Wandering Domains in Spherically Complete Fields...Wandering Domains in non-Archimedean ﬁelds Theorem (Benedetto, 2002) There exist inﬁnitely many a 2Cp such that Pa(z) = (1 a)zp+1

Wandering Domains in SphericallyComplete Fields

Eugenio Trucco

Universidad Austral de Chile

p-adic and Complex Dynamics, ICERM20120214

Complex Polynomial Dynamics

To understand the dynamics of a polynomial P ∈ C[z] ofdegree ≥ 2 we split the complex plain in two disjoint sets.That is

C = J (P) t F(P).

The Julia set J (P) is a compact set where thedynamics is chaotic.

The Fatou set F(P) is an open set where thedynamics is regular.

Fatou Components

If U ⊆ F(P) is a connected component of the Fatou set(Fatou component), then P(U ) is itself a Fatoucomponent.

Classification Theorem (Fatou, Julia, Siegel, Herman)

There exist only 3 types of fixed Fatou components:

Attracting basins.

Parabolic basins.

Siegel disks.

Are there Wandering Fatou components?

Question:

Is there a Fatou component U such that

Pn(U ) 6= Pm(U )

for all n 6= 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, everyFatou component is preperiodic.

Are there Wandering Fatou components?

Question:

Is there a Fatou component U such that

Pn(U ) 6= Pm(U )

for all n 6= 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, everyFatou component is preperiodic.

Non-Archimedean Fields

Examples

Cp, the completion of an algebraic closure of Qp.

L, the completion of the field of formal Puiseux serieswith coefficients in C.

Main difference between L and Cp

The residue field of L is C and it has characteristic 0.

The residue field of Cp is Fap and it has characteristic

p > 0.

Non-Archimedean Fields

Examples

Cp, the completion of an algebraic closure of Qp.

L, the completion of the field of formal Puiseux serieswith coefficients in C.

Main difference between L and Cp

The residue field of L is C and it has characteristic 0.

The residue field of Cp is Fap and it has characteristic

p > 0.

Non-Archimedean Dynamics

We will study the map

P : A1,anK −→ A1,an

K

where A1,anK 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 anattracting 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 aperiodic point in U .

Non-Archimedean Dynamics

We will study the map

P : A1,anK −→ A1,an

K

where A1,anK 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 anattracting 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 aperiodic point in U .

Question

Question

Are there wandering Fatou components?

Wandering Domains in non-Archimedean fields

Theorem (Benedetto, 2002)There exist infinitely many a ∈ Cp such that

Pa(z) = (1− a)zp+1 + azp ∈ Cp[z]

has a wandering Fatou component.

The parameters a are transcendentals over Qp.

There exists wild ramification. That is, there exists aclosed ball B ⊆ Cp with no critical points of P, suchthat the degree of the map

P : B −→ P(B)

is p.

Wandering Domains in non-Archimedean fields

Theorem (Benedetto, 2002)There exist infinitely many a ∈ Cp such that

Pa(z) = (1− a)zp+1 + azp ∈ Cp[z]

has a wandering Fatou component.

The parameters a are transcendentals over Qp.

There exists wild ramification. That is, there exists aclosed ball B ⊆ Cp with no critical points of P, suchthat the degree of the map

P : B −→ P(B)

is p.

No Wandering Domains Theorems

Theorem (Benedetto, 1998)Let K be a finite extension of Qp and let P ∈ K [z] have norecurrent wild critical points in its Julia set. Then theFatou set of P has no wandering components.

Theorem (T, 2010)Let P ∈ K [z] be a tame polynomial of degree ≥ 2. Then theFatou set of P has no wandering components.

Spherically Complete Fields

DefinitionA field is spherically complete if any nested sequence ofclosed balls has nonempty intersection

Examples

R is a spherically complete field.

Qp is a spherically complete field.

Cp is not a spherically complete field.

Remark

Any field K has a spherical completion denote by K̂ .



Examples




Remark




Examples




Remark


Tame Polynomials

DefinitionWe say that P ∈ K [z] is a tame polynomial if its criticalset is a finite tree.

Main Theorem

TheoremLet α ∈ K such that |α| > 1. Let d ≥ 3 such that

Pλ = α(z − 1)d(z + 1) + λ

is a tame polynomial. There exists a parameter in K̂ \ Ksuch that Pλ has a wandering Fatou component.

If char(K̃) = 0 then d ≥ 3.

If char(K̃) = p > 0 then 2 < p and p - d.

RemarkThere is no wild ramification.

Main Theorem


Pλ = α(z − 1)d(z + 1) + λ





Main Theorem


Pλ = α(z − 1)d(z + 1) + λ





Roughly Speaking...

If the field K is big enough then there exist polynomialshaving a wandering Fatou component.

Technical assumption on K

Technical assumption on K

K is a field of characteristic 0, algebraically closed andcomplete with respect to a nontrivial non-Archimedeanabsolute value.Moreover, there exists a discrete valued subfield k ⊆ Ksuch that ka is dense in K .

The Berkovich Affine Line

The Berkovich Line

A1,anK 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 6∈ |K×|).Type IV points correspond to nested sequences ofclosed balls with empty intersection.

The Berkovich Affine Line

The Berkovich Line

A1,anK 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 6∈ |K×|).Type IV points correspond to nested sequences ofclosed balls with empty intersection.

Action of P in A1,anK

Extension of P

If xB ∈ A1,anK the P(xB) = xP(B).

Local degree

The local degree of P at xB is the degree of the mapP : B → P(B). In our case, for tame polynomials, the localdegree at a point xB coincides with

1 +∑

ω∈Crit(P)∩B

multiplicity of ω.

Fatou and Julia sets on A1,anK

The filled Julia set of P is

K(P) ={x ∈ A1,an

K | {Pn(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) := A1,anK \ J (P).

Theorem (Rivera-Letelier)The Julia set of P is a non empty, compact and totallyinvariant set. Moreover J (P) = J (Pn).Furthermore, J (P) is the closure of the repelling periodicpoints in A1,an

K .



K(P) ={x ∈ A1,an





K .



K(P) ={x ∈ A1,an





K .



K(P) ={x ∈ A1,an





K .

Lemma 0

Lemma 0P has a wandering Fatou component if and only if thereexists a nonpreperiodic type II or type III point in itsJulia set.

The Family

Let α ∈ K such that |α| > 1. Let d ≥ 3 such that

Pλ(z) = α(z − 1)d(z + 1) + λ

is a tame polynomial

The critical points of Pλ are 1 and ω = 1−d1+d with local

degrees d and 2 respectively.

We are interested in the parameters such that 1 ∈ K(Pλ)and ω escapes.

There exits exactly one c ∈ J (Pλ) ∩ Crit(Pλ).

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

xg


x

xg


xP(x)

xg

Pλ(xg)


c

xg

Pλ(xg)


c P(c)

xg

Pλ(xg)

The Tableau

c

The Tableau

c P(c)

The Tableau

c P(c) P2(c)

The Tableau

c P(c) P2(c)

The Tableau

c P(c) P2(c)

The Fibonacci Tableau F

The Tableau Rules

Proposition (Branner and Hubbard, 1992; Harris, 1999)

Every tableau of a cubic polynomial satisfies the followingtableau rules. Conversely, every marked grid satisfyingthese rules can be realized as the critical tableau for acubic polynomial.

1 If Tm,n is critical then so is Ti,n for 0 ≤ i < m.

2 If Tm,n is critical, then for j, i ≥ 0, j + i ≤ m, Tj,n+i iscritical if and only if Tj,i is also critical.

3 Suppose Tm,n is critical but Tm+1,n is not. Supposealso that Tm−`,n+` is critical but that Tm−i,n+i , 0 < i < `,is not. If Tm−`+1,` is critical, then Tm−`+1,n+` is not.

4 If T1,m is critical, and T1,1 and T2,m are not, thenT1,m+1 is critical.

Hyperbolic distance

For x1 ≺ x2 ∈ A1,anK \ K we define the hyperbolic distance

between x1, x2 by

dH(x1, x2) = log(diam(x2))− log(diam(x1)).

Lemma (Rivera-Letelier, 2002)Let P ∈ K [z] be a polynomial and consider x1 ≺ x2 inA1,an

K \ K . If the local degree of P at y is η for all y ∈ ]x1, x2[then

dH(P(x1),P(x2)) = η · dH(x1, x2).

Key Lemma 1

Key Lemma 1

Recall that c ∈ J (Pλ) ∩ Crit(Pλ).

LemmaIf the Fibonacci tableau is the tableau of Pλ then dH(xg, c)

is finite, in particular c ∈ A1,anK \ K

The Family and the Parameters


Pλ(z) = α(z − 1)d(z + 1) + λ

is a tame polynomial.

For any n ≥ 1 we consider the level n parameter set

Λn = {λ ∈ K | Pnλ (λ) ∈ K(Pλ)}.

The Family and the Parameters


Pλ(z) = α(z − 1)d(z + 1) + λ

is a tame polynomial.

For any n ≥ 1 we consider the level n parameter set

Λn = {λ ∈ K | Pnλ (λ) ∈ K(Pλ)}.

Realization

F0

Realization

F1

Realization

F2

Realization

F3

Realization

F4

Realization

F5

Realization

F6

Realization

F7

Realization

F8

Realization

F9

Realization

F10

Realization

F11

Realization

F12

Key Lemma 2

Lemma (Level n parameters)

Let n > 0. There exists a parameter λn ∈ Λn ⊆ K suchthat the level n tableau of Pλn is Fn .

Key Lemma 3

P-D LemmaLet ∆n ⊆ Λn be a level n parameter ball. Then

Lλn (λ) = ∆n

for any λ ∈ ∆n .

The Final Ingredient

Structural Theorem, (T, 2010)Let P ∈ K [z] be a tame polynomial of degree ≥ 2. Thenone of the following statements hold:

1 J (P) ⊆ K .2 There exist finitely many type II periodic points

x1, . . . , xm contained in A1,anK \ K such that

J (P) \ K = GO(x1) t · · · tGO(xm),

whereGO(x) := {y ∈ A1,an

K | Pn(y) = x for some n ∈ N}.

Main Theorem


Pλ = α(z − 1)d(z + 1) + λ


Proof of the Main Theorem

(Realization) For all n ≥ 1 there exist parametersλn ∈ K such that Fn is its level n tableau of Pλ.

(P-D) A parameter such that F is its tableau belongsto⋂

Ln(Pλn (c)).

(Structural)+(Distance).⋂

Ln(Pλn (c)) = ∅.Conclusion: There exists a parameter λ ∈ K̂ \ K suchthat Pλ ∈ K̂ [z] has a nonpreperiodic type II point inits Julia set




Ln(Pλn (c)).






Ln(Pλn (c)).


Ln(Pλn (c)) = ∅.

Conclusion: There exists a parameter λ ∈ K̂ \ K suchthat Pλ ∈ K̂ [z] has a nonpreperiodic type II point inits Julia set




Ln(Pλn (c)).



Remarks

RemarkThis proof works with any admissible aperiodic tableausuch that dH(L0, c) is finite.

RemarkIt is possible to pass from K to a field extension K /K suchthat (⋂

Ln(Pλ(c)))∩K 6= ∅.

Remarks

RemarkThis proof works with any admissible aperiodic tableausuch that dH(L0, c) is finite.

RemarkIt is possible to pass from K to a field extension K /K suchthat (⋂

Ln(Pλ(c)))∩K 6= ∅.

Documents

Wandering Domains in Spherically Complete Fields...Wandering Domains in non-Archimedean ﬁelds Theorem (Benedetto, 2002) There exist inﬁnitely many a 2Cp such that Pa(z) = (1 a)zp+1