UNIFORM A PRIORI BOUNDS FOR NEUTRAL RENORMALIZATION. DZMITRY DUDKO AND MIKHAIL LYUBICH

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UNIFORM A PRIORI BOUNDS

FOR NEUTRAL RENORMALIZATION.

DZMITRY DUDKO AND MIKHAIL LYUBICH

Abstract. We prove uniform “pseudo-Siegel” a priori bounds for Siegel disks of bounded type

that give a uniform control of oscillations of their boundaries in all scales. As a consequence,

we construct the Mother Hedgehog controlling the postcritical set for any quadratic polynomial

with a neutral periodic point and show that this hedgehog has a star-like structure. Pseudo-Siegel

bounds imply uniform a priori bounds of the Sector Renormalization, which gives an opportunity

to extend Siegel/Pacman Renormalization Theory and Near-Parabolic Renormalization Theory to

all near-neutral quadratic polynomials. Various applications beyond quadratic polynomials are also

underway.

Contents

1. Introduction 2

Part 1. Rotational geometry 12

2. Preparation 12

3. Near-Rotation Systems 25

4. Parabolic fjords 30

Part 2. Pseudo-Siegel disks and Snakes 42

5. Pseudo-Siegel disks 42

6. Snakes 57

7. Welding of b

Zm+1 and parabolic fjords 69

Part 3. Covering and Calibration lemmas 75

8. Covering and Lair Lemmas 75

9. The Calibration Lemma 83

Part 4. Conclusions 87

10. Proof of the main result 87

11. Mother Hedgehogs and uniform quasi-conformality of b

Z92

Appendix A. Degeneration of Riemann surfaces 97

References 105

arXiv:2210.09280v2 [math.DS] 30 Dec 2024

2 DZMITRY DUDKO AND MIKHAIL LYUBICH

1. Introduction

Local dynamics near a neutral ﬁxed point, and a closely related dynamical theory of circle homeo-

morphisms, is a classical story going back to Poincar´e, Fatou, and Julia. It followed up in the next two

decades with breakthroughs by Denjoy (1932) and Siegel (1942) on the linearization of circle diﬀeomor-

phisms and local maps, respectively. In the 1950-60s, the Kolmogorov–Arnold–Moser (KAM) theory

emerged resulting in the reenvision of the near-rotation phenomenon in mathematics and physics. In

the mid-1970s, Renormalization Ideas were introduced into Dynamics by Feigenbaum, Coullet and

Tresser and led, in particular, to numerous conjectures in Low-Dimensional Dynamics including the

nature of the KAM small divisor problem.

The theory of analytic circle diﬀeomorphisms and local theory for neutral holomorphic germs and

quadratic polynomials received an essentially complete treatment in the second half of the last century

in the work by Arnold (in the KAM framework), Bruno, Herman, Yoccoz, and Perez-Marco. About

at the same time (1980–90s) a global and semi-local theory for neutral Siegel quadratic polynomials

fθ:z7→ e2πiθ z+z2with rotation numbers θof bounded type was designed on the basis of the Douady-

Ghys surgery. Renormalization Theory of Siegel maps, also initiated by physicists, was mathematically

Renormalization of bounded type was established in the framework of Pacman Renormalization [DLS]

providing tools to study near-Siegel maps [DL1].

A remarkable progress in understanding Near-Neutral Complex Dynamics came in the 2000s, when

Inou and Shishikura established uniform a priori bounds for quadratic polynomials fθwith rotation

numbers of high type (near-parabolic perturbative regime). This theory found numerous applica-

tions, from constructing examples of Julia sets of positive area by Buﬀ-Cheritat (2000s, [BC]) and

Avila-Lyubich (2010s, [AL2])) to a complete description, for high type rotation numbers, of the topo-

logical structure of the Mother Hedgehogs that capture the semi-local dynamics of neutral quadratic

polynomials (Shishikura-Yang, Cheraghi (2010s)). See §1.3 and §1.5 for a more detailed historical

account.

In this paper, we lay down a foundation for the renormalization theory based upon almost-invariant

objects, called pseudo-Siegel disks, that “hide” various geometric irregularities; compare with Item b

Z

in §1.5. In the almost invariant framework, we will establish uniform “pseudo-Siegel” a priori bounds

for neutral quadratic polynomials fθwith artbitrary rotation numbers and show that the postcritical

set of every such fθis “small”: it lies within the compact Mother Hedgehog Hθso that the restriction

fθ:Hθýis a homeomorphism. Pseudo-Siegel bounds can be transferred into other forms of a priori

bounds [DL2, DLL] making the theory more complete and compatible with the previous developments;

see §1.2 for the summary.

Ideas of the current paper helped to conﬁrm the MLC at the classical Feigenbaum parameter [DL4]

– historically, the most important parameter for the development of the Dynamical Renormalization

Theory; see Remark 1.8. Applications to McMullen’s problem on Sierpinski hyperbolic components

and to the construction of Herman curves have already appeared in [DLu, Lim]. More applications of

the theory developed in the current paper are underway; see §1.4.

Our proof is based upon analysis of degenerating Siegel disks of bounded type. The Near-Degenerate

Regime and its principles, in the quadratic-like renormalization context, were originally designed by

Jeremy Kahn [K], with a key analytic tool, the Covering Lemma, appeared in [KL1]. They serve

as an entry point for our paper. One of the major subtleties of our situation is that Siegel disks

of bounded type do not have uniformly bounded qc-geometry since they may develop long fjords in

all scale. (Otherwise, Cremer and even parabolic points would not have existed.) To deal with this

UNIFORM A PRIORI BOUNDS FOR NEUTRAL RENORMALIZATION 3

Figure 1. Diﬀerent types of Siegel disks: golden-ratio (top-left), near-Basilica

(bottom-left), near-cauliﬂower (bottom-right), near-1/4-Rabbit (top-right).

problem, we design a regularization machinery of ﬁlling-in the fjords to gradually turn Siegel disks

Zfinto almost-invariant pseudo-Siegel disks b

fn≥−1so that b

Z−1

fare uniform qc disks. Every b

is almost invariant up to qn+1 iterations so that \

n≥−1b

f=Hfis the Mother Hedgehog controlling

the postcritical set. Moreover, b

Zk(n+1)

fbecome uniform qc disks b

Z−1

fn+1 after applying the (n+ 1)-fold

sector renormalization f7→ fn+1; see Item (V) in §1.2.

We remark in conclusion that the resulting pseudo-Siege bounds are the ﬁrst non-perturbative a

priori bounds dealing with non-locally connected (e.g., Cremer) Julia sets – this removes arguably the

main conceptual obstacle towards the full MLC; see §1.5 for a detailed summary of the developments

in the direction of the MLC conjecture.

1.1. Results. Due to the Douady-Ghys surgery, for a Siegel map f=fθof bounded type, the

dynamics on the Siegel disk Z, all the way up to its boundary ∂Z, is qc conjugate to the rigid rotation

by θ, which provides us with the rotation combinatorial model for f|∂Z.

Let pn/qnbe the continued fraction approximands for θ, so for any x∈∂Z,fqnxare the closest

combinatorial returns of orb xback to x. A combinatorial interval I≡In

f(x)⊂∂Z of level nis the

combinatorially shortest interval bounded by xand fqnx. For a combinatorial interval I⊂∂Z, we

let e

I= 3I⊃Ibe the enlargement of Iby two attached combinatorial intervals.

Given a combinarorial interval I⊂∂Z, let us consider the family F+

3(I) of curves γ⊂b

C∖Z

connecting Ito points of ∂Z ∖e

I. The external modulus W+

3(I) is the extremal width (i.e., the inverse

of the extremal length) of the family F+

λ(I).

Uniform Bounds Theorem 1.1. There exists an absolute constant Ksuch that W+

3(I)≤Kfor

all Siegel quadratic polynomials f=fθof bounded type and all combinatorial intervals I=In

f(x).

4 DZMITRY DUDKO AND MIKHAIL LYUBICH

Ahull Q⊂Cis a compact connected full set. The Mother Hedgehog [Chi] for a neutral polynomial

fθis an invariant hull containing both the ﬁxed point 0 and the critical point c0(f):=−e2πiθ/2.

Mother Hedgehog Theorem 1.2. Any neutral quadratic polynomial f=fθ,θ̸∈ Q, has a Mother

Hedgehog Hf∋c0(f)such that f:Hf→Hfis a homeomorphism.

The last theorem is a consequence of the following result:

Quasidisk Approximation Theorem 1.3. There exists an absolute constant Ksuch that for any

Siegel quadratic polynomial fof bounded type there exists a K-quasidisk b

Zf⊃Zfsuch that f|b

Zfis

injective.

1.2. Variations of uniform bounds. Let us formulate a more technical statement representing the

main result of the paper. Let pn/qn≈θbe the best approximations of an irrational θ∈R/Zstarting

with q0= 1. Then the fqn

θ≡fqn[θ]

θare the closest combinatorial returns enumerated so that fq0

θ=f.

Let us introduce the following concept:

Deﬁnition 1.4 (Pseudo-Siegel bounds).Consider a quadratic polynomials fθ:z7→ e2πiθz+z2. We

say that fθhas pseudo-Siegel bounds if it has a sequence of disks

(1.1) b

Z−1≡b

Z−1

θ⊇b

θ⊇ ··· ⊇ Hθ∋0, f |b

θis injective, \

m≥−1b

θ=Hθ,

where Hθ≡Hfθis the Mother Hedgehog from Theorem 1.2, such that (see also §11.0.2)

(A) b

θis a pseudo-Siegel disk almost invariant under fqm+1

θ;

(B) ∂b

θhas a nest of tilings Tb

θwith essentially bounded geometry independent of m, see

Remark 11.8.

If the geometric bounds of Tb

θin (B) are independent of θ, then we say that fθhas uniform

pseudo-Siegel bounds.

The main outcome of the paper is the establishment of uniform pseudo-Siegel bounds for bounded

type rotation numbers. Since the bounds are uniform, they persist for all rotation numbers; see

Theorem 1.3 for illustration. Pseudo-Siegel disks b

θcan be viewed as matings of rotational and

unicritical circle dynamics; see §11.0.2. We remark that uniform bounds in (B) can only be stated for

almost-invariant objects because the inner geometry of b

Zmcan degenerate or even disappear in the

limit.

In [DL2], we transfer uniform pseudo-Siegel bounds to other forms of precompactness used in

Siegel/Pacman Renormalization Theory and in Near-Parabolic Renormalization Theory and clarify

the behavior of pseudo-Siegel disks b

θfor m≥0 at θof unbounded type.

Sector Bounds Theorem 1.5 ([DL2]).There exists a compact Sector Renormalization operator

Rsec for all neutral quadratic polynomials.

Moreover, Properties (I) – (VI) stated below hold.

The Sector Renormalization was originally designed to study Local Dynamics around an indiﬀerent

ﬁxed point [D1, §3.1], [Yo] with various ideas going back to the theory of Ecalle-Voronin’s invariants.

For high type rotation numbers, Inou and Shishikura [IS] extended the sector renormalization into

a semi-local compact operator with a geometric control of the critical orbit; see §1.3 for more de-

tails. Often, sectorial bounds are stated in the framework of cylinder renormalization which is more

canonical, [Ya1].

Sectorial Renormalization constructed in [DL2] comes along with the following structure:

UNIFORM A PRIORI BOUNDS FOR NEUTRAL RENORMALIZATION 5

(I) (appropriately speciﬁed) pseudo-Siegel disks b

θdepend, in the C0-topology of

RiemannMapθ:b

C\D→b

C\int( b

θ),∞ 7→ ∞,17→ c0(θ),

uniformly continuously in the topology coming from the compactiﬁcation

θ∈Θ:=θ= [0; a1, a2, . . . ]|ai∈N≥1∪ {∞}⊃Θ:= [R\Q]/Z.

We remark that Θ is a natural compactiﬁcation of the set of irrational rotation numbers Θ. Prop-

erty (I) implies:

(II) b

θexist by continuity for all θ∈Θ and are almost invariant under fqm+1[θ]

θas in (A) and (B),

where fqm+1[θ]

θm≥−1evolves into Lavaurs-Epstein towers [La, Ep] at θ∈Θ\Θ.

(III) the Mother Hedgehog Hθdepends uniformly continuous on θ∈Θ in the Hausdorﬀ topology.

If θ∈Θ\Θ, then Hθis the Mother Hedgehog of the associated Lauvars-Epstein tower.

(IV) The set of closest returns fqn[θ]

θn≥0θ∈Θ

is precompact up to linear conjugacy at the critical value in the following sense: for n(i)→ ∞,

any sequence fqn(i)[θ(i)]

θ(i)has a converging subsequence fqn(it)[θ(it)]

θ(it)→F, where F:WF→Cis

aσ-proper transcendental map. The convergence is uniform on compact subsets of WF.

Transcendental limit as in (IV) were previously established for bounded type parameters Θbnd in [McM1,

Theorem 8.1] and for the unstable manifolds associated with Θbnd in [DLS, Theorem 5.1]. The lat-

ter became a central tool in Pacman Renormalization Theory [DLS, DL1] in bypassing a technical

challenge that semi-local maps under consideration are not branched coverings. We believe that Tran-

scendental Dynamics as in (IV) will allow us to extend Pacman Renormalization Theory (starting with

the hyperbolicity result) to all rotation numbers; see also §1.4 and Remark 1.7.

A key ingredient for Properties (I) – (IV) is the uniform continuity of b

Z−1[Rm

secfθ] in the spirit of

Theorem 1.3; the key relation between b

fθand the sector renormalization operator Rsec:

(V) Under the renormalization change of variables representing fθ⇝fn,θ =Rn

secfθ, where θ∈Θ,

the pseudo-Siegel disk b

Zm+k(n)

fθbecomes b

fn,θ – a pseudo-Siegel disk almost invariant under

fqm+1

n,θ . Here k(n, θ)≥nrefers to the count of “non-negligible” (with respect to Rsec) renormal-

ization levels. Moreover, the above Properties (I) – (IV) with appropriate adjustments hold

for the sector renormalizations {fn,θ}of neutral quadratic polynomials.

The sectorial renormalization change of variables in Theorem 1.5 is expanding in the angular

direction. Since the “full” non-escaping set of sectororial changes of variables is Hθ, we deduce

(compare with [RRRS]) that

(VI) For every θ∈(R\Q)/Z, there is a set Qθ⊆R/Zinvariant under the rotation ϕ7→ ϕ+θsuch

that the Mother Hedgehog Hθis star-like (or a bouquet of curves):

Hfθ=[

ϕ∈Qθ

Iϕwith f(Iϕ) = Iϕ+θ,

where Iϕis a closed simple arc emerging from the α-ﬁxed point. The arcs Iϕare pairwise

disjoint away from the α. If Qθ̸=R/Z, then α(f) is a Cremer point. If Qθ=R/Z, then

Zf:= int Hfis the Siegel disk of f.

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UNIFORMAPRIORIBOUNDSFORNEUTRALRENORMALIZATION.DZMITRYDUDKOANDMIKHAILLYUBICHAbstract.Weproveuniform“pseudo-Siegel”aprioriboundsforSiegeldisksofboundedtypethatgiveauniformcontrolofoscillationsoftheirboundariesinallscales.Asaconsequence,weconstructtheMotherHedgehogcontrollingthepostcriticalsetforanyqua...

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