Wandering Domains in Spherically Complete Fields

Transcription

1 Wandering Domains in Spherically Complete Fields Eugenio Trucco Universidad Austral de Chile p-adic and Complex Dynamics, ICERM

2 Complex Polynomial Dynamics To understand the dynamics of a polynomial P C[z] of degree 2 we split the complex plain in two disjoint sets. That is C = J (P) F(P). The Julia set J (P) is a compact set where the dynamics is chaotic. The Fatou set F(P) is an open set where the dynamics is regular.

3 Fatou Components If U F(P) is a connected component of the Fatou set (Fatou component), then P(U) is itself a Fatou component. Classification Theorem (Fatou, Julia, Siegel, Herman) There exist only 3 types of fixed Fatou components: Attracting basins. Parabolic basins. Siegel disks.

4 Are there Wandering Fatou components? Question: Is there a Fatou component U such that P n (U) P m (U) for all n m and not in the basin of a periodic orbit?

5 Are there Wandering Fatou components? Question: Is there a Fatou component U such that P n (U) P m (U) for all n m and not in the basin of a periodic orbit? No Wandering Domains Theorem (Sullivan, 1985) Let P C[z] be a polynomial of degree 2. Then, every Fatou component is preperiodic.

6 Non-Archimedean Fields Examples C p, the completion of an algebraic closure of Q p. L, the completion of the field of formal Puiseux series with coefficients in C.

7 Non-Archimedean Fields Examples C p, the completion of an algebraic closure of Q p. L, the completion of the field of formal Puiseux series with coefficients in C. Main difference between L and C p The residue field of L is C and it has characteristic 0. The residue field of C p is F a p and it has characteristic p > 0.

8 Non-Archimedean Dynamics We will study the map P : A 1,an K A 1,an K where A 1,an K denotes the Berkovich affine line over K.

9 Non-Archimedean Dynamics We will study the map P : A 1,an K A 1,an K where A 1,an K denotes the Berkovich affine line over K. Classification Theorem (Rivera-Letelier, 2000) There exist only 3 types of fixed Fatou components: U in an immediate basin of attraction of an attracting periodic point in K. U is a component of the domain of quasi-periodicity. Every point in U belongs to a basin of attraction of a periodic point in U.

10 Question Question Are there wandering Fatou components?

11 Wandering Domains in non-archimedean fields Theorem (Benedetto, 2002) There exist infinitely many a C p such that P a (z) = (1 a)z p+1 + az p C p [z] has a wandering Fatou component.

12 Wandering Domains in non-archimedean fields Theorem (Benedetto, 2002) There exist infinitely many a C p such that P a (z) = (1 a)z p+1 + az p C p [z] has a wandering Fatou component. The parameters a are transcendentals over Q p. There exists wild ramification. That is, there exists a closed ball B C p with no critical points of P, such that the degree of the map is p. P : B P(B)

13 No Wandering Domains Theorems Theorem (Benedetto, 1998) Let K be a finite extension of Q p and let P K[z] have no recurrent wild critical points in its Julia set. Then the Fatou set of P has no wandering components. Theorem (T, 2010) Let P K[z] be a tame polynomial of degree 2. Then the Fatou set of P has no wandering components.

14 Spherically Complete Fields Definition A field is spherically complete if any nested sequence of closed balls has nonempty intersection

15 Spherically Complete Fields Definition A field is spherically complete if any nested sequence of closed balls has nonempty intersection Examples R is a spherically complete field. Q p is a spherically complete field. C p is not a spherically complete field.

16 Spherically Complete Fields Definition A field is spherically complete if any nested sequence of closed balls has nonempty intersection Examples R is a spherically complete field. Q p is a spherically complete field. C p is not a spherically complete field. Remark Any field K has a spherical completion denote by K.

17 Tame Polynomials Definition We say that P K[z] is a tame polynomial if its critical set is a finite tree.

18 Main Theorem Theorem Let α K such that α > 1. Let d 3 such that P λ = α(z 1) d (z + 1) + λ is a tame polynomial. There exists a parameter in K \ K such that P λ has a wandering Fatou component.

19 Main Theorem Theorem Let α K such that α > 1. Let d 3 such that P λ = α(z 1) d (z + 1) + λ is a tame polynomial. There exists a parameter in K \ K such that P λ has a wandering Fatou component. If char( K) = 0 then d 3. If char( K) = p > 0 then 2 < p and p d.

20 Main Theorem Theorem Let α K such that α > 1. Let d 3 such that P λ = α(z 1) d (z + 1) + λ is a tame polynomial. There exists a parameter in K \ K such that P λ has a wandering Fatou component. If char( K) = 0 then d 3. If char( K) = p > 0 then 2 < p and p d. Remark There is no wild ramification.

21 Roughly Speaking... If the field K is big enough then there exist polynomials having a wandering Fatou component.

22 Technical assumption on K Technical assumption on K K is a field of characteristic 0, algebraically closed and complete with respect to a nontrivial non-archimedean absolute value. Moreover, there exists a discrete valued subfield k K such that k a is dense in K.

23 The Berkovich Affine Line The Berkovich Line is the set of bounded multiplicative semi norms on K[T ] extending the absolute value of K. A 1,an K

24 The Berkovich Affine Line The Berkovich Line A 1,an K is the set of bounded multiplicative semi norms on K[T ] extending the absolute value of K. Type I points correspond to points in K. Type II points correspond to closed balls (r K ). Type III points correspond to irrational balls (r K ). Type IV points correspond to nested sequences of closed balls with empty intersection.

25 Action of P in A 1,an K Extension of P If x B A 1,an K the P(x B ) = x P(B). Local degree The local degree of P at x B is the degree of the map P : B P(B). In our case, for tame polynomials, the local degree at a point x B coincides with 1 + multiplicity of ω. ω Crit(P) B

26 Fatou and Julia sets on A 1,an K The filled Julia set of P is K(P) = { x A 1,an K {P n (x)} is a compact set. }

27 Fatou and Julia sets on A 1,an K The filled Julia set of P is K(P) = { x A 1,an K {P n (x)} is a compact set. } The Julia set of P, denoted by J (P), is K(P).

28 Fatou and Julia sets on A 1,an K The filled Julia set of P is K(P) = { x A 1,an K {P n (x)} is a compact set. } The Julia set of P, denoted by J (P), is K(P). The Fatou set of P, is F(P) := A 1,an K \ J (P).

29 Fatou and Julia sets on A 1,an K The filled Julia set of P is K(P) = { x A 1,an K {P n (x)} is a compact set. } The Julia set of P, denoted by J (P), is K(P). The Fatou set of P, is F(P) := A 1,an K Theorem (Rivera-Letelier) \ J (P). The Julia set of P is a non empty, compact and totally invariant set. Moreover J (P) = J (P n ). Furthermore, J (P) is the closure of the repelling periodic points in A 1,an K.

30 Lemma 0 Lemma 0 P has a wandering Fatou component if and only if there exists a nonpreperiodic type II or type III point in its Julia set.

31 The Family Let α K such that α > 1. Let d 3 such that is a tame polynomial P λ (z) = α(z 1) d (z + 1) + λ The critical points of P λ are 1 and ω = 1 d 1+d degrees d and 2 respectively. with local We are interested in the parameters such that 1 K(P λ ) and ω escapes. There exits exactly one c J (P λ ) Crit(P λ ).

32 The level points and the convex closure of J (P) x g

33 The level points and the convex closure of J (P) x g x

34 The level points and the convex closure of J (P) P λ (x g ) x g P(x) x

35 The level points and the convex closure of J (P) P λ (x g ) x g c

36 The level points and the convex closure of J (P) P λ (x g ) x g c P(c)

37 The Tableau c

38 The Tableau c P(c)

39 The Tableau c P(c) P 2 (c)

40 The Tableau c P(c) P 2 (c)

41 The Tableau c P(c) P 2 (c)

42 The Fibonacci Tableau F

43 The Tableau Rules Proposition (Branner and Hubbard, 1992; Harris, 1999) Every tableau of a cubic polynomial satisfies the following tableau rules. Conversely, every marked grid satisfying these rules can be realized as the critical tableau for a cubic polynomial. 1 If T m,n is critical then so is T i,n for 0 i < m. 2 If T m,n is critical, then for j, i 0, j + i m, T j,n+i is critical if and only if T j,i is also critical. 3 Suppose T m,n is critical but T m+1,n is not. Suppose also that T m l,n+l is critical but that T m i,n+i, 0 < i < l, is not. If T m l+1,l is critical, then T m l+1,n+l is not. 4 If T 1,m is critical, and T 1,1 and T 2,m are not, then T 1,m+1 is critical.

44 Hyperbolic distance For x 1 x 2 A 1,an K \ K we define the hyperbolic distance between x 1, x 2 by d H (x 1, x 2 ) = log(diam(x 2 )) log(diam(x 1 )). Lemma (Rivera-Letelier, 2002) Let P K[z] be a polynomial and consider x 1 x 2 in A 1,an K \ K. If the local degree of P at y is η for all y ]x 1, x 2 [ then d H (P(x 1 ), P(x 2 )) = η d H (x 1, x 2 ).

45 Key Lemma 1

46 Key Lemma 1 Recall that c J (P λ ) Crit(P λ ). Lemma If the Fibonacci tableau is the tableau of P λ then d H (x g, c) is finite, in particular c A 1,an K \ K

47 The Family and the Parameters Let α K such that α > 1. Let d 3 such that is a tame polynomial. P λ (z) = α(z 1) d (z + 1) + λ

48 The Family and the Parameters Let α K such that α > 1. Let d 3 such that is a tame polynomial. P λ (z) = α(z 1) d (z + 1) + λ For any n 1 we consider the level n parameter set Λ n = {λ K P n λ (λ) K(P λ)}.

49 Realization F 0

50 Realization F 1

51 Realization F 2

52 Realization F 3

53 Realization F 4

54 Realization F 5

55 Realization F 6

56 Realization F 7

57 Realization F 8

58 Realization F 9

59 Realization F 10

60 Realization F 11

61 Realization F 12

62 Key Lemma 2 Lemma (Level n parameters) Let n > 0. There exists a parameter λ n Λ n K such that the level n tableau of P λn is F n.

63 Key Lemma 3 P-D Lemma Let n Λ n be a level n parameter ball. Then for any λ n. L λ n(λ) = n

64 The Final Ingredient Structural Theorem, (T, 2010) Let P K[z] be a tame polynomial of degree 2. Then one of the following statements hold: 1 J (P) K. 2 There exist finitely many type II periodic points x 1,..., x m contained in A 1,an K \ K such that J (P) \ K = GO(x 1 ) GO(x m ), where GO(x) := {y A 1,an K P n (y) = x for some n N}.

65 Main Theorem Theorem Let α K such that α > 1. Let d 3 such that P λ = α(z 1) d (z + 1) + λ is a tame polynomial. There exists a parameter in K \ K such that P λ has a wandering Fatou component.

66 Proof of the Main Theorem (Realization) For all n 1 there exist parameters λ n K such that F n is its level n tableau of P λ.

67 Proof of the Main Theorem (Realization) For all n 1 there exist parameters λ n K such that F n is its level n tableau of P λ. (P-D) A parameter such that F is its tableau belongs to L n (P λn (c)).

68 Proof of the Main Theorem (Realization) For all n 1 there exist parameters λ n K such that F n is its level n tableau of P λ. (P-D) A parameter such that F is its tableau belongs to L n (P λn (c)). (Structural)+(Distance). L n (P λn (c)) =.

69 Proof of the Main Theorem (Realization) For all n 1 there exist parameters λ n K such that F n is its level n tableau of P λ. (P-D) A parameter such that F is its tableau belongs to L n (P λn (c)). (Structural)+(Distance). L n (P λn (c)) =. Conclusion: There exists a parameter λ K \ K such that P λ K[z] has a nonpreperiodic type II point in its Julia set

70 Remarks Remark This proof works with any admissible aperiodic tableau such that d H (L 0, c) is finite.

71 Remarks Remark This proof works with any admissible aperiodic tableau such that d H (L 0, c) is finite. Remark It is possible to pass from K to a field extension K /K such that ( Ln (P λ (c))) K.

72