A parameter space of cubic Newton maps with parabolics · a quasiconformal map, to one of the cubic Newton maps of the family (1), not on the whole of C^ but on some neighborhoods

A parameter space of cubic Newton maps with

parabolics

Khudoyor Mamayusupov

NRU Higher School of Economics

Topological methods in dynamics and related topicsNizhny Novgorod, January 4, 2019

Khudoyor Mamayusupov NRU Higher School of Economics

A parameter space of cubic Newton maps with parabolics

Main Object

Cubic Newton maps with a parabolic fixed point at infinity.The Newton map of an entire function g : C→ C is themeromorphic function Ng : C→ C defined by

Ng (z) := z − g(z)/g ′(z).

The Newton maps for the family of entire functions of the formg(z) = (z2 + a)ez , parametrized by a single complex number a 6= 0,is the family given by the following cubic rational maps

fa(z) = z − z2 + a

z2 + 2z + a. (1)



Denote f ◦n the n-th iterate of f .For a periodic point f ◦n(z) = z of period n ≥ 1 the numberλ = (f ◦n)′(z) is called the multiplier of the orbit{z , f (z), . . . , f ◦n−1(z)}.

Definition (Classification of fixed points)

A periodic point f ◦n(z) = z is called

attracting if |λ| < 1, in particular,

superattracting if λ = 0

repelling if |λ| > 1

indifferent if |λ| = 1, in this case let λ = e2πiθ

• rationally indifferent (also called parabolic) if θ ∈ Q• irrationally indifferent if θ 6∈ Q



The Basin of Attraction

Definition (The Basin of Attraction)

Let ξ be an attracting or parabolic fixed point of a rational map f .The basin A(ξ) of ξ is

int{z ∈ C : limn→∞

f ◦n(z) = ξ},

the interior of the set of starting points z which eventually convergeto ξ under iteration.The immediate basin A◦(ξ) of ξ is the forward invariant connectedcomponent of the basin.

Since Julia sets are connected for Newton maps, the basincomponents are simply connected.



The Julia set I



The Julia set II



Cubic Newton maps

The fixed points of fa are:

1 the roots of z2 + a = 0, which are superattracting, and

2 a point at infinity, which is a parabolic of multiplier +1 with oneparabolic attracting petal.

The critical points of fa are:

1 the roots of z2 + a = 0, and

2 the roots of z2 + 4z + a + 2 = 0

3 (Special case) If a = 1 then z = −1 is the only pole which isalso a critical point



Motivation

1 Every cubic rational map has four fixed points counted withmultiplicities.

2 If the two of the fixed points are attracting and the other twocoincide creating a fixed point of multiplicity 2 then this must bea parabolic fixed point with the multiplier +1.

3 Sending this point to infinity by a Mobius map, conformalchange of variable, we obtain a map which will be conjugate, viaa quasiconformal map, to one of the cubic Newton maps of thefamily (1), not on the whole of C but on some neighborhoods oftheir Julia sets. This can be done by a standard holomorphicsurgery to make the two attracting fixed points superattracting.



Components of the parameter space

• Every parabolic fixed point attracts a critical point, then

• We have only one critical point that has a “free” dynamics, callthis a free critical point.

Definition (Stable components)

The components of the parameter space of the cubic Newton mapsof the family (1) for which the free critical point belongs toattracting basins and the basin of the parabolic fixed point at ∞ arecalled stable components.

We study topological properties of these components.



Figure: The a-parameter plane of cubic Newton maps fa with twozoom-ins. Yellow – the free critical point belongs to the basin of ∞. Grey– the free critical point belongs to the basins of the two superattractingfixed points. Some centers (pcm) and half-centers (pcnm) are shown.



Main Theorem

Theorem

1 Every stable component is a topological open disk containing aunique center, which is a cubic postcritically minimal Newtonmap.

2 The main parabolic component: the free critical point belongs tothe immediate basin of ∞, the quasiconformal conjugacy classesof maps are of the following three types: type-I -homeomorphicto an open vertical strip, type-II -an analytic arc, type-III -apoint, which is the center or a cubic postcritically non-minimalNewton map.

3 The boundary of every such a type-I class in H consists oftype-II arcs meeting at type-III points.



Minimal critical orbit relationsCritical orbit relations - Type-III Newton maps

Let c1 and c2 be critical points of a rational function f . We say thatc1 and c2 are in a critical orbit relation if f ◦m(c1) = f ◦n(c2) for somenon-negative integers m and n, if c1 = c2 we require m 6= n.

Definition (Minimal and non-minimal critical orbit relations)

Let fa be a cubic Newton map of the form (1) and let c1 ∈ U1 andc2 ∈ U2 be its critical points with U1 and U2 the connectedcomponents of the basin of the parabolic fixed point at ∞. Assumef ◦ma (U1) = U2 with minimal such m ≥ 0. We say that c1 and c2 arein a minimal critical orbit relation if f ◦ma (c1) = c2 with the same m. If

f◦(m+n)a (c1) = f ◦ka (c2) with n > 0 and k ≥ 0 then we say that c1 andc2 are in non-minimal critical orbit relations.



Post-Critically Minimal Newton map

Definition (Postcritically minimal and Postcritically non-minimalcubic Newton maps)

A cubic Newton map fa of the form (1) is called postcriticallyminimal -pcm (postcritically non-minimal -npcm)

1 if its Fatou set consists of superattracting basins and theparabolic basin of ∞ and

2 if its free critical point is in a minimal critical orbit relation (anon-minimal critical orbit relation) with the other critical pointof fa in the parabolic basin of ∞.



As the critical orbit relations are given by a system of algebraicequations for the parameter a, these pcm and npcm Newton mapsform a discrete set in stable components. Thus a small perturbationdestroys the relation, thus these type of maps are rigid.Let us assume that the immediate basin U = A◦(∞) contains thetwo critical points c1 and c2 of fa and let ψU : U → D be a Riemannmap of U with ψU(c1) = 0. Denote w = ψU(c2) ∈ D the image ofthe second critical point. Then the conjugation ψU ◦ fa ◦ ψ−1

U is aBlaschke product on the unit disk D, denote it by B(z). TheBlaschke product B(z) then has critical points at 0 and w and as aself map of D its degree is 3.Denote by H the set of such an fa in the parameter space. We wantto show that it is a topological disk.



• It has a very rich structure because of interaction of the twocritical points.

• The idea is to construct a map from H to some model space,which will be parametrized by a coordinate in the unit disk.

• As a model space we take the space of cubic Blaschke productswith normalizations.



Automorphisms of the unit disk D, Aut(D)

Every conformal automorphism τ ∈Aut(D) of the unit disk D is afractional linear transformation of the form

τ(z) = γ · z − a

1− az, (2)

for some constants |a| < 1 and |γ| = 1. For γ = 1−a1−a , let us denote by

βa(z) =1− a

1− a· z − a

1− az, (3)

the unique automorphism of D sending a to the origin and normalizedto fix z = 1.



For given d + 1 complex numbers (constants) a1, a2, . . . , ad+1 in Dand a constant |γ| = 1, we define a Blaschke product of degree d + 1by the following product

f (z) = γ · βa1(z) · βa2(z) · · · βad+1(z).

1 Every proper holomorphic map f of the unit disk D has a finitedegree and f is the restriction of a Blaschke product on D of thesame degree.

2 Blaschke products are determined uniquely by the constants

γ = f (1) and {a1, a2, . . . , ad+1} = f −1({0}).3 It is clear that the poles of f are at 1/a1, 1/a2, . . . , 1/ad+1.

We are only interested in Blaschke products f normalized to fix 1,then these maps are invariant under the conjugation by the inversionz 7→ 1/z .



This symmetry yields the following.

1 If ξ is a fixed point of f then 1/ξ is also a fixed point.

2 If c is a critical point then 1/c is also a critical point.

As a rational map of C a Blaschke product f of degree d + 1 has

1 d + 1 roots,

2 d + 1 poles, and

3 2d critical points in C counted with their multiplicities.

4 d + 1 fixed points in C counted with their multiplicities.

If there is a fixed point in D then it is simple and is the only fixedpoint in D. In this case the fixed point is attracting and it attractsevery point of D under the iterations of the map.If there is no fixed point in D then there exists a fixed point in S1

that is attracting or parabolic, and every point in the complement ofS1 is attracted to this fixed point.



The critical points ⇐⇒ their Blaschke products.

Theorem (Heins)

Let c1, c2, . . . , cd be d (not necessarily distinct) points in D. Thenthere is a unique Blaschke product f of degree d + 1 with f (0) = 0and f (1) = 1 and having c1, c2, . . . , cd as its critical points.Moreover, if g is any other Blaschke product of degree d + 1 withcritical points c1, c2, . . . , cd , then there is a τ ∈ Aut(D) such that

τ ◦ g = f .



The cubic Blaschke products that we consider f (1) = 1, butf (0) 6= 0.Additionally f ′(1) = 1, and f ′(0) = 0.Moreover, we want that around z = 1 our map has the form1 + (z − 1) + A(z − 1)3 + o((z − 1)3), for a complex A 6= 0. It meansthat f ′′(1) = 0.

1 By theorem of Heins, every such a cubic Blaschke product isuniquely obtained by the position of its second critical pointw ∈ D.

2 ∀w ∈ D there exists a cubic Blaschke product normalized asabove and such that 0 and w are its critical points in D.

Let us call this the model space and denote by B3.



Let M =M(B3) -the moduli space of B3, the conformal conjugacyclasses of maps from B3.Denote a class [B] then [B1] = [B2] ⇐⇒ ∃ β ∈ Aut(D) ,∀z

β ◦ B1(z) = B2 ◦ β(z). (4)

Then β(1) = 1, normalized.Moreover, β({0,w1}) = {0,w2}, the critical points of B1 map to thecritical points of B2.If β(0) = 0 then β ≡ id and B1 ≡ B2.For the other case: β(0) = w2 and β(w1) = 0.Then β(z) = βw1(z) = 1−w1

1−w1

z−w1

1−w1zand β(0) = −w1

1−w1

1−w1= w2.

The latter is a necessary and sufficient condition for B1 and B2 to beconformally conjugate.



The Equivalence

If w1 is real then w2 = −w1. If w1 = vi is a pure imaginary for

v 2 < 1 then w2 = 2v2

1+v2 − v(1−v2)1+v2 i .

Let z = (x , y) and w = (u, v). Then (x , y) ∼ (u, v) ⇐⇒(u, v) = (− x(1−x)2−y2(2−x)

(1−x)2+y2 ,− y(1−(x2+y2))(1−x)2+y2 ).

Thus M = D/ ∼, the quotient of D by this equivalence relation.



For ∀fa ∈ H corresponds an element in M =M(B3), and thus thestable component H can be identified with the moduli space M.Indeed, for fa in H , let φ1 be the Riemann map of the parabolicimmediate basin A◦(∞) such that φ1(c1) = 0 and letφ1(c2) = w1 ∈ D. Denote B1 = φ1 ◦ fa ◦ φ−1

1 , which is a Blaschkeproduct in B3.Consider φ2 the Riemann map of A◦(∞) such that φ2(c2) = 0 andlet φ2(c1) = w2 ∈ D.Then the conjugacy B2 = φ2 ◦ fa ◦ φ−1

2 is also a Blaschke product.B1 and B2 are conjugate to each other as we havefa = φ−1

1 ◦ B1 ◦ φ1 = φ−12 ◦ B2 ◦ φ2,

denote φ = φ2 ◦ φ−11

then φ ◦ B1 = B2 ◦ φ.



Main Parabolic Component H



Type-I and Type-II classes inM = H

Demonstration of proof:For a ∈ H , let Pa be the maximal attracting petal in A◦(∞):

1 ∃c1 ∈ ∂Pa, a critical point of fa

2 Pa is open, forward invariant, simply connected

3 fa is injective on Pa

4 ∃ a conformal map φa : Pa → Hr , the right half plane

5 φa(fa(z)) = φa(z) + 1, the Abel equation.



Demonstration of Proof

Assume that and denote by w ′ = fa(c2) ∈ Pa \ fa(Pa). Then on Hr

we have 0 < Reφa(w ′) ≤ 1 and Imφa(w ′) ∈ R.The idea is to change the position of the critical value on the verticalstrip.Let x0 + y0i = φ(w ′), for the case w ′ /∈ ∂P , it is easy to check that

`(h,t) : (x , y) 7→{

( x0+hx0

x , y + tx0x), 0 ≤ x ≤ x0,

((1− h1−x0

)x + h1−x0

, y + t1−x0

(1− x)), x0 ≤ x ≤ 1,

is a quasiconformal homeomorphism of the strip 0 ≤ Re z ≤ 1parametrized by z = x + iy .

The real dilatation: |∂z l∂z l

(z)| =√h2+t2√

(2x0+h)2+t2at 0 ≤ Re z < x0 and

|∂z l∂z l

(z)| =√h2+t2√

(2(1−x0)−h)2+t2at x0 ≤ Re z ≤ 1.



1 Extend `(h,t)(x , y) continuously to the right half plane by thetranslation z 7→ z + 1.

2 Pulling it back by the Fatou coordinate φa define an almostcomplex structure on the petal Pa

3 by the dynamics of fa, pull it back to A(∞)

4 extend it by zero (the standard complex structure) to the rest ofthe plane, we obtain a Beltrami form on C with the maximaldilatation equal to that of `(h,t).



By Measurable Riemann Mapping theorem there exists aquasiconformal map Φ(h,t) on C that solves the Beltrami equation.Then Φ(h,t) ◦ fa ◦ Φ−1

(h,t) is a Newton map in H and has requiredconditions.



Thank you.



Documents

A parameter space of cubic Newton maps with parabolics · a quasiconformal map, to one of the cubic Newton maps of the family (1), not on the whole of C^ but on some neighborhoods