polynomial matings and rational maps with cluster cyclestsharland/harvard.pdfγ2(−θ). the maps fi...

Polynomial Matings and Rational Maps with ClusterCycles

Thomas Sharland

Mathematics InstituteUniversity of Warwick

4th April 2012Harvard Informal Seminar

Tom Sharland (University of Warwick) Mating and Clustering 4th April 2012 1 / 27

Outline

1 IntroductionStandard DefinitionsMatingsThurston Equivalence

2 ClusteringCombinatorial data

3 ResultsThurston EquivalenceFixed Cluster pointsPeriod 2 cluster cycle resultsHigher Degrees

Definitions.

Let f : C → C be a rational map.

The Julia set J(f) is the closure of the set of repelling periodicpoints of f .

The Fatou set F (f) is C \ J(f).

If f is a polynomial

The point ∞ is a superattracting fixed point.

The filled Julia set is K(f) = z ∈ C | fn(z) 9 ∞, so thatJ(f) = ∂K(f)

In this talk, we will generally assume that f has a (finite)superattracting periodic cycle of period p > 1.

Definitions.

Let f : C → C be a rational map.

The Julia set J(f) is the closure of the set of repelling periodicpoints of f .

The Fatou set F (f) is C \ J(f).

If f is a polynomial

The point ∞ is a superattracting fixed point.

The filled Julia set is K(f) = z ∈ C | fn(z) 9 ∞, so thatJ(f) = ∂K(f)

In this talk, we will generally assume that f has a (finite)superattracting periodic cycle of period p > 1.

Böttcher’s theorem and external rays

There exists an analytic conjugacy φ between f on C \K(f) and themap z 7→ zd on C \ D.

φ is asymptotic to the identity at ∞.

The set Rθ = φ−1re2πiθ | r ∈ (1,∞) is called the external ray ofangle θ.

If J(f) is locally connected, the point γ(θ) = limr→1 φ−1(re2πiθ)

exists for all θ and belongs to J(f).We have the identities f(Rθ) = Rdθ and f(γ(θ)) = γ(dθ).

The points βk = γ(k/(d− 1)), k = 0, 1, . . . , d− 2 are fixed points onJ(f).If α ∈ J(f) is another fixed point and α = γ(θ), thenα = γ(dθ), γ(d2θ), . . .

An exampleDouady’s Rabbit

Douady’s rabbit with the external rays of angle p/7.

Formal MatingsNew maps from old

Denote fi(z) = zd + ci. Define C = C ∪ ∞ · e2πis | s ∈ R/Z.

Extend f1 and f2 to the boundary circle at infinity, e.g.f1(∞ · e2πis) = ∞ · e2dπis.

Define S2f1,f2

= Cf1 ⊎ Cf2/(∞ · e2πis, f1) ∼ (∞ · e−2πis, f2).

The formal mating is a branched covering f1 ⊎ f2 : S2f1,f2

→ S2f1,f2

f1 ⊎ f2 = f1 on Cf1

f1 ⊎ f2 = f2 on Cf2

Define S2f1,f2

= Cf1 ⊎ Cf2/(∞ · e2πis, f1) ∼ (∞ · e−2πis, f2).

→ S2f1,f2

f1 ⊎ f2 = f1 on Cf1

f1 ⊎ f2 = f2 on Cf2

Define S2f1,f2

= Cf1 ⊎ Cf2/(∞ · e2πis, f1) ∼ (∞ · e−2πis, f2).

→ S2f1,f2

f1 ⊎ f2 = f1 on Cf1

f1 ⊎ f2 = f2 on Cf2

Topological Matings

Denote fi(z) = zd + ci with filled Julia set Ki = K(fi).Define a new map f1 ⊥⊥ f2 on K1 ⊥⊥ K2.

Take the disjoint union of K1 and K2.K1 ⊥⊥ K2 is the quotient space formed by identifying γ1(θ) withγ2(−θ).The maps fi on the Ki fit together to form a new map f1 ⊥⊥ f2.

We say f1 and f2 are mateable if this quotient K1 ⊥⊥ K2 is a sphere.

Topological Matings

Obstructed matings

Let f1 be the rabbit polynomial and f2 the anti-rabbit polynomial.

Obstructed matings

The quotient space created by the ray equivalence class is nothomeomorphic to the sphere.

So f1 ⊥⊥ f2 is not equivalent to a rational map.

The same holds for any mating of f1 with a map in the anti-rabbitlimb.

DefinitionA multicurve Γ = γ1, . . . , γn of F is called a Levy cycle if fori = 1, 2, . . . , n, the curve γi−1 is homotopic (rel PF ) to a componentγ′i−1 of F−1(γi) and the map F : γ′i → γi is a homeomorphism.

Obstructed matings

The quotient space created by the ray equivalence class is nothomeomorphic to the sphere.

So f1 ⊥⊥ f2 is not equivalent to a rational map.

The same holds for any mating of f1 with a map in the anti-rabbitlimb.

DefinitionA multicurve Γ = γ1, . . . , γn of F is called a Levy cycle if fori = 1, 2, . . . , n, the curve γi−1 is homotopic (rel PF ) to a componentγ′i−1 of F−1(γi) and the map F : γ′i → γi is a homeomorphism.

Thurston’s Theorem

Let F : Σ → Σ and F : Σ → Σ be postcritically finiteorientation-preserving branched self-coverings of topological2-spheres. An equivalence is given by a pair of orientation-preservinghomeomorphisms (Φ,Ψ) from Σ to Σ such that

Φ|PF= Ψ|PF

Φ F = Ψ F

Φ and Ψ are isotopic via a family of homeomorphisms t 7→ Φtwhich is constant on PF .

Theorem (Thurston)Let F : Σ → Σ be a postcritically finite branched cover with hyperbolicorbifold. Then F is equivalent to a rational map if and only if F has noThurston obstructions. This rational map is unique up to Möbiustransformation. In particular, any equivalence between two rationalmaps is realised by a Möbius transformation.

Thurston’s Theorem

Let F : Σ → Σ and F : Σ → Σ be postcritically finiteorientation-preserving branched self-coverings of topological2-spheres. An equivalence is given by a pair of orientation-preservinghomeomorphisms (Φ,Ψ) from Σ to Σ such that

Φ|PF= Ψ|PF

Φ F = Ψ F

Φ and Ψ are isotopic via a family of homeomorphisms t 7→ Φtwhich is constant on PF .

Theorem (Thurston)Let F : Σ → Σ be a postcritically finite branched cover with hyperbolicorbifold. Then F is equivalent to a rational map if and only if F has noThurston obstructions. This rational map is unique up to Möbiustransformation. In particular, any equivalence between two rationalmaps is realised by a Möbius transformation.

Simplifying Thurston’s criterion

In general it is difficult to find Thurston obstructions. Levy cyclessimplify the search.

F has a Levy cycle ⇒ F has a Thurston obstruction.

In the bicritical case: F has a Thurston obstruction ⇒ F has aLevy cycle.

Theorem (Rees, Shishikura, Tan)

In the bicritical case, if [α1] 6= [α2], K1 ⊥⊥ K2 is homeomorphic to S2

and we can give this sphere a unique conformal structure to makef1 ⊥⊥ f2 a holomorphic degree d rational map.

Simplifying Thurston’s criterion

In general it is difficult to find Thurston obstructions. Levy cyclessimplify the search.

F has a Levy cycle ⇒ F has a Thurston obstruction.

In the bicritical case: F has a Thurston obstruction ⇒ F has aLevy cycle.

Theorem (Rees, Shishikura, Tan)

In the bicritical case, if [α1] 6= [α2], K1 ⊥⊥ K2 is homeomorphic to S2

and we can give this sphere a unique conformal structure to makef1 ⊥⊥ f2 a holomorphic degree d rational map.

Clusters (Definition)

Let F be a bicritical rational map such that the two critical points belongto the attracting basins of two disjoint (super)attracting periodic orbitsof the same period.Clustering is the condition that the critical orbit Fatou componentsgroup together to form a periodic cycle...

The dynamics on each Fatou component can be conjugated usingBöttcher’s theorem.

Internal rays

The 0 internal ray is fixed under the first return map.

If the 0 internal rays meet at a point c, and this point is periodic,we say c is a cluster point for F .

e.g. Rabbit ⊥⊥ Airplane.

Internal rays

Example

The Julia set for Rabbit ⊥⊥ Airplane(and Airplane ⊥⊥ Rabbit!).

Another example

The Julia set for a map with a period two cluster cycle.

Combinatorial data

Restrict attention to the period one and two cases.We can describe a cluster in simple combinatorial terms.

1 (The period of the critical cycles n.)2 The combinatorial rotation number ρ.3 The critical displacement δ.

The combinatorial data of the cluster will be the pair (ρ, δ).

Combinatorial data

Clusters and Thurston Equivalence

Theorem (S., 2010)If F and G are bicritical rational maps of the same degree and

F and G have fixed cluster points and are of degree d

F and G are quadratic with a period 2 cluster cycle

then F and G have the same combinatorial data if and only if they areThurston equivalent.

In particular, this means that a rational map is entirely determined byits behaviour inside the cluster cycle.The theorem is not true in the case where F and G have degree d ≥ 3and a period two cluster cycle (An example is given in joint work withAdam Epstein.)

Proof Outline

In both cases we use the following method. let XF and XG be theunion of the stars of the clusters of F and G.

1 There exists φ : XF → XG which conjugates the dynamics.2 Extend φ to a map Φ: C → C.3 Construct a new map Φ such that Φ F = G Φ.4 Modify Φ so that it is equal to φ on XF .5 Modify this new Φ so that it is isotopic to Φ rel XF .

Proof Outline

A bit more detail. . .

(C,XF )

(Φ,φ)

F%%JJJJJJJJJ

(C \ Ω, ∂Ω)ηFoo

(Ψ,ψ)

(C,XF )

(Φ,φ)

(C \ Ω, ∂Ω)ηFoo

(Ψ,ψ)

(C,XG) (C \ Ω, ∂Ω)ηG

(C,XG)

G99ttttttttt

(C \ Ω, ∂Ω)ηG

Ω is either D (fixed case) or an annulus (period 2 case).

Period 1 Results

A rabbit is any map with a “star-shaped” Hubbard tree. They belong tohyperbolic components which bifurcate from the (unique) period onecomponent in the Multibrot set.

Period 1 Results

LemmaIf F = f1 ⊥⊥ f2 has a fixed cluster point, then precisely one of the fi isa rabbit.

LemmaAll combinatorial data can be realised (in precisely 2d− 2 ways), savefor the case with δ = 1 or δ = 2n − 1.

The rotation number is fixed by the rotation number of the α-fixed pointfor the rabbit. The critical displacement is determined by the choice ofthe complementary map.

Period 1 Results

Period 2 (quadratic) Results

A bi-rabbit is a map bifurcating off the period 2 component. . .

. . . but there is another component of the same period in the wake ofthe bi-rabbit!

The rays landing on the period 2 cycle of the bi-rabbit. . .

. . . have the same angles as those landing on the period 2 cycle of thesecondary map.

A bi-rabbit is a map bifurcating off the period 2 component.

Theorem (S., 2009)If F = f1 ⊥⊥ f2 has a period two cluster cycle, one of the fi is either abi-rabbit or a secondary map which lies in the limb of the bi-rabbit.

Lemma (S., 2009)All combinatorial data can be realised (in at least two ways).

Theorem (S., 2010)The cases δ = 1 and δ = 2n − 1 can be constructed from mating withthe secondary map.

Example

The mating of these two components. . .

This is a well-known example. . .Tom Sharland (University of Warwick) Mating and Clustering 4th April 2012 20 / 27

Example

. . . is equivalent to the mating of these two components

This is a well-known example. . .Tom Sharland (University of Warwick) Mating and Clustering 4th April 2012 20 / 27

Higher degrees

Consider the degree d Multibrot set Md. It has d− 1 period twohyperbolic components.

Joint work with Epstein

We can mate the corresponding period two polynomials (if they are notcomplex conjugate) to get a degree d rational map with a period twocluster cycle. Each cluster contains precisely one periodic Fatoucomponent from each critical orbit.

The rotation number must be 0/1

The displacement must be 1

But not all maps arising from these matings are equivalent! We caneven write the maps in normal form as

Fa(z) =zd − a

zd − 1,

where a 6= 1 is a (d− 1)th root of unity.

Joint work with Epstein

All the maps look like this. . .

. . . and in degree d, there are d− 2 configurations which can occur andwhich are not equivalent.

A schematic proof

We can describe the Julia sets of the maps schematically

We have a modified notion of displacement. . . and this helps usclassify the maps.Tom Sharland (University of Warwick) Mating and Clustering 4th April 2012 24 / 27

News just in. . .

Figure:(10

63, 17

)⊥⊥

63, 34

63, 35

)⊥⊥

63, 56

63, 53

)⊥⊥

63, 20

These all have period 3 cluster cycles with the same intrinsic data. Butthey certainly don’t all look the same!

Summary

In simple cases, the combinatorial data of a cluster completelydefines a rational map.

Period 3 - experimental pictures suggest not?

Period 1 and period 2 cases are very similar, but with anincreased level of complexity for the period 2 case.

“Non-trivial” shared matingsDifferent combinatorial dataSimple Thurston classification only works in the quadratic case forperiod 2

Combinatorics of the matings

Outlook

How to completely classify the period two case?Positions of fixed points - how can we describe this combinatorially?Joint work with Epstein may help describe the new combinatorics.

Cluster cycles of period ≥ 3.More “secondary maps”Early results suggest far more complexity in the descriptions of thematings

What about a more general notion of clustering (not restricted tothe bicritical case)?

Parameter space (cf. discussion in Buff, Écalle and Epstein for theparabolic case)?

Thanks for listening!

Outlook

How to completely classify the period two case?Positions of fixed points - how can we describe this combinatorially?Joint work with Epstein may help describe the new combinatorics.

Cluster cycles of period ≥ 3.More “secondary maps”Early results suggest far more complexity in the descriptions of thematings

What about a more general notion of clustering (not restricted tothe bicritical case)?

Parameter space (cf. discussion in Buff, Écalle and Epstein for theparabolic case)?

Thanks for listening!

polynomial matings and rational maps with cluster cyclestsharland/harvard.pdfγ2(−θ). the maps fi...

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