ sd−1 = q represents a way of the permutation on q triangles 4, 4, . . . , 4 and d − 2 sticks |, |, . . . , |. This completes the proof. We use Maple to find out all possible degree 3 5-rotation cycles, see Appendix. In fact, we demostrate the algorithm in Algorithm 6.1. One may follow the procedure to find out what he needs.. 13.

(21) 4. Critical Portraits. In this section, we assume Standing Hypothesis 2.1 and the following hypothesis. Standing Hypothesis 4.1. Assume f is a monic polynomial of degree d ≥ 2 and every critical point of f must be a landing point of at least one external ray. Definition 4.1. Two subsets A, B ⊂ R/Z are unlinked if there are two disjoint components U, V of R/Z such that A ⊂ U, B ⊂ V . Definition 4.2. A critical portrait of f is a finite set Θ = {Θ1 , Θ2 , . . . , Θs } consisting of those finite types of all critical points of f satisfying three conditions: (1) Every external ray Rt , where t ∈ Θj for some j, must land at the corresponding critical point ωj . (2) For any two x, y ∈ Θj , x = y (mod d1 ). (3) For any j = 1, 2, . . . , s, #Θj = µj + 1 where µj is the multiplicity of ωj . From the Böttcher conformal isomorphism, we see that the Θj are disjoint and pairwise unlinked. And this contributes to our next step. Lemma 4.1. Given a critical portrait Θ = {Θ1 , Θ2 , . . . , Θs } of f , the following conditions hold: C1 The Θj are disjoint and pairwise unlinked. C2 For any two x, y ∈ Θj , x = y (mod d1 ). C3 For any j = 1, 2, . . . , s, #Θj ≥ 2 and. s X. (#Θj − 1) = d − 1.. j=1. Moreover, we call any finite collection of finite sets of angles which satisfies these three conditions a formal critical portrait. Before we proceed to the Fisher’s Main Theorem, we illustrate how to write down the itinerary of t. Given a formal critical portrait Θ = {Θ1 , Θ2 , . . . , Θs }, we draw all the external rays and the landing points. This divide the complex plane d non-overlapping S into S S regions, say T Ω1 , Ω2 , . . . , Ωd , and the Julia set J = J1 J2 · · · Jd where Jj = J Ωj for j = 1, 2, . . . , d. 14.

(22) S We define an equivalence relation ∼ on R/Z \ sj=1 Θj by t ∼ t0 if and only if {t, t0 } , Θ1 , . . . , Θs are pairwise unlinked. It is straightforward that we get the equivalence classes L1 , L2 , . . . , Ld . By the itinerary of t, we mean a sequence, p1 , p2 , . . . , of positive integers between 1 and d where fdn (t) = dn t mod Z ∈ Lpn for each n. Lemma 4.2. Each Li or even Li has length d1 . Proof. For each Li , the closure Li has at most finite points which belong to. s [. Θj. j=1. P more than Li . From C2, ` (Li ) = kdi and di=1 ` (Li ) = 1 implies k1 = k2 = · · · = kd = 1. Moreover, only one Li is disconnected. Lemma 4.3. For each Li , there is one and only one continuous monotone degree one circle map φi : R/Z → R/Z where φi (t) ≡ dt (mod Z) for any t ∈ Li and φi is constant on every component of R/Z \ Li . Proof. Since ` Li = d1 and the boundary points of Li satisfies C2, then we can define the φi by drawing a line with slope d and horizontal lines between the adjacent critical angles outside Li . This construction is unique. There are two more necessarty conditions for a critical portrait: C4 For any θ ∈. Ss. j=1. Θj , θ is strictly preperiodic.. C5 θ ∈ Θi , θ0 ∈ Θj with i 6= j, then θ and θ0 do not have any itinerary in common. In Section §4 of [P], we see an example for a critical portrait of a polynomial with C1-C3 but not C4. That is why we do not put the five properties together. Furthermore, Fisher’s work asserts that a formal critical portrait satisfying C4 is realized by a polynomial whose Julia set is connected if and only if the portrait satisfies C5. We will state the following two conclusions Lemma 4.4 [GM] and Theorem 4.1 [BFH] without proof. Lemma 4.4. A formal critical portrait satisfying C4 is given, then for any s, t ∈ S 1 , the external rays Rs , Rt lands at a common point of J if and only if for any n ≥ 0, there is pn with 1 ≤ pn ≤ d such that sn , tn ∈ Lpn . And each Jj contains a unique fixed point of f .. 15.

(23) Theorem 4.1. Suppose a formal critical portrait Θ sastifying C4 and C5 is given. Then there is one and only one polynomial f which has connected J and realizes Θ. Furthermore, f is critically pre-periodic, i.e., every orbit of critical point is strictly pre-periodic, and f has d repelling fixed points. Lemma 4.5. Assume that a formal critical portrait sastisfies C4. For each associated map φj , we have (a) ρ (φj ) is rational, say. pj . qj. (b) Every periodic point is either repelling or ultra-attracting. (c) These two kinds of periodic points alternate around the circle , and the number of orbits of each kind is between 1 and d − 1. (d) Every point of R/Z is either periodic or pre-periodic. Proof. (a) This follows from that the endpoints of Lj are strictly preperiodic under φj . (b) Clearly, every periodic point is either repelling, ultra-attracting , or repelling on one side and ultra-attracting on the other side. But the endpoints of each Lj are strictly preperiodic. This completes our second conclusion. (c) Note that Φqj − pj crosses from above the diagonal to below at every ultraattracting periodic point and it crosses from below the diagonal to above at every repelling periodic point. Note that each interval of constancy of φj has at most an orbit of one ultra-attracting periodic points. This shows the third assertion. (d) For the last assertion, clearly every point of R/Z is either periodic or not periodic. For those non-periodic points, their orbits tend asymptotically to an one-sided attracting periodic orbit by Theorem 3.2.. 16.

(24) 5. Fixed Point Portraits. In this section, we assume Standing Hypothesis 2.1. Definition 5.1. With Standing Hypothesis 2.1, given a fixed point z of f , let T = T (f, z) = {t ∈ Q/Z |Rt lands at z } denote the rational type of z. If T = ∅, then we say that z is rationally invisible; otherwise, z is called rationally visible. By the fixed point portrait of f , we mean the set consisting of the rational types of all rationally visible fixed points. Lemma 5.1. A fixed point z of f is rationally visible if and only if it is either repelling or parabolic. Proof. The sufficient condition is a direct consequence of Theorem 2.8. Assume that z is rationally visible. Pick t in T (f, z) which is periodic. There is a positive integer m such that Rt is a fixed ray of f m . It follows that z is a repelling or parabolic fixed point of f m from Lemma 2.4. This completes the proof of the necessary condition. Lemma 5.2. The rational type of a rationally visible fixed point z of f is a ratation set of S 1 . Proof. Clearly, there is at least a periodic ray landing at z. It follows that T (f, z) is finite from Lemma 2.2. In addition, they have common period, say q. Note that all q-cycles of f are rotation cycles with rotation numbers pq and, from the orbit of smallest, the circle is divided into q sectors, and the proof is completed. Lemma 5.3. Let f be the monic polynomial obtained from Theorem 4.1 and z1 , z2 , . . . , zd be the repelling fixed points of f with zj ∈ Jj . Then T (f, zj ) is the set consisting of all the repelling periodic points of the associated map φj . Proof. From Lemma 4.4 and Theorem 4.1, we may rearrangement the d repelling fixed points such that the itinerary jjjjj · · · of zj ∈ Jj . Hence T (f, zj ) is the set consisting of all periodic points of φj on Lj which are repelling. Theorem 5.1. Let P = {T1 , T2 , . . . , Tk } be the fixed point portrait of f , then the following four conditions hold: P1 Every Tj is a rational rotation set. P2 The Tj are disjoint and pairwise unlinked. 17.

(25) P3. [ ρ(Tj )=0. Tj = 0,. d−2 1 ,..., . d−1 d−1. P4 i 6= j with ρ (Ti ) 6= 0 and ρ (Tj ) 6= 0, then there exists a Tl with ρ (Tl ) = 0 such that Tl seperates Ti , Tj . P1 is a direct consequence of Lemma 5.2. It is straightforward that P2 and P3 holds. As for P4, we omit the proof and refer to [GM]. Now, we consider the converse statement with exactly d nonempty subsets of Q/Z which are called a candidate of fixed point portrait. Is there a polynomial of degree d whose fixed point portrait is the given one? d−2 0 , . . . , ,T We start with the so-called elementary fixed point portrait P0 = d−1 d−1 and illustrate how a critical portrait is generated from P0 . Like in Section 3, say T = {t0 < t1 < · · · < tkq−1 } with rotation number pq . For . those i with ω(Ai ) > 0, choose θi in Ai such that θi + hd |h = 0, 1, . . . , ω(Ai ) is contained in Ai and θi belongs to the backward orbit of a fixed point in Ai . Furthermore, we require (ti , θi ] is disjoint from any pq -rotation cycles and any points of the j j form d−1 − hd where d−1 ∈ Ai and 0 ≤ h ≤ ω(Ai ). List those i whose weight is larger than one and relabel them as follows: i1 < i2 < · · · < im . Let Θj denote the angles obtained from Aij . We remark that there might be many choices, and we might not have uniqueness. Lemma 5.4. The portrait Θ = {Θ1 , Θ2 , . . . , Θm } as constructed above sastisfies C1-C5, and is realized by a unique critically preperiodic polynomial f of degree d by Theorem 4.1. Proof. C1 follows since each Aij is non-overlapping, C2,C3 and C4 follows by the form we pick from Aij , and C5 follows from the fact that each equivalence class contains precisely one fixed point type of P0 . Theorem 5.2. A collection P = {T1 , T2 , . . . , Td } of exactly d nonempty subsets of Q/Z is the fixed point portrait of some critically pre-periodic polynomial of degree d if and only if the following four conditions P1-P4 in Theorem 5.1 hold. Proof. The necessary condition is done by Theorem 5.1. d−2 0 In the elementary case, assume P = , . . . , d−1 , T , we have a critd−1 ically pre-periodic polynomial f of degree d in Lemma 5.4. It turns to examine the fixed point portrait is indeed it follows the fixed point por d−2the given one. Again, 0 0 0 0 trait P = , . . . , d−1 , T where T ⊃ T from the fact each equivalence d−1 class contains precisely an element of P. 18.

(26) Assume T 0 6= T , we may have t0 ∈ T 0 \ T , then let A0i denote the rightmost arc of the arc Ai generated by T with `(A0i ) < d1 . Note that fdq (A0i ) = A0i and ` (fdq (A0i )) > `(A0i ), then this makes a contradiction. For the general case, say P = {T1 , T2 , . . . , Td }, we still define the equivalence d [ relation ∼ on R/Z \ Ti where t ∼ s means ∀i = 1, 2, . . . , d, t, s belong to i=1. the same component of R/Z \ Ti . Assume U1 , U2 , . . . , Um are equivalent classes of d [ R/Z \ Ti . i=1. From P4, there is at most one Ti with nonzero rotation number intersecting the boundary of each Uj , and there is at least one or otherwise, we have fixed point types less than d. From the discussion above, we see that each Uj is an arc of R/Z \ Ti or the remaining set removed a finite number of intervals with endpoints fixed points from that arc for corresponding i, including degenerated intervals. Also, define the weight of Uj , denoted by ω(Uj ), to be the number of missing intervals. As the proof in the elementary case, we construct a critical portrait Θ = {Θ1 , Θ2 , . . . , Θs } from these i where ω(Uj ) > 0. We see that Θ satisfies C1-C5 and we get a critically pre-periodic polynomial f to realize Θ. Each Uj has a well-defined rotation number, and the fixed point portrait P 0 = {T10 , T20 , . . . , Td0 } has the property Ti ⊆ Ti0 . As the proof in the elementary case, we have Ti = Ti0 .. 19.

(27) 6. Classification of Degree 3 Fixed Point Portraits. We consider what fixed point portraits generated from critical portraits with C1-C5 as d = 3. P Since sj=1 (#Θj − 1) = 2, then there are two possible cases that either Θ has only one element or Θ has two elements.. The former case: Say, Θ = θ, θ + 13 , θ + 23 nwith θ ∈ 0, 31 . o The resulted fixed point portrait will be T = {0} , 21 , T pq , (s1 , q) where p is rational and strictly between 0 and 1. Moreover, its deployment sequence is q either (0, q) or (q, q). θ P n o 1 p 1 0<θ< 6 {0} , 2 , T q , (q, q) n o 1 p 1 1 <θ< 3 {0} , 2 , T q , (0, q) 6 The latter case:1 Say, Θ = {Θ 1 , Θ2 }. . . 1 A = α, α + 3 (α ∈ 0, 3 ), B = β, β + 31 (β ∈ 13 , 23 ), C = γ, γ − 23 (γ ∈ 23 , 1 ). Θ1 and Θ2 can be composed of two of A, B, C. Θ = {A, B} (Implicitly, 0 < α < α + 31n < β < β + 13 < 23 .) o 1 p p0 1 1 2 0 If 0 < α < 6 and 2 < β < 3 , then T = 0, 2 , T q ; (q, q) , T q0 ; (0, q ) . n o If 0 < α < 61 and α + 13 < β < 21 , then T = {0} , 21 , T pq ; (q, q) . n o 1 p 1 1 If 6 < α < 3 , then T = {0} , 2 , T q ; (0, q) . α. β. 0<α<. 1 6. 0<α<. 1 6. 1 6. 1 3. <α<. 1 2. <β<. n. 2 3. α+. 1 3. <β<. 1 2. α+. 1 3. <β<. 2 3. 0,. 1 2. ,T. . P. p , (q, q) q. T. . o. p0 , (0, q 0 ) q0. n o 1 p {0} , 2 , T q , (q, q) n o {0} , 21 , T pq , (0, q). Θ = {A, C} (Implicitly,n0 < γ − 23 < α < α + 13o < 32 < γ < 1.) . If 0 < α < 16 , then T = {0} , 12 , T pq ; (q, q) . n o If 16 < α < 13 , then T = {0} , 21 , T pq ; (s1 , kq) where 0 ≤ s1 ≤ q, as k = 1 ; 0 ≤ s1 ≤ 2q and s1 is odd as k = 2.. 20.

(28) P n o 1 p 0<α< Independent {0} , 2 , T q , (q, q) n o 1 p 1 1 < α < 3 Independent {0} , 2 , T q , (s1 , kq) 6 where 0 ≤ s1 ≤ q, as k = 1; 0 ≤ s1 ≤ 2q and s1 is odd as k = 2. + 13 < γ < 1.) Θ = {B, C} (Implicitly,n0 < γ − 32 < 13< β < βo If 12 < β < 23 , then T = {0} , 12 , T pq ; (0, q) . n o If 31 < β < 12 , then T = {0} , 21 , T pq ; (s1 , kq) where 0 ≤ s1 ≤ q, as k = 1 ; 0 ≤ s1 ≤ 2q and s1 is odd as k = 2. β γ P n o 1 p 1 1 Independent {0} , < β < , T , (s , kq) 1 3 2 2 n q o . p 1 1 2 < β < Independent {0} , , T , (0, q) 2 3 2 q where 0 ≤ s1 ≤ q, as k = 1; 0 ≤ s1 ≤ 2q and s1 is odd as k = 2. Now we will demonstrate how to use Maple to obtain pq -rotation cycles. We provide an algorithm as follows: α. γ. 1 6. 21.

(29) Input: Set degree d,period q, c = dq − 1, arrays v, w with length c, q, resp.. Output: Classify All possible pq -rotation cycles with periods. Let f : θ 7−→ dθ mod c, x = 2c ,and v be the zero vector in Rc . For each i Let w be an undefined vector in Rq with first coordinate i. if v[i] = 0 then set j = 1, v[i] = 1, w[1] = i For each j w[j] = f (w[j − 1]) v[w[j]] = 1 if w[1] < w[2] < · · · < w[q] then set p = 1 end elif w[1] < w[ q+3 ] < · · · < w[ q+1 ] then 2 2 set p = 2 end .. . elif w[1] < w[q] < · · · < w[2] then set p = q − 1 end else set p =None, end print w, p a = [w[1], w[2], . . . , w[q]] ;z =sort(a); if p =None then print p end elif z[1] < x, z[2] < x, . . . , z[q] < x then set s = q end elif z[1] < x, z[2] < x, . . . , z[q − 1] < x, z[q] > x then set s = q − 1 end .. . elif z[1] < x, z[2] > x, . . . , z[q] > x then set s = 1 end else set s = 0, end print z, s end Algorithm 6.1: The algorithm for degree d pq -rotation cycles with Maple.

(30) The following code is executed in Maple. Program 6.1 A program for degree 3 p5 -rotation cycles with Maple d := 3; q := 5; c := dˆq-1; x:=c/2; f := proc (x) options operator, arrow; ‘mod‘(d*x, c) end v := array(1 .. c); v[1] := 0; for i from 2 to c do v[i] := v[i-1] end do; for i to c-1 do w := array(1 .. q); if v[i] = 0 then j := 1;w[1] := i; v[i] := 1; for j from 2 to q do w[j] := f(w[j-1]); v[w[j]] := 1 end if w[1] < w[2] < w[3] < w[4] < w[5] ,then p := 1; elif w[1] < w[4] < w[2] < w[5] < w[3],then p := 2; elif w[1] < w[3] < w[5] < w[2] < w[4],then p := 3; elif w[1] < w[5] < w[4] < w[3] < w[2],then p := 4; elif w[1] = w[2 = w[3]= w[4]= w[5],then p := 5; else p := None; end if: print(w,’p’=p); a:=[ w[1], w[2], w[3], w[4], w[5]]; z:=sort(a); if p=None then print(None); elif z[1]<x and z[2]<x and z[3]<x and z[4]<x and z[5]<x, elif z[1]<x and z[2]<x and z[3]<x and z[4]<x and z[5]>x, elif z[1]<x and z[2]<x and z[3]<x and z[4]>x and z[5]>x, elif z[1]<x and z[2]<x and z[3]>x and z[4]>x and z[5]>x, elif z[1]<x and z[2]>x and z[3]>x and z[4]>x and z[5]>x, else s := 0; end if: print(z,’deployment sequence’=(s,5)); end if end do. proc;. do;. then then then then then. s s s s s. := := := := :=. 5; 4; 3; 2; 1;.

(31) Appendix This is the table of degree 3 p5 -rotation cyles. RN\ DS (0,5) (1,5) 1/5 {122, 124, 130, 148, 202} {41, 123, 127, 139, 175} 2/5 {131, 149, 151, 205, 211} {50, 140, 150, 178, 208} 3/5 {152, 158, 212, 214, 232} {71, 155, 185, 213, 223} 4/5 {161, 215, 233, 239, 241} {80, 188, 224, 236, 240} RN\ DS (3,5) (4,5) 1/5 {5, 15, 45, 135, 163} {2, 6, 18, 54, 162} 2/5 {20, 56, 60, 168, 180} {19, 29, 57, 87, 171} 3/5 {61, 65, 101, 183, 195} {34, 64, 92, 102, 192} 4/5 {76, 106, 116, 200, 228} {67, 103, 115, 119, 201} RN: Rotation Number; DS: Deployment Sequence.. (2,5) {14, 42, 126, 136, 166} {47, 59, 141, 177, 181} {62, 74, 182, 186, 222} {79, 107, 197, 227, 237} (5,5) {1, 3, 9, 27, 81} {10, 28, 30, 84, 90} {31, 37, 91, 93, 111} {40, 94, 112, 118, 120}. References [AFB] Alan F. Beardon, Iteration of Rational Functions, Springer (1991). [BFH] B. Bielefeld, Y. Fisher, J. H. Hubbard, The dynamic of a kinetic activatorinhibitor system, Journal of the American Mathematical Society 5 (1992), 721–762. [CR] Clark Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC, 1995, 49-52. [D] Robert L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison Wesley 2nd Ed. (1991). [G] Lisa R. Goldberg, Fixed points of polynomial maps. Part I. Rotation subsets ´ Norm. Sup. 4 (1992), 679-685. of the circles, Ann. scient. Ec. [GM] Lisa R. Goldberg and John Milnor, Fixed points of polynomial maps. Part II. ´ Norm. Sup. 4 (1993), 51-98. Fixed point portraits, Ann. scient. Ec. [J]. John Milnor, Dynamics in One Complex Variable, Princeton University Press 3rd Ed. (2006).. 24.

(32) [P] Alfredo Poirier, On Post Critically Finite Polynomials. Part One: Critical Portraits, (1993), 11-15.. 25.

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