Parabolic Components in Cubic Polynomial Slice Per 11 Runze Zhang April 2022

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Parabolic Components in Cubic Polynomial Slice P er1(1)

Runze Zhang

April 2022

Abstract

We prove that the boundary of every parabolic component in the cubic polynomial

slice P er1(1) is a Jordan curve by adapting the technique of para-puzzles presented in

[10]. We also give a global description of the connected locus C1: it is the union of two

main parabolic components and the limbs attached on their boundaries.

Contents

1 Introduction 1

2 Parametrisation of parabolic components 4

2.1 Basic topological descriptions of H∞and Hn.................. 5

2.2 Describing the special locus I........................... 7

2.3 Parametrisation of Hn,n≥0 ........................... 10

3 Graphs and puzzles 14

3.1 Dynamicalrays................................... 14

3.2 Parameter rays and landing properties . . . . . . . . . . . . . . . . . . . . . . 16

3.3 Parameter graphs and puzzles . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 Dynamical graphs and puzzles . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Local connectivity of ∂Hn26

4.1 Misiurewicz parabolic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2 Nonrenormalizablecase.............................. 28

4.3 Renormalizablecase ................................ 30

4.4 Globaldescriptions................................. 32

1 Introduction

Consider the space of unitary cubic polynomials ﬁxing the origin 0:

fa,λ(z) = z3+az2+λz, (a, λ)∈C2.

arXiv:2210.14305v1 [math.DS] 25 Oct 2022

Let Ja,λ denote the Julia set of fa,λ. The connected locus, as the analogue to the Mandelbrot

set for the quadratic case, is deﬁned by

C={(a, λ)∈C2;Ja,λ is connected}

={(a, λ)∈C2; both critical points of fa,λ do not escape to ∞}.

One of the attempts, proposed by Milnor, to study Cis to restrict ourselves to slices of

polynomials by ﬁxing λ, that is, to consider

P er1(λ) = {fa,λ;a∈C}∼

and its corresponding connected locus Cλ={a∈C;Ja,λ is connected}.

When |λ|<1, 0 is always a (super-)attracting ﬁxed point. Deﬁne

Hλ={a∈C; both critical points of fa,λ are attracted by 0}.

This is the union of hyperbolic components of adjacent and capture type, proposed by Milnor

in [6], which are of special interest in this situation. Adjacent means that two critical points

belong to the immediate basin of 0; capture means that one in contained in the immediate

basin of 0 while the other not (but still attracted by 0). It has been proved in [2], [10]

independently by D. Faught and P. Roesch that the boundary of every component of H0is

a Jordan curve. Combining this with the following result of Petersen-Tan we get that the

boundary of every component of Hλis a Jordan curve for |λ|<1:

Theorem 1.0.1 ([14]).There is a dynamically deﬁned holomorphic motion

A:D×(C\ H0)−→ C,(λ, a)7→ A(λ, a)

such that fa,0is hybrid conjugated to fA(λ,a),λ and A(λ, C\ H0) = C\ Hλ.

In this paper we investigate what happens when λ= 1 (for simplicity we will omit the

multiplier λin the corresponding notations appearing in the rest of the article). Now 0

becomes a parabolic ﬁxed point for the family P er1(1), degenerated (which means that it

has two attracting axis) if and only if a= 0.

For a∈C∗, let Ba(0) be the (unique) parabolic basin and B∗

a(0) the immediate parabolic

basin associated to 0, which is deﬁned to be the unique component of Ba(0) containing an

attracting petal. One can deﬁne the analogue to Hλ, λ ∈D:

H={a∈C∗; Both critical points are contained in Ba(0)}.

A connected component of His called a parabolic component. In this article we study

the boundaries of parabolic components and mainly show that

Theorem A. The boundary of every parabolic component is a Jordan curve.

As an analogue to the adjacent and capture hyperbolic components, we deﬁne

Deﬁnition (adjacent and capture parabolic components).Hhas a natural decomposition

H=Sn≥0Hn, where H0consists of the parameters asuch that both critical points belongs

to B∗

a(0) and Hn, n ≥1consists of parameters such that one critical point is in B∗

a(0) while

the other belongs to f−n−1

a(B∗

a(0)) \f−n

a(B∗

a(0)). A connected component of H0is called a

parabolic component of adjacent type1(or sometimes called a main parabolic component),

that of Hn, n ≥1is called a parabolic component of capture type.

Using the technique of puzzles, we also give a complete description of landing property

of external rays for parameters on ∂H0\ {0}:

Theorem B. Let a0∈∂H0\ {0}. Then if a0is renormalizable2, there are exactly two

external rays R∞(η),R∞(η0)landing which bound a copy of Mandelbrot set attached at

a0; otherwise there is only one external ray R∞(t)landing. Moreover if a0is Misiurewicz

parabolic3, then tsatisﬁes ∃m, 3mt= 1.

This permits us to deﬁne wakes at renormalizable parameters on ∂H0\ {0}to be the

region between the two landing rays which contains the Mandelbrot set copy attached at

a0. The limbs attached at a0∈∂H0\ {0}is deﬁned to be the intersection of C1and the

closure of the corresponding wake if a0is renormalizable and otherwise to be a0. The wakes

and limbs at parameter 0 are deﬁned manually, see Deﬁnition 3.2.6. As a direct corollary of

Theorem B, we get the following global picture for C1:

Theorem C. The connected locus C1can be written as the union of H0and the limbs

attached on its boundary.

Figure 1: The ”Butterﬂy” C1.His in green. The adjacent type H0consists of the two wings

of the butterﬂy. There are two copies of parabolic Mandelbrot sets (in black) attached at

the origin, which correspond to parabolic-like maps whose renormalized map has connected

Julia set (see L. Lomonaco [4], [5]). The small patches in green attached at the tips are of

capture type.

1We will show that there are only two components of this type. See Corollary 2.3.4.

2refer to Deﬁnition 3.4.7.

3refer to Deﬁnition 2.3.5.

Notice that C1,Hn, n ≥0 are symmetric with respect to x, y−axis since fa(z) = fa(z),

−fa(−z) = f−a(z) and therefore the dynamics of f±aand facan be identiﬁed. So essentially,

it suﬃces to study the parameters in S, where S={x+iy;x>0, y>0}.

This paper is organized as follows: in Section 2 we parametrize the parabolic components

by locating the free critical value in the basin of the ”parabolic model” z2+1

4. A diﬀerence

from work in the super-attracting case [10] is that, a priori the free critical point is not

marked out, so we need to ﬁnd out ﬁrst which critical point appears always on the boundary

of the maximal petal. Section 3 is devoted to the construction of parameter puzzles as well

as dynamical puzzles. Loosely speaking, parameter puzzles are just ”copies” of dynamical

puzzles via parametrizations. Conversely they partition the parameter space into pieces

on which the dynamical puzzles remain ”stable”. Moreover one should be careful when

dealing with the landing properties of rational parameter rays since a= 0 might appear

on the boundary of connected components of Hwhile at a= 0 we lose the stability of

repelling petals at the origin. The proof of Theorem A, essentially the local connectivity

of the boundary of parabolic components U, will be proceeded in Section 4. Parameters

on ∂Uare divided into three classes: Misiurewicz parabolic type, non-renormalizable type

and renormalizable type. The ﬁrst type corresponds to the ”cusps” on which no copy of

Mandelbrot set is attached, which do not appear for the boundary of components of H0,

and we will deal with it by puzzles without internal rays (the Qndeﬁned at the beginning

of subsection 3.3). The other two cases left are parallel to those in the super-attracting

slice P er1(0) (see [10]) and the strategy of the proofs there is adapted. The basic idea is to

transfer the ”shrinking property” of dynamical puzzles to that of para-puzzles via two key

lemmas 4.2.2 and 4.2.5. In Subsection 4.4 we prove Theorem B and Theorem C.

Acknowledgements. I am grateful to my advisor Pascale Roesch for suggesting me writ-

ing this down and for her reviews and comments on this article. I am also grateful to Arnaud

Ch´eritat for his computer program to create pictures of connected locus and Julia sets.

2 Parametrisation of parabolic components

We ﬁrst recall here the following classical result of dynamics associated to parabolic ﬁxed

point:

Proposition. Let R:ˆ

C−→ ˆ

Cbe a rational function and R(0) = 0,R0(0) = 1. Let B∗be

an immediate basin of 0, then there exists

•a unique semi-conjugacy φ:B∗−→ Cup to translation such that φ(R(z)) = φ(z) + 1.

•a unique simply connected open set Ω⊂B∗whose boundary contains 0 and a critical

point, such that it is sent conformally by φonto some right-half plane. Ωis called the

maximal petal.

Holomorphic dependence of Fatou coordinate. From the proof of the existence of

Fatou coordinate (cf. [7]) one may deduce easily holomorphic dependence of Fatou coordi-

nate for an analytic family in the following sense: let Ra:C−→ Cbe an analytic family

(parametrized by a∈Λ⊂C, Λ open) with Ra(0) = 0, R0

a(0) = e2πi p

q. Write Taylor expan-

sion near 0: Rq

a(z) = z+ω(a)zq+1 +o(zq+1). Suppose that at some a0,ω(a0)6= 0. Then

for every attracting (resp. repelling) axis of Ra0, there exists

•a small neighborhood Da0of a0; a topological disk Vatt ={x+iy;x>c −b|y|} (resp.

Vrep ={x+iy;x<c −b|y|}), where the constants b, c are positive and do not depend

on a; a family of attracting (resp. repelling) petals (Pa)a∈B(a0)

•a family of Fatou coordinates (φa)a∈Da0of Rasuch that φa:Pa−→ Vatt (resp. Vrep)

is conformal and φ−1

ais analytic in a.

An immediate consequence from this and the λ-Lemma is that φ−1

a◦φa0(parametrized by

a∈Ua0) deﬁnes a dynamical holomorphic motion of Pa0such that φ−1

a◦φa0(Pa0) = Pa.

A choice of inverse branch of critical points. One can compute explicitly the two

critical points of fa:

˜c+(a) = −a+√a2−3

3,˜c−(a) = −a−√a2−3

where the inverse branch √·is deﬁned on C\R−. Hence ˜c±(a) are continuous for a∈C\Z

where Z= [−√3,√3] ∪iR. If we deﬁne

c+(a) = (˜c+(a), a ∈H\(0,√3]

˜c−(a), a ∈ −H\[−√3,0) c−(a) = (˜c−(a), a ∈H\(0,√3]

˜c+(a), a ∈ −H\[−√3,0) (1)

where H={z=x+iy;x>0}is the right-half plan, then a7→ c±(a) can be extended

continuously C\[−√3,√3]. We will use this choice of inverse branch for the rest of the

paper. For a∈C\[−√3,√3], let v±(a) = fa(c±(a)) be the two corresponding critical

values. An elementary calculation gives the following asymptotic formulas near ∞:

c−(a) = −2a

3+O(1

a), v−(a) = 4a3

27 +O(a).(2)

2.1 Basic topological descriptions of H∞and Hn

Deﬁne H∞=C\ C1to be the complement of the connected locus. We ﬁrst prove that:

I. H∞is simply connected. Clearly H∞is open since being attracted by ∞is an open

property. Combining the asymptotic formula (2) of v−(a) with the following lemma, we get

immediately that H∞has a connected component containing a neighborhood of ∞on which

c−(a) escapes to ∞:

Lemma 2.1.1. There exists M>0such that for all asatisfying |a|>M, one has

{z;|z|>|a|2} ⊂ Ba(∞),

where Ba(∞)is the attracting basin of ∞.

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ParabolicComponentsinCubicPolynomialSlicePer1(1)RunzeZhangApril2022AbstractWeprovethattheboundaryofeveryparaboliccomponentinthecubicpolynomialslicePer1(1)isaJordancurvebyadaptingthetechniqueofpara-puzzlespresentedin[10].WealsogiveaglobaldescriptionoftheconnectedlocusC1:itistheunionoftwomainparabolic...

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