Non-archimedean Dynamics in Dimension One: Lecture...

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Non-archimedean Dynamics in Dimension One: Lecture 3 Robert L. Benedetto Amherst College Arizona Winter School Still Sunday, π Day, 2010

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Page 1: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Non-archimedean Dynamics in Dimension One:Lecture 3

Robert L. BenedettoAmherst College

Arizona Winter School

Still Sunday, π Day, 2010

Page 2: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Revisiting the Quadratic Polynomial Example

φ(z) = z2 + az ∈ CK [z ].

Page 3: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Revisiting the Quadratic Polynomial Example

φ(z) = z2 + az ∈ CK [z ].

◮ If |a| ≤ 1, then φ(ζ(0, 1)) = ζ(0, 1), with φ = z2 + az , andhence with degζ(0,1) φ = 2.

Page 4: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Revisiting the Quadratic Polynomial Example

φ(z) = z2 + az ∈ CK [z ].

◮ If |a| ≤ 1, then φ(ζ(0, 1)) = ζ(0, 1), with φ = z2 + az , andhence with degζ(0,1) φ = 2.

Then φ has good reduction, because deg φ = deg φ.

Page 5: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Revisiting the Quadratic Polynomial Example

φ(z) = z2 + az ∈ CK [z ].

◮ If |a| ≤ 1, then φ(ζ(0, 1)) = ζ(0, 1), with φ = z2 + az , andhence with degζ(0,1) φ = 2.

Then φ has good reduction, because deg φ = deg φ.

Each residue class x is mapped to the residue class φ(x).

Page 6: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Revisiting the Quadratic Polynomial Example

φ(z) = z2 + az ∈ CK [z ].

◮ If |a| ≤ 1, then φ(ζ(0, 1)) = ζ(0, 1), with φ = z2 + az , andhence with degζ(0,1) φ = 2.

Then φ has good reduction, because deg φ = deg φ.

Each residue class x is mapped to the residue class φ(x).

So Fφ,Ber = P1Ber r {ζ(0, 1)}, and Jφ,Ber = {ζ(0, 1)}.

Page 7: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Revisiting the Quadratic Polynomial Example

φ(z) = z2 + az ∈ CK [z ].

◮ If |a| ≤ 1, then φ(ζ(0, 1)) = ζ(0, 1), with φ = z2 + az , andhence with degζ(0,1) φ = 2.

Then φ has good reduction, because deg φ = deg φ.

Each residue class x is mapped to the residue class φ(x).

So Fφ,Ber = P1Ber r {ζ(0, 1)}, and Jφ,Ber = {ζ(0, 1)}.

◮ If |a| > 1, then Jφ,Ber = Jφ ⊆ P1(CK ) is the same Cantor setas before.

Then Fφ,Ber = P1Ber r Jφ, all points of which are attracted to

∞ under iteration.

Page 8: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Revisiting Hsia’s Cubic Polynomial Example

φ(z) = az3 + z2 + bz + c , where 0 < |a| < 1, and |b|, |c | ≤ 1.

Page 9: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Revisiting Hsia’s Cubic Polynomial Example

φ(z) = az3 + z2 + bz + c , where 0 < |a| < 1, and |b|, |c | ≤ 1.

Then φ(z) = z2 + bz + c , so that φ maps ζ(0, 1) to itself withmultiplicity 2.

Page 10: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Revisiting Hsia’s Cubic Polynomial Example

φ(z) = az3 + z2 + bz + c , where 0 < |a| < 1, and |b|, |c | ≤ 1.

Then φ(z) = z2 + bz + c , so that φ maps ζ(0, 1) to itself withmultiplicity 2.

So ζ(0, 1) ∈ Jφ,Ber is a (Type II) repelling fixed point.

Page 11: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Revisiting Hsia’s Cubic Polynomial Example

φ(z) = az3 + z2 + bz + c , where 0 < |a| < 1, and |b|, |c | ≤ 1.

Then φ(z) = z2 + bz + c , so that φ maps ζ(0, 1) to itself withmultiplicity 2.

So ζ(0, 1) ∈ Jφ,Ber is a (Type II) repelling fixed point.

φ maps each residue class x other than ∞ to its image under φ.

Page 12: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Revisiting Hsia’s Cubic Polynomial Example

φ(z) = az3 + z2 + bz + c , where 0 < |a| < 1, and |b|, |c | ≤ 1.

Then φ(z) = z2 + bz + c , so that φ maps ζ(0, 1) to itself withmultiplicity 2.

So ζ(0, 1) ∈ Jφ,Ber is a (Type II) repelling fixed point.

φ maps each residue class x other than ∞ to its image under φ.

Since none of them ever hits ∞, they are all contained in FBer.

Page 13: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Revisiting Hsia’s Cubic Polynomial Example

φ(z) = az3 + z2 + bz + c , where 0 < |a| < 1, and |b|, |c | ≤ 1.

Then φ(z) = z2 + bz + c , so that φ maps ζ(0, 1) to itself withmultiplicity 2.

So ζ(0, 1) ∈ Jφ,Ber is a (Type II) repelling fixed point.

φ maps each residue class x other than ∞ to its image under φ.

Since none of them ever hits ∞, they are all contained in FBer.

However, φ maps the residue class ∞ onto all of P1Ber. The Julia

set Jφ,Ber is scattered through this residue class.

Page 14: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Revisiting Hsia’s Cubic Polynomial Example

φ(z) = az3 + z2 + bz + c , where 0 < |a| < 1, and |b|, |c | ≤ 1.

Then φ(z) = z2 + bz + c , so that φ maps ζ(0, 1) to itself withmultiplicity 2.

So ζ(0, 1) ∈ Jφ,Ber is a (Type II) repelling fixed point.

φ maps each residue class x other than ∞ to its image under φ.

Since none of them ever hits ∞, they are all contained in FBer.

However, φ maps the residue class ∞ onto all of P1Ber. The Julia

set Jφ,Ber is scattered through this residue class.

Recall that the classical Julia set Jφ was not compact; but ofcourse the Berkovich Julia set Jφ,Ber must be compact.

Page 15: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Revisiting Hsia’s Cubic Polynomial Example

φ(z) = az3 + z2 + bz + c , where 0 < |a| < 1, and |b|, |c | ≤ 1.

Then φ(z) = z2 + bz + c , so that φ maps ζ(0, 1) to itself withmultiplicity 2.

So ζ(0, 1) ∈ Jφ,Ber is a (Type II) repelling fixed point.

φ maps each residue class x other than ∞ to its image under φ.

Since none of them ever hits ∞, they are all contained in FBer.

However, φ maps the residue class ∞ onto all of P1Ber. The Julia

set Jφ,Ber is scattered through this residue class.

Recall that the classical Julia set Jφ was not compact; but ofcourse the Berkovich Julia set Jφ,Ber must be compact.

In particular, that sequence β1, β2, . . . (of preimages of therepelling fixed point α) accumulates at ζ(0, 1) ∈ Jφ,Ber.

Page 16: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Components of the (Berkovich) Fatou Set

TheoremLet φ(z) ∈ CK (z) be a rational function of degree d ≥ 2,with (Berkovich) Fatou set Fφ,Ber and Julia set Jφ,Ber.

Page 17: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Components of the (Berkovich) Fatou Set

TheoremLet φ(z) ∈ CK (z) be a rational function of degree d ≥ 2,with (Berkovich) Fatou set Fφ,Ber and Julia set Jφ,Ber.

Let U ⊆ Fφ,Ber be a connected component of Fφ,Ber, and letx ∈ U. Then

Page 18: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Components of the (Berkovich) Fatou Set

TheoremLet φ(z) ∈ CK (z) be a rational function of degree d ≥ 2,with (Berkovich) Fatou set Fφ,Ber and Julia set Jφ,Ber.

Let U ⊆ Fφ,Ber be a connected component of Fφ,Ber, and letx ∈ U. Then

◮ U is the union of all connected Berkovich affinoids containingx and contained in Fφ,Ber.

Page 19: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Components of the (Berkovich) Fatou Set

TheoremLet φ(z) ∈ CK (z) be a rational function of degree d ≥ 2,with (Berkovich) Fatou set Fφ,Ber and Julia set Jφ,Ber.

Let U ⊆ Fφ,Ber be a connected component of Fφ,Ber, and letx ∈ U. Then

◮ U is the union of all connected Berkovich affinoids containingx and contained in Fφ,Ber.

◮ φ(U) is a connected component of Fφ,Ber.

Page 20: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Components of the (Berkovich) Fatou Set

TheoremLet φ(z) ∈ CK (z) be a rational function of degree d ≥ 2,with (Berkovich) Fatou set Fφ,Ber and Julia set Jφ,Ber.

Let U ⊆ Fφ,Ber be a connected component of Fφ,Ber, and letx ∈ U. Then

◮ U is the union of all connected Berkovich affinoids containingx and contained in Fφ,Ber.

◮ φ(U) is a connected component of Fφ,Ber.

◮ φ−1(U) is a disjoint union of at most d connectedcomponents V1, . . . ,Vℓ of Fφ,Ber.

Page 21: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Components of the (Berkovich) Fatou Set

TheoremLet φ(z) ∈ CK (z) be a rational function of degree d ≥ 2,with (Berkovich) Fatou set Fφ,Ber and Julia set Jφ,Ber.

Let U ⊆ Fφ,Ber be a connected component of Fφ,Ber, and letx ∈ U. Then

◮ U is the union of all connected Berkovich affinoids containingx and contained in Fφ,Ber.

◮ φ(U) is a connected component of Fφ,Ber.

◮ φ−1(U) is a disjoint union of at most d connectedcomponents V1, . . . ,Vℓ of Fφ,Ber.

Each Vi maps di -to-1 onto U, for some di ≥ 1,and d1 + · · · + dℓ = d.

Page 22: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Periodic Fatou Components

DefinitionLet φ ∈ CK (z) be a rational function of degree d ≥ 2 with Fatouset Fφ,Ber.

Page 23: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Periodic Fatou Components

DefinitionLet φ ∈ CK (z) be a rational function of degree d ≥ 2 with Fatouset Fφ,Ber.Let U ⊆ Fφ,Ber be a connected component of the Fatou set, andsuppose that φm(U) = U for some (minimal) integer m ≥ 1.

Page 24: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Periodic Fatou Components

DefinitionLet φ ∈ CK (z) be a rational function of degree d ≥ 2 with Fatouset Fφ,Ber.Let U ⊆ Fφ,Ber be a connected component of the Fatou set, andsuppose that φm(U) = U for some (minimal) integer m ≥ 1.

◮ We say U is an indifferent component if the mappingφm : U ։ U is one-to-one.

Page 25: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Periodic Fatou Components

DefinitionLet φ ∈ CK (z) be a rational function of degree d ≥ 2 with Fatouset Fφ,Ber.Let U ⊆ Fφ,Ber be a connected component of the Fatou set, andsuppose that φm(U) = U for some (minimal) integer m ≥ 1.

◮ We say U is an indifferent component if the mappingφm : U ։ U is one-to-one.

◮ We say U is an attracting component if there is an attractingperiodic point x ∈ U of period m, and if lim

n→∞φmn(ζ) = x for

all ζ ∈ U.

Page 26: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Periodic Fatou Components

DefinitionLet φ ∈ CK (z) be a rational function of degree d ≥ 2 with Fatouset Fφ,Ber.Let U ⊆ Fφ,Ber be a connected component of the Fatou set, andsuppose that φm(U) = U for some (minimal) integer m ≥ 1.

◮ We say U is an indifferent component if the mappingφm : U ։ U is one-to-one.

◮ We say U is an attracting component if there is an attractingperiodic point x ∈ U of period m, and if lim

n→∞φmn(ζ) = x for

all ζ ∈ U.

A connected component of Fφ,Ber that is not preperiodic is calleda wandering domain.

Page 27: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Rivera-Letelier’s Classification Theorem

Theorem (Rivera-Letelier, 2000)

Let φ ∈ CK (z) be a rational function of degree d ≥ 2 with Fatouset Fφ,Ber.

Let U ⊆ Fφ,Ber be a connected component of the Fatou set.

Page 28: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Rivera-Letelier’s Classification Theorem

Theorem (Rivera-Letelier, 2000)

Let φ ∈ CK (z) be a rational function of degree d ≥ 2 with Fatouset Fφ,Ber.

Let U ⊆ Fφ,Ber be a connected component of the Fatou set.

Then exactly one of the following three possibilities occurs.

Page 29: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Rivera-Letelier’s Classification Theorem

Theorem (Rivera-Letelier, 2000)

Let φ ∈ CK (z) be a rational function of degree d ≥ 2 with Fatouset Fφ,Ber.

Let U ⊆ Fφ,Ber be a connected component of the Fatou set.

Then exactly one of the following three possibilities occurs.

1. Some iterate φn(U) is an indifferent periodic component.

Page 30: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Rivera-Letelier’s Classification Theorem

Theorem (Rivera-Letelier, 2000)

Let φ ∈ CK (z) be a rational function of degree d ≥ 2 with Fatouset Fφ,Ber.

Let U ⊆ Fφ,Ber be a connected component of the Fatou set.

Then exactly one of the following three possibilities occurs.

1. Some iterate φn(U) is an indifferent periodic component.

2. Some iterate φn(U) is an attracting periodic component.

Page 31: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Rivera-Letelier’s Classification Theorem

Theorem (Rivera-Letelier, 2000)

Let φ ∈ CK (z) be a rational function of degree d ≥ 2 with Fatouset Fφ,Ber.

Let U ⊆ Fφ,Ber be a connected component of the Fatou set.

Then exactly one of the following three possibilities occurs.

1. Some iterate φn(U) is an indifferent periodic component.

2. Some iterate φn(U) is an attracting periodic component.

3. U is a wandering domain.

Page 32: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Example: Good Reduction

Recall that φ has good reduction if when we write φ(z) =f (z)

g(z)where f , g ∈ O[z ] satisfy

◮ (f , g) = 1,

◮ at least one coefficient of f and/or g is a unit (i.e., |a| = 1),

Page 33: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Example: Good Reduction

Recall that φ has good reduction if when we write φ(z) =f (z)

g(z)where f , g ∈ O[z ] satisfy

◮ (f , g) = 1,

◮ at least one coefficient of f and/or g is a unit (i.e., |a| = 1),

then deg(f /g) = deg φ.

Page 34: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Example: Good Reduction

Recall that φ has good reduction if when we write φ(z) =f (z)

g(z)where f , g ∈ O[z ] satisfy

◮ (f , g) = 1,

◮ at least one coefficient of f and/or g is a unit (i.e., |a| = 1),

then deg(f /g) = deg φ.

In that case, Jφ,Ber = {ζ(0, 1)}, and each residue class is a Fatoucomponent.

Page 35: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Example: Good Reduction

Recall that φ has good reduction if when we write φ(z) =f (z)

g(z)where f , g ∈ O[z ] satisfy

◮ (f , g) = 1,

◮ at least one coefficient of f and/or g is a unit (i.e., |a| = 1),

then deg(f /g) = deg φ.

In that case, Jφ,Ber = {ζ(0, 1)}, and each residue class is a Fatoucomponent.

The components map to each other as dictated by φ := f /g actingon P1(k).

Page 36: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Example: Good Reduction

Recall that φ has good reduction if when we write φ(z) =f (z)

g(z)where f , g ∈ O[z ] satisfy

◮ (f , g) = 1,

◮ at least one coefficient of f and/or g is a unit (i.e., |a| = 1),

then deg(f /g) = deg φ.

In that case, Jφ,Ber = {ζ(0, 1)}, and each residue class is a Fatoucomponent.

The components map to each other as dictated by φ := f /g actingon P1(k).

An n-periodic residue class DBer(a, 1) is attracting if and only if

φ has a critical point among{

a, φ(a), . . . , φn−1

(a)}

.

Page 37: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Example: Polynomials

If φ(z) ∈ Cv [z ] with deg φ ≥ 2 is a polynomial, then the Fatoucomponent W containing ∞ is fixed and attracting.

Page 38: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Example: Polynomials

If φ(z) ∈ Cv [z ] with deg φ ≥ 2 is a polynomial, then the Fatoucomponent W containing ∞ is fixed and attracting.

If φ is not of potentially good reduction,

Page 39: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Example: Polynomials

If φ(z) ∈ Cv [z ] with deg φ ≥ 2 is a polynomial, then the Fatoucomponent W containing ∞ is fixed and attracting.

If φ is not of potentially good reduction, thenW is not a disk.Instead, it is of Cantor type.

Page 40: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Example: Polynomials

If φ(z) ∈ Cv [z ] with deg φ ≥ 2 is a polynomial, then the Fatoucomponent W containing ∞ is fixed and attracting.

If φ is not of potentially good reduction, thenW is not a disk.Instead, it is of Cantor type.

That is, let V0 = P1(CK ) r DBer(a, r) ⊆ Fφ be the largest openP1

Ber-disk containing ∞.

Page 41: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Example: Polynomials

If φ(z) ∈ Cv [z ] with deg φ ≥ 2 is a polynomial, then the Fatoucomponent W containing ∞ is fixed and attracting.

If φ is not of potentially good reduction, thenW is not a disk.Instead, it is of Cantor type.

That is, let V0 = P1(CK ) r DBer(a, r) ⊆ Fφ be the largest openP1

Ber-disk containing ∞.

V1 := φ−1(V0) ) V0 is a non-disk open affinoid, with at least twoends outside (the unique) end of V0.

Page 42: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Example: Polynomials

If φ(z) ∈ Cv [z ] with deg φ ≥ 2 is a polynomial, then the Fatoucomponent W containing ∞ is fixed and attracting.

If φ is not of potentially good reduction, thenW is not a disk.Instead, it is of Cantor type.

That is, let V0 = P1(CK ) r DBer(a, r) ⊆ Fφ be the largest openP1

Ber-disk containing ∞.

V1 := φ−1(V0) ) V0 is a non-disk open affinoid, with at least twoends outside (the unique) end of V0.

V2 := φ−1(V1) ) V1 is a non-disk open affinoid; with at least twoends outside each end of V1.

Page 43: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Example: Polynomials

If φ(z) ∈ Cv [z ] with deg φ ≥ 2 is a polynomial, then the Fatoucomponent W containing ∞ is fixed and attracting.

If φ is not of potentially good reduction, thenW is not a disk.Instead, it is of Cantor type.

That is, let V0 = P1(CK ) r DBer(a, r) ⊆ Fφ be the largest openP1

Ber-disk containing ∞.

V1 := φ−1(V0) ) V0 is a non-disk open affinoid, with at least twoends outside (the unique) end of V0.

V2 := φ−1(V1) ) V1 is a non-disk open affinoid; with at least twoends outside each end of V1.

etc. In the end, W =⋃

n≥0

Vn.

Page 44: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Rivera-Letelier’s Classification: Continued

Theorem (Rivera-Letelier, 2000)

Let φ ∈ CK (z) be a rational function of degree d ≥ 2 with Fatouset Fφ,Ber, and let U ⊆ Fφ,Ber be a periodic connected componentof the Fatou set.

Page 45: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Rivera-Letelier’s Classification: Continued

Theorem (Rivera-Letelier, 2000)

Let φ ∈ CK (z) be a rational function of degree d ≥ 2 with Fatouset Fφ,Ber, and let U ⊆ Fφ,Ber be a periodic connected componentof the Fatou set.

1. If U is indifferent, then U is a rational open connectedaffinoid, and φ permutes the (finitely many) boundary pointsof U.

Page 46: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Rivera-Letelier’s Classification: Continued

Theorem (Rivera-Letelier, 2000)

Let φ ∈ CK (z) be a rational function of degree d ≥ 2 with Fatouset Fφ,Ber, and let U ⊆ Fφ,Ber be a periodic connected componentof the Fatou set.

1. If U is indifferent, then U is a rational open connectedaffinoid, and φ permutes the (finitely many) boundary pointsof U. The boundary points are all type II periodic Julia points.

Page 47: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Rivera-Letelier’s Classification: Continued

Theorem (Rivera-Letelier, 2000)

Let φ ∈ CK (z) be a rational function of degree d ≥ 2 with Fatouset Fφ,Ber, and let U ⊆ Fφ,Ber be a periodic connected componentof the Fatou set.

1. If U is indifferent, then U is a rational open connectedaffinoid, and φ permutes the (finitely many) boundary pointsof U. The boundary points are all type II periodic Julia points.

2. If U is attracting, then U is either a rational open disk or adomain of Cantor type.

Page 48: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Rivera-Letelier’s Classification: Continued

Theorem (Rivera-Letelier, 2000)

Let φ ∈ CK (z) be a rational function of degree d ≥ 2 with Fatouset Fφ,Ber, and let U ⊆ Fφ,Ber be a periodic connected componentof the Fatou set.

1. If U is indifferent, then U is a rational open connectedaffinoid, and φ permutes the (finitely many) boundary pointsof U. The boundary points are all type II periodic Julia points.

2. If U is attracting, then U is either a rational open disk or adomain of Cantor type.For an open disk, the unique boundary point is a type IIrepelling periodic (Julia) point.

Page 49: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Rivera-Letelier’s Classification: Continued

Theorem (Rivera-Letelier, 2000)

Let φ ∈ CK (z) be a rational function of degree d ≥ 2 with Fatouset Fφ,Ber, and let U ⊆ Fφ,Ber be a periodic connected componentof the Fatou set.

1. If U is indifferent, then U is a rational open connectedaffinoid, and φ permutes the (finitely many) boundary pointsof U. The boundary points are all type II periodic Julia points.

2. If U is attracting, then U is either a rational open disk or adomain of Cantor type.For an open disk, the unique boundary point is a type IIrepelling periodic (Julia) point.For Cantor type, the boundary is uncountable and containedin the Julia set.

Page 50: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Rivera-Letelier’s Classification: Continued

Theorem (Rivera-Letelier, 2000)

Let φ ∈ CK (z) be a rational function of degree d ≥ 2 with Fatouset Fφ,Ber, and let U ⊆ Fφ,Ber be a periodic connected componentof the Fatou set.

1. If U is indifferent, then U is a rational open connectedaffinoid, and φ permutes the (finitely many) boundary pointsof U. The boundary points are all type II periodic Julia points.

2. If U is attracting, then U is either a rational open disk or adomain of Cantor type.For an open disk, the unique boundary point is a type IIrepelling periodic (Julia) point.For Cantor type, the boundary is uncountable and containedin the Julia set. The boundary can include points of type I,type II, or type IV.

Page 51: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Rivera-Letelier’s Classification: Continued

Theorem (Rivera-Letelier, 2000)

Let φ ∈ CK (z) be a rational function of degree d ≥ 2 with Fatouset Fφ,Ber, and let U ⊆ Fφ,Ber be a periodic connected componentof the Fatou set.

1. If U is indifferent, then U is a rational open connectedaffinoid, and φ permutes the (finitely many) boundary pointsof U. The boundary points are all type II periodic Julia points.

2. If U is attracting, then U is either a rational open disk or adomain of Cantor type.For an open disk, the unique boundary point is a type IIrepelling periodic (Julia) point.For Cantor type, the boundary is uncountable and containedin the Julia set. The boundary can include points of type I,type II, or type IV.(Maybe also type III? Requires a wandering domain withcertain properties.)

Page 52: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Example: A Non-disk Indifferent Component

φ(z) =1

1 − z+ π2z =

π2z2 − π2z − 1

z − 1∈ CK [z ], with 0 < |π| < 1.

Page 53: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Example: A Non-disk Indifferent Component

φ(z) =1

1 − z+ π2z =

π2z2 − π2z − 1

z − 1∈ CK [z ], with 0 < |π| < 1.

Then φ has a repelling fixed point at ∞ (|multiplier| = |π|−2 > 1),

Page 54: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Example: A Non-disk Indifferent Component

φ(z) =1

1 − z+ π2z =

π2z2 − π2z − 1

z − 1∈ CK [z ], with 0 < |π| < 1.

Then φ has a repelling fixed point at ∞ (|multiplier| = |π|−2 > 1),

but ζ(0, 1) is an indifferent fixed point, with φ(z) =1

1 − z.

Page 55: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Example: A Non-disk Indifferent Component

φ(z) =1

1 − z+ π2z =

π2z2 − π2z − 1

z − 1∈ CK [z ], with 0 < |π| < 1.

Then φ has a repelling fixed point at ∞ (|multiplier| = |π|−2 > 1),

but ζ(0, 1) is an indifferent fixed point, with φ(z) =1

1 − z.

Note that φ3(z) = z , with ∞ 7→ 0 7→ 1 7→ ∞.

Page 56: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Example: A Non-disk Indifferent Component

φ(z) =1

1 − z+ π2z =

π2z2 − π2z − 1

z − 1∈ CK [z ], with 0 < |π| < 1.

Then φ has a repelling fixed point at ∞ (|multiplier| = |π|−2 > 1),

but ζ(0, 1) is an indifferent fixed point, with φ(z) =1

1 − z.

Note that φ3(z) = z , with ∞ 7→ 0 7→ 1 7→ ∞.

It is not hard to check that

φ maps

ζ(0, |π|−1) 7→ ζ(0, |π|) with multiplicity 2,

ζ(0, |π|) 7→ ζ(1, |π|) with multiplicity 1,

ζ(1, |π|) 7→ ζ(0, |π|−1) with multiplicity 1,

so these three type II points form a repelling cycle of period 3.

Page 57: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Example: A Non-disk Indifferent Component

φ(z) =1

1 − z+ π2z =

π2z2 − π2z − 1

z − 1∈ CK [z ], with 0 < |π| < 1.

Then φ has a repelling fixed point at ∞ (|multiplier| = |π|−2 > 1),

but ζ(0, 1) is an indifferent fixed point, with φ(z) =1

1 − z.

Note that φ3(z) = z , with ∞ 7→ 0 7→ 1 7→ ∞.

It is not hard to check that

φ maps

ζ(0, |π|−1) 7→ ζ(0, |π|) with multiplicity 2,

ζ(0, |π|) 7→ ζ(1, |π|) with multiplicity 1,

ζ(1, |π|) 7→ ζ(0, |π|−1) with multiplicity 1,

so these three type II points form a repelling cycle of period 3.

It’s also easy to check that φ maps the open connected affinoid

U := DBer(0, |π|−1) r(

DBer(0, |π|) ∪ DBer(1, |π|))

bijectively onto itself.

Page 58: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

No Wandering Domains

DefinitionLet φ ∈ CK (z) be a rational function, and let x ∈ P1(CK ).

◮ x is recurrent under φ if x ∈⋃

n≥1 φn(x).

Page 59: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

No Wandering Domains

DefinitionLet φ ∈ CK (z) be a rational function, and let x ∈ P1(CK ).

◮ x is recurrent under φ if x ∈⋃

n≥1 φn(x).

◮ x is a wild critical point of φ if the multiplicity degx φ of φ atx is divisible by the residue characteristic of CK .

Page 60: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

No Wandering Domains

DefinitionLet φ ∈ CK (z) be a rational function, and let x ∈ P1(CK ).

◮ x is recurrent under φ if x ∈⋃

n≥1 φn(x).

◮ x is a wild critical point of φ if the multiplicity degx φ of φ atx is divisible by the residue characteristic of CK .

Theorem (RB, 1998)

Let K be a locally compact non-archimedean field, with CK thecompletion of an algebraic closure of K.

Page 61: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

No Wandering Domains

DefinitionLet φ ∈ CK (z) be a rational function, and let x ∈ P1(CK ).

◮ x is recurrent under φ if x ∈⋃

n≥1 φn(x).

◮ x is a wild critical point of φ if the multiplicity degx φ of φ atx is divisible by the residue characteristic of CK .

Theorem (RB, 1998)

Let K be a locally compact non-archimedean field, with CK thecompletion of an algebraic closure of K. (Note: char k = p > 0.)

Page 62: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

No Wandering Domains

DefinitionLet φ ∈ CK (z) be a rational function, and let x ∈ P1(CK ).

◮ x is recurrent under φ if x ∈⋃

n≥1 φn(x).

◮ x is a wild critical point of φ if the multiplicity degx φ of φ atx is divisible by the residue characteristic of CK .

Theorem (RB, 1998)

Let K be a locally compact non-archimedean field, with CK thecompletion of an algebraic closure of K. (Note: char k = p > 0.)

Let φ ∈ K (z) be a rational function of degree d ≥ 2 with classicalJulia set Jφ,I and Berkovich Fatou set Fφ,Ber.

Page 63: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

No Wandering Domains

DefinitionLet φ ∈ CK (z) be a rational function, and let x ∈ P1(CK ).

◮ x is recurrent under φ if x ∈⋃

n≥1 φn(x).

◮ x is a wild critical point of φ if the multiplicity degx φ of φ atx is divisible by the residue characteristic of CK .

Theorem (RB, 1998)

Let K be a locally compact non-archimedean field, with CK thecompletion of an algebraic closure of K. (Note: char k = p > 0.)

Let φ ∈ K (z) be a rational function of degree d ≥ 2 with classicalJulia set Jφ,I and Berkovich Fatou set Fφ,Ber.

Suppose that either

◮ char K = p and Jφ,I contains no wild critical points, or

◮ char K = 0 and Jφ,I contains no wild recurrent critical points.

Page 64: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

No Wandering Domains

DefinitionLet φ ∈ CK (z) be a rational function, and let x ∈ P1(CK ).

◮ x is recurrent under φ if x ∈⋃

n≥1 φn(x).

◮ x is a wild critical point of φ if the multiplicity degx φ of φ atx is divisible by the residue characteristic of CK .

Theorem (RB, 1998)

Let K be a locally compact non-archimedean field, with CK thecompletion of an algebraic closure of K. (Note: char k = p > 0.)

Let φ ∈ K (z) be a rational function of degree d ≥ 2 with classicalJulia set Jφ,I and Berkovich Fatou set Fφ,Ber.

Suppose that either

◮ char K = p and Jφ,I contains no wild critical points, or

◮ char K = 0 and Jφ,I contains no wild recurrent critical points.

Then Fφ,Ber has no wandering domains.

Page 65: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

A Power Series Lemma

LemmaLet a ∈ C×

K , set r = |a|, and let 0 < s < r .

Let f (z) = c0 + cdzd + · · · ∈ CK [[z ]] converge on D(0, r), andassume that |cn|r

n < |dcd |rd for all n > d ≥ 1.

Page 66: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

A Power Series Lemma

LemmaLet a ∈ C×

K , set r = |a|, and let 0 < s < r .

Let f (z) = c0 + cdzd + · · · ∈ CK [[z ]] converge on D(0, r), andassume that |cn|r

n < |dcd |rd for all n > d ≥ 1.

(In particular, f has no critical points in D(0, r) except maybe atz = 0; and in positive characteristic, z = 0 is not wild.)

Page 67: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

A Power Series Lemma

LemmaLet a ∈ C×

K , set r = |a|, and let 0 < s < r .

Let f (z) = c0 + cdzd + · · · ∈ CK [[z ]] converge on D(0, r), andassume that |cn|r

n < |dcd |rd for all n > d ≥ 1.

(In particular, f has no critical points in D(0, r) except maybe atz = 0; and in positive characteristic, z = 0 is not wild.)

If p|d, assume s < |p|1/(p−1)r .

Page 68: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

A Power Series Lemma

LemmaLet a ∈ C×

K , set r = |a|, and let 0 < s < r .

Let f (z) = c0 + cdzd + · · · ∈ CK [[z ]] converge on D(0, r), andassume that |cn|r

n < |dcd |rd for all n > d ≥ 1.

(In particular, f has no critical points in D(0, r) except maybe atz = 0; and in positive characteristic, z = 0 is not wild.)

If p|d, assume s < |p|1/(p−1)r . Thendiam

(

f (D(a, s)))

diam(

f (D(0, r))) = |d |

s

r.

Page 69: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

A Power Series Lemma

LemmaLet a ∈ C×

K , set r = |a|, and let 0 < s < r .

Let f (z) = c0 + cdzd + · · · ∈ CK [[z ]] converge on D(0, r), andassume that |cn|r

n < |dcd |rd for all n > d ≥ 1.

(In particular, f has no critical points in D(0, r) except maybe atz = 0; and in positive characteristic, z = 0 is not wild.)

If p|d, assume s < |p|1/(p−1)r . Thendiam

(

f (D(a, s)))

diam(

f (D(0, r))) = |d |

s

r.

Idea of Proof. Rewrite f (z) as a power series

∞∑

n=0

bn(z − a)n

centered at a,

Page 70: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

A Power Series Lemma

LemmaLet a ∈ C×

K , set r = |a|, and let 0 < s < r .

Let f (z) = c0 + cdzd + · · · ∈ CK [[z ]] converge on D(0, r), andassume that |cn|r

n < |dcd |rd for all n > d ≥ 1.

(In particular, f has no critical points in D(0, r) except maybe atz = 0; and in positive characteristic, z = 0 is not wild.)

If p|d, assume s < |p|1/(p−1)r . Thendiam

(

f (D(a, s)))

diam(

f (D(0, r))) = |d |

s

r.

Idea of Proof. Rewrite f (z) as a power series

∞∑

n=0

bn(z − a)n

centered at a, and use |cn|rn < |dcd |r

d to show that

|b1|s = |dcdad−1|s > |bn|sn for all n ≥ 2.

Page 71: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

A Power Series Lemma

LemmaLet a ∈ C×

K , set r = |a|, and let 0 < s < r .

Let f (z) = c0 + cdzd + · · · ∈ CK [[z ]] converge on D(0, r), andassume that |cn|r

n < |dcd |rd for all n > d ≥ 1.

(In particular, f has no critical points in D(0, r) except maybe atz = 0; and in positive characteristic, z = 0 is not wild.)

If p|d, assume s < |p|1/(p−1)r . Thendiam

(

f (D(a, s)))

diam(

f (D(0, r))) = |d |

s

r.

Idea of Proof. Rewrite f (z) as a power series

∞∑

n=0

bn(z − a)n

centered at a, and use |cn|rn < |dcd |r

d to show that

|b1|s = |dcdad−1|s > |bn|sn for all n ≥ 2.

So diam(

f (D(a, s)))

= |dcd |rd−1s, and

diam(

f (D(0, r)))

= |cd |rd .

Page 72: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Sketch of Proof of No Wandering DomainsChange coordinates so that Jφ ⊆ D(0, 1).Extend K to include all critical points of φ and some point of asupposed wandering domain U.

Page 73: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Sketch of Proof of No Wandering DomainsChange coordinates so that Jφ ⊆ D(0, 1).Extend K to include all critical points of φ and some point of asupposed wandering domain U.

Note that φn(U) is a disk, and does not contain any critical points,for all n big enough.

Page 74: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Sketch of Proof of No Wandering DomainsChange coordinates so that Jφ ⊆ D(0, 1).Extend K to include all critical points of φ and some point of asupposed wandering domain U.

Note that φn(U) is a disk, and does not contain any critical points,for all n big enough.

For any n ≥ 0, let Vn ) φn(U) be a slightly larger disk.

Then Vn intersects Jφ,Ber, so the forward iterates of Vn get big.

Page 75: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Sketch of Proof of No Wandering DomainsChange coordinates so that Jφ ⊆ D(0, 1).Extend K to include all critical points of φ and some point of asupposed wandering domain U.

Note that φn(U) is a disk, and does not contain any critical points,for all n big enough.

For any n ≥ 0, let Vn ) φn(U) be a slightly larger disk.

Then Vn intersects Jφ,Ber, so the forward iterates of Vn get big.

By the no wild (recurrent) Julia critical hypothesis, there is aradius R > 0 so that φm(Vn) has to get up to radius at least Rbefore it can contain any (or more than M) wild critical points.

Page 76: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Sketch of Proof of No Wandering DomainsChange coordinates so that Jφ ⊆ D(0, 1).Extend K to include all critical points of φ and some point of asupposed wandering domain U.

Note that φn(U) is a disk, and does not contain any critical points,for all n big enough.

For any n ≥ 0, let Vn ) φn(U) be a slightly larger disk.

Then Vn intersects Jφ,Ber, so the forward iterates of Vn get big.

By the no wild (recurrent) Julia critical hypothesis, there is aradius R > 0 so that φm(Vn) has to get up to radius at least Rbefore it can contain any (or more than M) wild critical points.

By the power series lemma, φm+n(U) has to have radius at leastabout R (or |p|M

R).

Page 77: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Sketch of Proof of No Wandering DomainsChange coordinates so that Jφ ⊆ D(0, 1).Extend K to include all critical points of φ and some point of asupposed wandering domain U.

Note that φn(U) is a disk, and does not contain any critical points,for all n big enough.

For any n ≥ 0, let Vn ) φn(U) be a slightly larger disk.

Then Vn intersects Jφ,Ber, so the forward iterates of Vn get big.

By the no wild (recurrent) Julia critical hypothesis, there is aradius R > 0 so that φm(Vn) has to get up to radius at least Rbefore it can contain any (or more than M) wild critical points.

By the power series lemma, φm+n(U) has to have radius at leastabout R (or |p|M

R).

So U has infinitely many non-overlapping iterates of radiusbounded below and intersecting the compact set OK ,

Page 78: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Sketch of Proof of No Wandering DomainsChange coordinates so that Jφ ⊆ D(0, 1).Extend K to include all critical points of φ and some point of asupposed wandering domain U.

Note that φn(U) is a disk, and does not contain any critical points,for all n big enough.

For any n ≥ 0, let Vn ) φn(U) be a slightly larger disk.

Then Vn intersects Jφ,Ber, so the forward iterates of Vn get big.

By the no wild (recurrent) Julia critical hypothesis, there is aradius R > 0 so that φm(Vn) has to get up to radius at least Rbefore it can contain any (or more than M) wild critical points.

By the power series lemma, φm+n(U) has to have radius at leastabout R (or |p|M

R).

So U has infinitely many non-overlapping iterates of radiusbounded below and intersecting the compact set OK , acontradiction. QED

Page 79: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

There Can be Wandering Domains

One of the hypotheses of the No Wandering Domains result is thatφ is defined over a locally compact subfield of CK .

Page 80: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

There Can be Wandering Domains

One of the hypotheses of the No Wandering Domains result is thatφ is defined over a locally compact subfield of CK .

But if we relax that condition, we can find wandering domains.

TheoremLet CK have residue field k that is not algebraic over a finite field.

Page 81: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

There Can be Wandering Domains

One of the hypotheses of the No Wandering Domains result is thatφ is defined over a locally compact subfield of CK .

But if we relax that condition, we can find wandering domains.

TheoremLet CK have residue field k that is not algebraic over a finite field.

Then any φ(z) ∈ CK (z) with a type II Julia periodic point ζ haswandering domains “in the basin of attraction” of ζ.

Page 82: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

There Can be Wandering Domains

One of the hypotheses of the No Wandering Domains result is thatφ is defined over a locally compact subfield of CK .

But if we relax that condition, we can find wandering domains.

TheoremLet CK have residue field k that is not algebraic over a finite field.

Then any φ(z) ∈ CK (z) with a type II Julia periodic point ζ haswandering domains “in the basin of attraction” of ζ.

The wandering domains in question are just wandering residueclasses of ζ whose iterates avoid “bad”residue classes.

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No Other Wandering Domains

Theorem (RB, 2005)

Let K be a complete discretely valued non-archimedean field ofresidue characteristic zero, let CK be the completion of analgebraic closure of K, and let φ ∈ K (z) be a rational function ofdegree d ≥ 2.

Page 84: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

No Other Wandering Domains

Theorem (RB, 2005)

Let K be a complete discretely valued non-archimedean field ofresidue characteristic zero, let CK be the completion of analgebraic closure of K, and let φ ∈ K (z) be a rational function ofdegree d ≥ 2.Then φ has no wandering domains besides those in attractingbasins of periodic type II points.

Page 85: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

No Other Wandering Domains

Theorem (RB, 2005)

Let K be a complete discretely valued non-archimedean field ofresidue characteristic zero, let CK be the completion of analgebraic closure of K, and let φ ∈ K (z) be a rational function ofdegree d ≥ 2.Then φ has no wandering domains besides those in attractingbasins of periodic type II points.

Theorem (Trucco, 2009)

Let L be the field of Puiseux series over Q, and let φ ∈ L[z ] be apolynomial of degree d ≥ 2.

Page 86: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

No Other Wandering Domains

Theorem (RB, 2005)

Let K be a complete discretely valued non-archimedean field ofresidue characteristic zero, let CK be the completion of analgebraic closure of K, and let φ ∈ K (z) be a rational function ofdegree d ≥ 2.Then φ has no wandering domains besides those in attractingbasins of periodic type II points.

Theorem (Trucco, 2009)

Let L be the field of Puiseux series over Q, and let φ ∈ L[z ] be apolynomial of degree d ≥ 2.Then φ has no wandering domains besides those in attractingbasins of periodic type II points.

Page 87: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Subtler Wandering Domains

Even for Cp and other fields with residue field Fp, there can bewandering domains not associated with type II periodic points.

Page 88: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Subtler Wandering Domains

Even for Cp and other fields with residue field Fp, there can bewandering domains not associated with type II periodic points.

Theorem (RB, 2002)

Let CK have residue characteristic p > 0.

Page 89: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Subtler Wandering Domains

Even for Cp and other fields with residue field Fp, there can bewandering domains not associated with type II periodic points.

Theorem (RB, 2002)

Let CK have residue characteristic p > 0.Then there is a parameter a ∈ CK (in fact, a dense set of suchparameters in CK r D(0, 1)) such that

φa(z) := (1 − a)zp+1 + azp

has a wandering domain not in the attracting basin of a periodictype II point.

Page 90: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Subtler Wandering Domains

Even for Cp and other fields with residue field Fp, there can bewandering domains not associated with type II periodic points.

Theorem (RB, 2002)

Let CK have residue characteristic p > 0.Then there is a parameter a ∈ CK (in fact, a dense set of suchparameters in CK r D(0, 1)) such that

φa(z) := (1 − a)zp+1 + azp

has a wandering domain not in the attracting basin of a periodictype II point.

(Idea of Proof: see Project #4)

Page 91: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Other Ways to Break the Hypotheses

What if we stick to K locally compact?

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Other Ways to Break the Hypotheses

What if we stick to K locally compact?It is easy to force a wild critical point into the Julia set.

Example. 0 < |π| < 1, and φ(z) = π−1(zp+1 − zp) + 1, whichmaps 0 7→ 1 7→ 1, with 0 wild critical and 1 repelling fixed.

Page 93: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Other Ways to Break the Hypotheses

What if we stick to K locally compact?It is easy to force a wild critical point into the Julia set.

Example. 0 < |π| < 1, and φ(z) = π−1(zp+1 − zp) + 1, whichmaps 0 7→ 1 7→ 1, with 0 wild critical and 1 repelling fixed.

What about wild recurrent Julia critical points?

Page 94: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Other Ways to Break the Hypotheses

What if we stick to K locally compact?It is easy to force a wild critical point into the Julia set.

Example. 0 < |π| < 1, and φ(z) = π−1(zp+1 − zp) + 1, whichmaps 0 7→ 1 7→ 1, with 0 wild critical and 1 repelling fixed.

What about wild recurrent Julia critical points?

Theorem (Rivera-Letelier, 2005)

Let K be a complete non-archimedean field of residuecharacteristic p. Then there are polynomials φ ∈ K [z ] with wildrecurrent Julia critical points.

Proof. See project #4.

Page 95: Non-archimedean Dynamics in Dimension One: Lecture 3swc.math.arizona.edu/aws/2010/2010BenedettoLectureNotes3.pdfLecture 3 Robert L. Benedetto Amherst College Arizona Winter School

Other Ways to Break the Hypotheses

What if we stick to K locally compact?It is easy to force a wild critical point into the Julia set.

Example. 0 < |π| < 1, and φ(z) = π−1(zp+1 − zp) + 1, whichmaps 0 7→ 1 7→ 1, with 0 wild critical and 1 repelling fixed.

What about wild recurrent Julia critical points?

Theorem (Rivera-Letelier, 2005)

Let K be a complete non-archimedean field of residuecharacteristic p. Then there are polynomials φ ∈ K [z ] with wildrecurrent Julia critical points.

Proof. See project #4.

In both cases (char K = p > 0 with wild Julia critical points, orchar K = 0 with wild recurrent Julia critical points), we don’t knowwhether there can be wandering domains.