Puiseux series dynamics and leading monomials of escape regions Jan - - PowerPoint PPT Presentation

Puiseux series dynamics and leading monomials

• f escape regions

Jan Kiwi PUC, Chile Workshop on Moduli Spaces Associated to Dynamical Systems ICERM Providence, April 17, 2012

Parameter space

Monic centered cubic polynomials with marked critical points:

Parameter space

Monic centered cubic polynomials with marked critical points:

fa,v : z → (z − a)2(z + 2a) + v

where (a, v) ∈ C2.

Parameter space

Monic centered cubic polynomials with marked critical points:

fa,v : z → (z − a)2(z + 2a) + v

where (a, v) ∈ C2. After identification of (a, v) with (−a, −v) one obtains the moduli space of cubic polynomials with marked critical points.

Parameter space

Monic centered cubic polynomials with marked critical points:

fa,v : z → (z − a)2(z + 2a) + v

where (a, v) ∈ C2. After identification of (a, v) with (−a, −v) one obtains the moduli space of cubic polynomials with marked critical points. Critical points of fa,v are ±a. Critical value: v = fa,v(+a). Co-critical value −2a, since v = fa,v(+a) = fa,v(−2a).

Periodic Curves

For p ≥ 1, the periodic curve Sp consists of all (a, v) such that +a has period p under fa,v.

Periodic Curves

For p ≥ 1, the periodic curve Sp consists of all (a, v) such that +a has period p under fa,v. Sp ⊂ C2 is a smooth affine algebraic curve of degree dp where

• n|p dn = 3p−1.

Periodic Curves

For p ≥ 1, the periodic curve Sp consists of all (a, v) such that +a has period p under fa,v. Sp ⊂ C2 is a smooth affine algebraic curve of degree dp where

• n|p dn = 3p−1.

What is its topology?

Periodic Curves

For p ≥ 1, the periodic curve Sp consists of all (a, v) such that +a has period p under fa,v. Sp ⊂ C2 is a smooth affine algebraic curve of degree dp where

• n|p dn = 3p−1.

What is its topology? Is Sp irreducible?

Periodic Curves

For p ≥ 1, the periodic curve Sp consists of all (a, v) such that +a has period p under fa,v. Sp ⊂ C2 is a smooth affine algebraic curve of degree dp where

• n|p dn = 3p−1.

What is its topology? Is Sp irreducible? Bonifant,-,Milnor: the Euler characteristic is Sp is (2 − p)dp.

Periodic Curves

For p ≥ 1, the periodic curve Sp consists of all (a, v) such that +a has period p under fa,v. Sp ⊂ C2 is a smooth affine algebraic curve of degree dp where

• n|p dn = 3p−1.

What is its topology? Is Sp irreducible? Bonifant,-,Milnor: the Euler characteristic is Sp is (2 − p)dp. What is the Euler characteristic of the smooth compactification of Sp?

Periodic Curves

For p ≥ 1, the periodic curve Sp consists of all (a, v) such that +a has period p under fa,v. Sp ⊂ C2 is a smooth affine algebraic curve of degree dp where

• n|p dn = 3p−1.

What is its topology? Is Sp irreducible? Bonifant,-,Milnor: the Euler characteristic is Sp is (2 − p)dp. What is the Euler characteristic of the smooth compactification of Sp? Requires to compute the number Np of “escape regions”. (De Marco and Schiff, De Marco-Pilgrim).

Escape Regions

The connectedness locus C(Sp) =

• fa,v ∈ Sp | fn

a,v(−a) → ∞

• is compact.

Escape Regions

The connectedness locus C(Sp) =

• fa,v ∈ Sp | fn

a,v(−a) → ∞

• is compact.

The escape locus E(Sp) =

• fa,v ∈ Sp | fn

a,v(−a) → ∞

• is open and every connected component is unbounded.

Escape Regions

The connectedness locus C(Sp) =

• fa,v ∈ Sp | fn

a,v(−a) → ∞

• is compact.

The escape locus E(Sp) =

• fa,v ∈ Sp | fn

a,v(−a) → ∞

• is open and every connected component is unbounded.

A escape region U is a connected component of E(Sp): U is conformally isomorphic to punctured disk. The puncture is at ∞U.

Asymptotics of critical periodic orbit

For fa,v ∈ U, let

a0 = +a → a1 = v → · · · → ap−1 → a0.

Asymptotics of critical periodic orbit

For fa,v ∈ U, let

a0 = +a → a1 = v → · · · → ap−1 → a0.

Dynamical space picture. After conjugacy, fa,v(az)/a, we have:

Asymptotics of critical periodic orbit

For fa,v ∈ U, let

a0 = +a → a1 = v → · · · → ap−1 → a0.

Dynamical space picture. After conjugacy, fa,v(az)/a, we have:

aj =

      

a + o(a)

• r

−2a + o(a).

Asymptotics of critical periodic orbit

For fa,v ∈ U, let

a0 = +a → a1 = v → · · · → ap−1 → a0.

Dynamical space picture. After conjugacy, fa,v(az)/a, we have:

aj =

      

a + o(a)

• r

−2a + o(a). Bonifant and Milnor: Do the leading terms of aj − a, for j = 1, . . . , p − 1 determine U uniquely?

Leading monomials

There exists µ ≥ 1 and a local coordinate ζ for U near ∞ such that

a = 1

ζµ .

Leading monomials

There exists µ ≥ 1 and a local coordinate ζ for U near ∞ such that

a = 1

ζµ . Then,

aj − a a

= holomorphic(ζ) =

• k≥k0

ckζk =

• k≥k0

ck

1

a

k/µ .

Leading monomials

There exists µ ≥ 1 and a local coordinate ζ for U near ∞ such that

a = 1

ζµ . Then,

aj − a a

= holomorphic(ζ) =

• k≥k0

ckζk =

• k≥k0

ck

1

a

k/µ . Leading monomial

mj = ck0

1

a

k0/µ .

• Theorem. The leading monomial vector
• m = (m1, . . . , mp−1, 0)

determines the escape region uniquely.

• Theorem. The leading monomial vector
• m = (m1, . . . , mp−1, 0)

determines the escape region uniquely.

• Corollary. (Bonifant,-,Milnor) Assume that the leading

monomial vector of U is (c1a−k1/µ, . . . , cp−1a−kp−1/µ). Then, for all j,

aj ∈ Q(c1, . . . , cp−1)((a−1/µ)).

Equations

For an escape region U,

v = a1 = (+a or − 2a) + a ·

• k≥k0

ck

1

a

k/µ is a solution of

fp

a,v(+a) = +a

(*) in some extension of Q((1/a)).

Field

Put t instead of 1/a to get Q((t)) and study solutions of (*) in the algebraic closure of Q((t)): Qa ≪ t ≫= ∪Qa((t1/m)).

Field

Put t instead of 1/a to get Q((t)) and study solutions of (*) in the algebraic closure of Q((t)): Qa ≪ t ≫= ∪Qa((t1/m)). Qa ≪ t ≫ is endowed with |z =

• k≥k0

cktk/m| = e− ord0(z) = e−k0/m.

Field

Put t instead of 1/a to get Q((t)) and study solutions of (*) in the algebraic closure of Q((t)): Qa ≪ t ≫= ∪Qa((t1/m)). Qa ≪ t ≫ is endowed with |z =

• k≥k0

cktk/m| = e− ord0(z) = e−k0/m.

Complete it to get L.

Non-Archimedean problem

For ν ∈ L, consider ψν(z) = t−2(z − 1)(z + 2) + ν ∈ L[z].

Non-Archimedean problem

For ν ∈ L, consider ψν(z) = t−2(z − 1)(z + 2) + ν ∈ L[z]. ν is periodic parameter if ω+ = +1 is periodic under ψν.

Non-Archimedean problem

For ν ∈ L, consider ψν(z) = t−2(z − 1)(z + 2) + ν ∈ L[z]. ν is periodic parameter if ω+ = +1 is periodic under ψν. In this case: ω+ = +1 → ω+

1 = ν → · · · → ω+ p−1 → ω+.

Non-Archimedean problem

For ν ∈ L, consider ψν(z) = t−2(z − 1)(z + 2) + ν ∈ L[z]. ν is periodic parameter if ω+ = +1 is periodic under ψν. In this case: ω+ = +1 → ω+

1 = ν → · · · → ω+ p−1 → ω+.

It follows, ω+

j =

       +1 + c · tk/m + h.o.t. −2 + d · tℓ/n + h.o.t.

Non-Archimedean problem

For ν ∈ L, consider ψν(z) = t−2(z − 1)(z + 2) + ν ∈ L[z]. ν is periodic parameter if ω+ = +1 is periodic under ψν. In this case: ω+ = +1 → ω+

1 = ν → · · · → ω+ p−1 → ω+.

It follows, ω+

j =

       +1 + c · tk/m + h.o.t. −2 + d · tℓ/n + h.o.t. The corresponding leading monomial are

m(ω+

j − ω+) =

      

c · tk/m

−3 respectively.

• Theorem. The leading monomial vector
• m = (m1, . . . , mp−1, 0)

determines the periodic parameter ν uniquely.

Dynamical Space

The filled Julia set of ψν is

K(ψν) = {z ∈ L | ψn

ν(z) → ∞}.

Dynamical Space

The filled Julia set of ψν is

K(ψν) = {z ∈ L | ψn

ν(z) → ∞}.

|ν| ≤ 1 =⇒ ψn

ν(z) → ∞ when |z| > 1.

Dynamical Space

The filled Julia set of ψν is

K(ψν) = {z ∈ L | ψn

ν(z) → ∞}.

|ν| ≤ 1 =⇒ ψn

ν(z) → ∞ when |z| > 1.

Level 0 dynamical ball D0 = {|z| ≤ 1}.

Dynamical Space

The filled Julia set of ψν is

K(ψν) = {z ∈ L | ψn

ν(z) → ∞}.

|ν| ≤ 1 =⇒ ψn

ν(z) → ∞ when |z| > 1.

Level 0 dynamical ball D0 = {|z| ≤ 1}. Level n set {ψn

ν(z) ∈ D0}

is a disjoint union of finitely many dynamical balls of level n.

Dynamical Space

The filled Julia set of ψν is

K(ψν) = {z ∈ L | ψn

ν(z) → ∞}.

|ν| ≤ 1 =⇒ ψn

ν(z) → ∞ when |z| > 1.

Level 0 dynamical ball D0 = {|z| ≤ 1}. Level n set {ψn

ν(z) ∈ D0}

is a disjoint union of finitely many dynamical balls of level n. Each level n + 1 ball is contained in, and maps onto, a level n ball.

Branches

Maximal open balls in a closed ball D are parametrized by Qa. If ψν maps D onto D′ by degree d, then it induces a polynomial map of degree d in Qa.

Computing |ω+

j − ω+|.

• Fact. If B is a maximal open ball of a dynamical ball Dℓ of level ℓ,

then B contains at most one dynamical ball of level ℓ + 1.

Computing |ω+

j − ω+|.

• Fact. If B is a maximal open ball of a dynamical ball Dℓ of level ℓ,

then B contains at most one dynamical ball of level ℓ + 1.

• Consequence. If Dℓ(ω+) = Dℓ(ω+

j ) but Dℓ+1(ω+) Dℓ+1(ω+ j ),

then |ω+

j − ω+| is the diameter of Dℓ(ω+) = Dℓ(ω+ j ).

Parameter space

The level 0 parameter ball is D0 = {|ψν(ω+) = ν| ≤ 1}.

Parameter space

The level 0 parameter ball is D0 = {|ψν(ω+) = ν| ≤ 1}. The parameters of level n {|ψn−1

ν

(ω+)| ≤ 1} are a disjoint union of closed balls Dn called level n parameter balls.

Parameter ball, dynamical balls and branch dynamics

• Proposition. Assume Dℓ is a level ℓ parameter ball. Then:

Dℓ = Dℓ(ν) for all ν ∈ Dℓ.

Parameter ball, dynamical balls and branch dynamics

• Proposition. Assume Dℓ is a level ℓ parameter ball. Then:

Dℓ = Dℓ(ν) for all ν ∈ Dℓ. The level ℓ + 1 − j dynamical ball ψj

ν(Dℓ(ν)) is independent of

ν ∈ Dℓ.

Parameter ball, dynamical balls and branch dynamics

• Proposition. Assume Dℓ is a level ℓ parameter ball. Then:

Dℓ = Dℓ(ν) for all ν ∈ Dℓ. The level ℓ + 1 − j dynamical ball ψj

ν(Dℓ(ν)) is independent of

ν ∈ Dℓ. The ψν action on maximal open balls is also independent of ν ∈ Dℓ.

Parameter ball, dynamical balls and branch dynamics

• Proposition. Assume Dℓ is a level ℓ parameter ball. Then:

Dℓ = Dℓ(ν) for all ν ∈ Dℓ. The level ℓ + 1 − j dynamical ball ψj

ν(Dℓ(ν)) is independent of

ν ∈ Dℓ. The ψν action on maximal open balls is also independent of ν ∈ Dℓ. If pℓ is the smallest integer q such that ω+ ∈ ψq−1

ν

(Dℓ(ν)), then every periodic parameter in Dℓ has period at least pℓ.

Centers

• Proposition. There exists a unique ν in Dℓ which is periodic with

period pℓ.

Centers

• Proposition. There exists a unique ν in Dℓ which is periodic with

period pℓ.

• Proof. The map T : Dℓ → Dℓ defined by

T(ν) =

• ψpℓ−1

ν

|Dℓ(ν) −1 (ω+) is a strict contraction.

• Such ν is called the center of Dℓ.

Level ℓ + 1 correspondence

• Proposition. Let B be a maximal open ball of a parameter ball

Dℓ and consider any ν ∈ Dℓ.

B contains a parameter ball of level ℓ + 1

if and only if

B contains a dynamical ball of level ℓ + 1.

In this case, the level ℓ + 1 balls are unique.

Leading monomials determine parameter balls

Assume ν, ν′ have the same leading monomial vector.

Leading monomials determine parameter balls

Assume ν, ν′ have the same leading monomial vector. Let us prove by induction on ℓ. Dℓ(ν) = Dℓ(ν′)

Leading monomials determine parameter balls

Assume ν, ν′ have the same leading monomial vector. Let us prove by induction on ℓ. Dℓ(ν) = Dℓ(ν′) Take νℓ the center of level ℓ.

Leading monomials determine parameter balls

Assume ν, ν′ have the same leading monomial vector. Let us prove by induction on ℓ. Dℓ(ν) = Dℓ(ν′) Take νℓ the center of level ℓ. There is a unique maximal open ball B in Dℓ(νℓ) whose orbit is compatible with the leading monomial vector.

Leading monomials determine parameter balls

Assume ν, ν′ have the same leading monomial vector. Let us prove by induction on ℓ. Dℓ(ν) = Dℓ(ν′) Take νℓ the center of level ℓ. There is a unique maximal open ball B in Dℓ(νℓ) whose orbit is compatible with the leading monomial vector. That is, ν, ν′ belong to B Thus ν, ν′ belong to the unique level ℓ + 1 parameter ball contained in B. Hence, Dℓ+1(ν) = Dℓ+1(ν′).

Homework

Given a sequence D0 ⊃ D1 ⊃ · · · . With centers ν0, ν1, · · · . Which lie in

L0((t1/m0)) ⊂ L1((t1/m1)) ⊂ · · ·

Branner and Hubbard tell us how to compute mk.

Homework

Given a sequence D0 ⊃ D1 ⊃ · · · . With centers ν0, ν1, · · · . Which lie in

L0((t1/m0)) ⊂ L1((t1/m1)) ⊂ · · ·

Branner and Hubbard tell us how to compute mk. Compute Lk.