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

Non-archimedean Dynamics in Dimension One:Lecture 3

Robert L. BenedettoAmherst College

Arizona Winter School

Still Sunday, π Day, 2010

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.

φ(z) = z2 + az ∈ CK [z ].

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

φ(z) = z2 + az ∈ CK [z ].

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

φ(z) = z2 + az ∈ CK [z ].

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

φ(z) = z2 + az ∈ CK [z ].

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.

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.

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.

Periodic Fatou Components

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

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.

◮ 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.

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.

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),

◮ (f , g) = 1,

then deg(f /g) = deg φ.

◮ (f , g) = 1,

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

◮ (f , g) = 1,

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

◮ (f , g) = 1,

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

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,

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 =⋃

Rivera-Letelier’s Classification: Continued

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.

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.

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.

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.

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.

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.)

Example: A Non-disk Indifferent Component

φ(z) =1

1 − z+ π2z =

π2z2 − π2z − 1

z − 1∈ CK [z ], with 0 < |π| < 1.

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),

φ(z) =1

1 − z+ π2z =

π2z2 − π2z − 1

z − 1∈ CK [z ], with 0 < |π| < 1.

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

1 − z.

φ(z) =1

1 − z+ π2z =

π2z2 − π2z − 1

z − 1∈ CK [z ], with 0 < |π| < 1.

1 − z.

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

φ(z) =1

1 − z+ π2z =

π2z2 − π2z − 1

z − 1∈ CK [z ], with 0 < |π| < 1.

1 − z.

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.

φ(z) =1

1 − z+ π2z =

π2z2 − π2z − 1

z − 1∈ CK [z ], with 0 < |π| < 1.

1 − z.

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.

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 .

n≥1 φn(x).

Theorem (RB, 1998)

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

n≥1 φn(x).

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.)

n≥1 φn(x).

Theorem (RB, 1998)

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

n≥1 φn(x).

Theorem (RB, 1998)

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.

n≥1 φn(x).

Theorem (RB, 1998)

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.

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.

LemmaLet a ∈ C×

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

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

LemmaLet a ∈ C×

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

If p|d, assume s < |p|1/(p−1)r .

LemmaLet a ∈ C×

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

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

f (D(a, s)))

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

LemmaLet a ∈ C×

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

f (D(a, s)))

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

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

∞∑

bn(z − a)n

centered at a,

LemmaLet a ∈ C×

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

f (D(a, s)))

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

∞∑

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.

LemmaLet a ∈ C×

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

f (D(a, s)))

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

∞∑

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

f (D(0, r)))

= |cd |rd .

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

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

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

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.

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.

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 (RB, 2005)

Theorem (Trucco, 2009)

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

Theorem (RB, 2005)

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.

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.

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.

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)

Other Ways to Break the Hypotheses

What if we stick to K locally compact?

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?

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.

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.

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

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