Importance of domestic releases in urban sewer networks: case of alkylphenols, phthalates and parabens in the Paris agglomeration

. Then, we tried to identify ways to reduce the contamination of domestic wastewater. First the origin of the contamination was investigated through greywater and finally the impact of regulation and/or changing of practices on the contamination of wastewater were assessed.

Industrial vs. domestic fluxes at the Paris conurbation scale

Alkylphenols (NP: nonylphenols, OP: octylphenols), phthalates (DEP: diethylphthalate, DnBP: di-n-butylphthalate, BBP: butylbenzylphthalate, DEHP: di-(2-ethylhexyl)phthalate) and parabens (MeP: methylparaben, EtP: ethylparaben, PrP: propylparaben) were measured in industrial wastewater, domestic wastewater and in the influent of Parisian WWTP mixing both domestic and industrial wastewater. Figure 1 summarizes the levels of contamination measured for domestic water (measured at the head of the watershed), industrial water and water entering the WWTP. For all substances, domestic water is systematically more contaminated than industrial water. The calculated flows highlighted that the industrial contribution represents less than 3% of the incoming flow in WWTP for the studied substances. This low contribution from industry can be explained by the nature of the Ile-de-France watershed. Indeed, the Paris region is a highly urbanized but poorly industrialized watershed. The effort granted by the industries would therefore reduce by 3% the contributions to the sewer network. What levers of action to imagine for other sources, namely domestic sources? To answer this question, greywater and the impact of the changes in consumer practices and the impact of the regulation concerning substances were investigated. Greywater, the major source of alkylphenols, phthalates and parabens in domestic wastewater Domestic wastewater consists of greywater (80% by volume of water) and sewage (20% by volume of water). In the idea of finding levers of action for a reduction at the source of micropollutants, the study of greywater was prioritized because they are more important in water flow and it seems easier to modify our practices concerning them. We were therefore interested in the flows emitted by five different types of greywater: shower, washbasin, manual dishwashing, dishwasher and washing machine. Shower and washing machine effluents were the main contributor to the wastewater contamination. This study highlighted that the daily use of different products for domestic activities explained the levels observed in wastewater. Moreover, the concentration variability could be explained by different habits of consuming.

Evolution of concentration in wastewater following changing of industrial formulations

Concentrations of alkylphenols, phthalates and parabens were measured in the influent of the main WWTP of the Paris agglomeration in 2010 [START_REF] Bergé | Phthalates and alkylphenols in industrial and domestic effluents: Case of Paris conurbation (France)[END_REF][START_REF] Gasperi | First assessment of triclosan, triclocarban and paraben mass loads at a very large regional scale: Case of Paris conurbation (France)[END_REF]. These concentrations represent the reference levels of the impregnation of urban wastewater for these micropollutants since they were the first studies providing information on these molecules. The same sampling points were again studied in 2015 and 2016. The evolution of the concentrations measured in Parisian wastewater between these two periods is showed in Figure 2. Since 2010, paraben concentrations have decreased by a factor of between 6 for MeP and 3.5 for EtP and PrP. For the alkylphenols, the NP concentrations are 5 times lower whereas those of the OP have been divided by 10; the levels recorded in 2015 are now very often close to the limit of quantification (0.17 μg/L). For phthalates, DnBP stands out, the concentrations in 2015 are higher than those of reference by a factor of 5 whereas for the other congeners the levels measured are approximately 5 times lower. Thus, for all the molecules, except for DnBP, the concentrations observed in 2015 are lower than the reference levels. This trend, which seems to be general, is however a reflection of very contrasting situations from one family to another. Parabens were almost withdrawn from personal care products following a sanitary alert [START_REF] Darbre | Concentrations of parabens in human breast tumours[END_REF]) and substituted by other preservatives. This latter withdrawal was the response of cosmetic manufacturers although no new regulations have been issued to ban them.

Conclusion

This study focuses on the domestic origin of the contamination of wastewater. In the case of a heavily urbanized and scarcely industrialized conurbation, we highlighted that the contamination of domestic wastewater is predominant compared to industrial wastewater for three families of micropollutants: alkylphenols, phthalates and parabens. Then, the investigation of the quality of greywater allowed us to identify shower effluent and washing machine effluent as the main source of domestic wastewater contamination and so to identify ways to reduce it. Finally, the decrease of observed concentrations between 2010 and 2015 in the main trunks of the Paris conurbation, allowed us to understand the determinants (regulations and/or consuming practices) of these fluctuations in order to imagine levers for action by local authorities. The regulation of domestic activities seems to be central to promote a source reduction of micropollutants.

Figure 1 :

 1 Figure 1: Concentrations (µg/L) in domestic wastewater, industrial wastewater and in influent of WWTP

  Concerning alkylphenols and phthalates recent regulations limited their use in manufactured products (Regulation (EC) No 1907/2006, directive 2003/53/CE).

Figure 2 :

 2 Figure 2: Temporal evolution of the concentrations (µg/L) in influent of WWTP following modifications of industrial formulations or regulation concerning phthalates, alkylphenols and parabens


C Paijens

Origins of biocides in combined sewer overflows in urban areas

Keywords: Biocides, combined sewer overflows, stormwater, urban sources, wastewater

Context

Historically, pesticides used in agriculture have received much more attention than biocides used in urban areas. However, even if they are used in smaller quantities in cities, biocides loads in receiving waters from rural and urban areas can be in the same order of magnitude (Wittmer et al., 2011). Biocidal substances are largely used and hence contribute to a diffuse contamination of receiving waters. For example, they are used as disinfectant in hospitals and homes or as preservatives in cosmetics, personal care products or building materials (bitumen sheets, paints, renders, tiles, etc.). They are released into receiving waters via discharges of wastewater treatment plant (WWTP) effluents and also via stormwater (separate sewer systems) or combined sewer overflows (combined sewer systems) discharges during wet weather (Bollmann et al., 2014;Bucheli et al., 1998;Burkhardt et al., 2011). As they can have effect at very low concentration on aquatic organisms (Mohr et al., 2008), they are part of emerging concerns but they are not sufficiently monitored in the environment (Dulio et al., 2015) and in urban matrices. However, in order to reduce emissions in the aquatic environment, the identification of their sources is needed. Thus, the purpose of this study is to identify the different urban sources of 18 biocides in combined sewer overflows (CSOs).

Results and discussions

The results on dissolved fraction have shown a contamination of CSOs by the 18 targeted biocides. Concentrations are in the same order of magnitude for the two CSOs. They range from few ng/L to tens of ng/L for most of biocides but can exceed 100 ng/L (MIT, benzalkonium C16 and diuron) or even 1 µg/L (benzalkonium C12 and C14). First, compared to WWTP influents, concentrations of most biocides in CSOs are higher or in the same order of magnitude, which highlights that stormwater do not dilute biocides in wastewater but is also a source of biocides. Also, concentrations in CSOs can vary depending on the percentage of stormwater (figure 2). Three cases have been observed:

-Biocides for which stormwater substantially contributes to the contamination of CSOs. As shown on figure 2 (top charts), for carbendazim, diuron, OIT and terbutryn, the higher the percentage of stormwater is and the higher the concentrations are. Hence their major source in CSOs seems to be urban surfaces, and more specifically building materials (bitumen sheets, paints, renders, tiles, etc.) from which biocides have leached during rain events.

In that context, the Central Laboratory of Police Prefecture in partnership with the Laboratory of water, environment and urban systems (Leesu) has developed a method to simultaneously quantify 18 biocides, used in building materials but also in household products, at trace level (ng/L-µg/L) in diverse urban matrices. The analysed biocides are the following ones: benzisothiazolinone (BIT), chloromethylisothiazolinone (CMIT), dichlorooctylisothiazolinone (DCOIT), methylisothiazolinone (MIT), octylisothiazolinone (OIT), cybutryn, terbutryn, terbuthylazine, diuron, isoproturon, carbendazim, iodopropynyl butylcarbamate (IPBC), tebuconazole, thiabendazole, mecoprop and bezalkonium chlorides (C12-C16). First, the sample is filtered through 0.7 µm. Both fractions are then extracted: solid phase extraction for the dissolved fraction and microwaves assisted extraction for the particulate fraction. And both extracts are analysed by high-performance liquid chromatography coupled to tandemmass spectrometry.

Materials and methods

Flow-proportional composite samples have been collected in Paris conurbation (figure 1) from two combined sewer overflows (CSOs) and from the Seine Centre WWTP (900 000 connected inhabitants, influents and effluents are sampled) with the help of the Mairie de Paris and the Public service of sanitation in the metropolitan area of Paris (SIAAP). The percentage of stormwater in CSOs has been estimated thanks to the sample conductivity, assuming that CSOs are composed of both stormwater (80-150 µS.cm -1 ) and wastewater (1050-1170 µS.cm -1 ).

-Biocides for which the contribution of wastewater seems more important. For instance, BIT, CMIT and MIT have been detected and quantified in Clichy CSOs only when stormwater percentage was below 60%. These compounds are widely used in household products, cosmetics and personal care products and thus, these uses are their major source in CSOs. -Biocides for which no clear trend was observed (bottom charts on figure 2) and then both contributions are significant and cannot be discriminated.

Finally, annual mass loads of the 18 targeted biocides have been estimated in both CSOs and in the WWTP effluents of Seine Centre. Compared to WWTP effluents, CSOs seem to be a significant source of benzalkonium C12 and C14, CMIT, cybutryn, and OIT as their mass loads in CSOs are in the same order of magnitude or higher than in WWTP discharges. Mass loads of BIT, MIT, thiabendazole and benzalkonium C16 in CSOs represent less than 10% of mass loads in WWTP effluent. Thus, the contribution of CSOs to the contamination of surface waters seems to be minor. Mass loads of other biocides represent from 11 to 32% of those in WWTP effluents.

Figure 1 :

 1 Figure 1: Sampling sites


ARNAUD CH ÉRITAT Alexander Blokh

Lex Oversteegen

Vladlen Timorin

Anatole Katok

LOCATION OF SIEGEL CAPTURE POLYNOMIALS IN PARAMETER SPACES

Keywords: Mathematics Subject Classification. Primary 37F45, Secondary 37F10, 37F20, 37F50 Complex dynamics, Julia set, laminations, Siegel capture polynomial

A cubic polynomial with a marked fixed point 0 is called an IS-capture polynomial if it has a Siegel disk D around 0 and if D contains an eventual image of a critical point. We show that any IS-capture polynomial is on the boundary of a unique bounded hyperbolic component of the polynomial parameter space determined by the rational lamination of the map and relate IScapture polynomials to the cubic Principal Hyperbolic Domain and its closure.

Introduction

A complex polynomial P of any degree is said to be hyperbolic if all of its critical points belong to the basins of attracting or superattracting periodic cycles. The set of all hyperbolic polynomials in any particular parameter space is open. Components of this set are called hyperbolic components. The dynamics of hyperbolic complex polynomials is well understood. According to the famous Fatou conjecture [START_REF] Fatou | Sur les équations fonctionelles[END_REF], hyperbolic polynomials are dense in the parameter space of all complex polynomials. This explains why hyperbolic components play a prominent role in complex dynamics.

By a general result of J. Milnor [START_REF] Milnor | Hyperbolic components. With an appendix by A. Poirier[END_REF], every bounded hyperbolic component in the moduli space of degree d polynomials is an open topological cell of complex dimension d -1. Hence it is fair to say that the structure of such hyperbolic domains is known. However, in degrees greater than 2, the same cannot be said about the closures of hyperbolic components. Arguably, even in the case of the cubic Principal Hyperbolic Domain PHD 3 (defined as the subset of the cubic parameter space consisting of classes of polynomials with a unique (super)attracting fixed point and a Jordan curve Julia set), the description of its boundary has proved to be rather elusive. For example, in a recent paper by L. Petersen and T. Lei [START_REF] Petersen | Analytic coordinates recording cubic dynamics[END_REF] it is shown that the boundary of PHD 3 has a very intricate "fractal" structure that is not fully understood. Thus, understanding the boundaries of hyperbolic components, in particular understanding the boundary of PHD 3 , is an important open problem.

Qualitative changes in the dynamics of polynomials take place on the boundary of the connectedness locus. It is known that boundaries of bounded hyperbolic components are contained in the boundary of the entire connectedness locus. This provides an additional incentive for studying boundaries of hyperbolic components.

In our paper we consider these issues in the cubic case. More precisely, we consider the parameter space of cubic polynomials with a marked fixed point. The corresponding connectedness locus contains many complex analytic disks in its boundary. A typical example is provided by IS-capture polynomials, i.e., polynomials that have an invariant Siegel domain around the marked fixed point and a critical point which is eventually mapped into it. In this paper we study the dynamics of such polynomials and their location in the parameter space; below we briefly summarize our main results.

Summary of the main results. An IS-capture polynomial f belongs to the boundary of a unique bounded hyperbolic component with the same rational lamination as f . Moreover, f belongs to a complex analytic disk lying in the boundary of this hyperbolic component.

We also obtain some corollaries. In [START_REF] Blokh | The main cubioid[END_REF] it was proven that all polynomials from PHD 3 satisfy some simple conditions. We conjecture that these conditions are not only necessary but also sufficient for a polynomial to belong to the closure of PHD 3 . In the present paper we show that any polynomial satisfying the above mentioned conditions but not belonging to the closure of PHD 3 must be a polynomial of so-called queer type. This improves our earlier results [START_REF] Blokh | Laminations from the Main Cubioid[END_REF].

To state another corollary of our results, we remind the reader about Brjuno numbers.

The following is a classical result by A. D. Brjuno [START_REF] Brjuno | Analytic forms of differential equations[END_REF].

t:br Theorem 1.2 ( [START_REF] Brjuno | Analytic forms of differential equations[END_REF]). If a is an irrationally indifferent fixed point of a polynomial f with multiplier e 2πiθ and θ ∈ B, then the point a is a Siegel fixed point.

Another classical result, due to J.-C. Yoccoz, states that in the quadratic case Theorem 1.2 is sharp.

t:yoc Theorem 1.3 ( [START_REF] Yoccoz | Petits Diviseurs en Dimension[END_REF]). In the situation of Theorem 1.2, if f is quadratic and θ / ∈ B is not a Brjuno number, then a is a Cremer fixed point of f . A conjecture by A. Douady states that Theorem 1.3 holds for higher degree polynomials too. We show that if a polynomial P does not belong to the closure of PHD 3 and has multiplier λ = e 2πiθ at its fixed point w, where θ is not a Brjuno number, then w is a Cremer fixed point of P . Thus, if a cubic counterexample to the Douady conjecture exists, it must be a polynomial from the boundary of PHD 3 .

Understanding the structure of the polynomial space in degree greater than two is an important problem in complex dynamics. Describing the location of IS-capture polynomials in the parameter space is a step which will, hopefully, allow one to study the boundaries of hyperbolic components and their mutual disposition, extending our knowledge about the cubic polynomial parameter space.

Detailed statement of the results

s:detailed

We write C for the plane of complex numbers. The Julia set of a polynomial f : C → C is denoted by J(f ), and the filled Julia set of f by K(f ). For quadratic polynomials, a crucial object of study is the Mandelbrot set M 2 . Let P c (z) be a quadratic polynomial defined by the formula P c (z) = z 2 + c. Clearly, 0 is the only critical point of the polynomial P c in C. By definition, c ∈ M 2 if the orbit of 0 under P c is bounded (points with unbounded orbits are said to escape). Equivalently, c ∈ M 2 if and only if the filled Julia set K(P c ) is connected. If c ∈ M 2 , then the set K(P c ) is a Cantor set.

By classes of polynomials we mean affine conjugacy classes. The class of f is denoted by [f ]. The parameters c of P c (z) are in one-to-one correspondence with classes of quadratic polynomials. A higher-degree analog of the set M 2 is the degree d connectedness locus M d , i.e., the set of classes of degree d polynomials f all of whose critical points do not escape or, equivalently, whose Julia set J(f ) is connected.

The structure of the Mandelbrot set is described in the seminal work of Thurston [START_REF] Thurston | The combinatorics of iterated rational maps[END_REF] (see also [START_REF] Douady | Étude dynamique des polynômes complexes I & II Publ[END_REF]). In particular, [START_REF] Thurston | The combinatorics of iterated rational maps[END_REF] gives a full description of how distinct hyperbolic components of M 2 are located with respect to each other and what kind of dynamics is exhibited by polynomials from their boundaries. However, for degrees d > 2 studying the set M d has proven to be a difficult task. Certain full dimensional parts of M d are well understood; e.g., results of [START_REF] Epstein | Geography of the cubic connectedness locus: intertwining surgery[END_REF][START_REF] Inou | Combinatorics and topology of straightening maps, I: Compactness and bijectivity[END_REF] allow to find copies of M 2 × M 2 or MK in M 3 (here MK is the set of pairs (c, z), where c ∈ M 2 and z ∈ K(P c )). However, the combinatorial structure of M d as a whole remains elusive.

The central and, arguably, the simplest part of the Mandelbrot set is the (quadratic) Principal Hyperbolic Domain denoted by PHD 2 . It is the set of all parameter values c such that the polynomial P c has an attracting fixed point. All these polynomials have Jordan curve Julia sets. The closure PHD 2 of PHD 2 consists of all parameter values c such that P c has a non-repelling fixed point. It is sometimes called the filled Main Cardioid. Its boundary Bd(PHD 2 ) is a plane algebraic curve, a cardioid called the Main Cardioid. As follows from the Douady-Hubbard parameter landing theorem and from the "no ghost limbs" theorem by Yoccoz [START_REF] Douady | Étude dynamique des polynômes complexes I & II Publ[END_REF][START_REF] Hubbard | Local connectivity of Julia sets and bifurcation loci: three theorems of Yoccoz[END_REF], the Mandelbrot set itself can be thought of as the union of PHD 2 and limbs, connected components of M 2 \ PHD 2 , parameterized by reduced rational fractions p/q ∈ (0, 1).

It is natural to consider analogs of the Main Cardioid for higher degree polynomials, in particular for cubic polynomials. This motivates our interest to the boundary of the cubic Principal Hyperbolic Domain PHD 3 defined as the set of classes of cubic polynomials that have an attracting fixed point and whose Julia set is a Jordan curve. A closely related set, the so-called Main Cubioid, was studied in a few recent papers [START_REF] Blokh | The main cubioid[END_REF][START_REF] Blokh | Complementary components to the cubic principal domain[END_REF][START_REF] Blokh | Laminations from the Main Cubioid[END_REF][START_REF] Blokh | Quadratic-like dynamics of cubic polynomials[END_REF]). In this framework an important task is to describe whether polynomials with certain dynamical properties belong to the boundary of the Main Cubioid. This is one of the problems addressed in the present paper.

Let us now concentrate on cubic polynomials. Let F be the space of polynomials f λ,b given by the formula

f λ,b (z) = λz + bz 2 + z 3 , λ ∈ C, b ∈ C.
The space F is adapted to studying polynomials with a marked fixed point. Any such polynomial is affinely conjugate to one from F under a conjugacy sending the marked fixed point to 0. All polynomials g ∈ F have 0 as a fixed point. Let the λ-slice F λ of F be the space of all polynomials g ∈ F with g (0) = λ. It is well known that two polynomials f λ,b and f λ,b are conjugate by a Möbius transformation M (z) that fixes 0 if and only if M (z) = ±z and b = ±b. We will deal with f ∈ F λ for some λ and consider only perturbations of f in F. Set F at = |λ|<1 F λ (the subscript at stands for attracting).1 Let us emphasize that F at is the family of polynomials from F that have the point 0 as an attracting fixed point. For each g ∈ F at , let A(g) be the immediate basin of attraction of 0. Denote by F nr the set of all polynomials f = f λ,b ∈ F such that 0 is non-repelling for f (so that |λ| 1).

Suppose that a is a fixed point of a polynomial f of any degree. Assume that f (a) = e 2πiθ where θ is irrational. Then a is said to be an irrationally indifferent fixed point. If f is linearizable (i.e., analytically conjugate to a rotation) in a neighborhood of a, the point a is called a Siegel fixed point. In this case the rotation in question is well defined and is the rotation by 2πθ so that θ is called the rotation number. Moreover, this is equivalent to the existence of an orientation preserving topological conjugacy between f in a neighborhood of a and the rotation by 2πθ of the unit disk. If a is a Siegel fixed point, the biggest neighborhood of a on which f is linearizable exists and is called the Siegel disk around a. If f is not linearizable in any neighborhood of a then the point a is called a Cremer fixed point. d:compotypes Definition 2.1 (Siegel captures). Suppose that a polynomial f ∈ F has a Siegel disk ∆(f ) around 0. If a critical point of f is eventually mapped to ∆(f ), then this critical point is denoted by ca(f ) (here "ca" stands for "captured"), and f is called an IS-capture polynomial, or simply an IS-capture (here "I" stands for "invariant" and "S" stands for "Siegel"). By [START_REF] Mañé | On a theorem of Fatou[END_REF], there exists a recurrent critical point re(f ) of f (here "re" stands for "recurrent") whose limit set contains Bd(∆(f )).

It follows that the critical points ca(f ) and re(f ) are well-defined and distinct (evidently, ca(f ) is not recurrent).

r:0-siegel Remark 2.2. Generically, maps in the family F have three fixed points. Any of these points, not only 0, could have a Siegel disk around it that captures a critical point. However, let us stress that we only speak of IS-captures when 0 is the Siegel fixed point whose Siegel disk captures a critical point.

In this paper, we study the location of IS-captures in F relative to hyperbolic components. An important role here is played by the set P • of all hyperbolic polynomials f ∈ F such that f ∈ F at and J(f ) is a Jordan curve. Equivalently, f ∈ F at belongs to P • if and only if A(f ), the immediate basin of attraction of 0, contains both critical points of f . Evidently, P • is open in F. To see that P • is one hyperbolic component of F, not only of F at , observe that polynomials f b,λ = z 3 + bz 2 + λz with |λ| = 1 are not hyperbolic and that by Corollary 4.9, the set P • is connected. d:princrit Definition 2.3. The set P • is called the principal hyperbolic component of F. We say that a hyperbolic polynomial f ∈ F at is an IAcapture polynomial (IA stands for Invariant Attracting) if a critical point of f , denoted by ω 2 (f ), is eventually mapped to A(f ) but does not lie in A(f ) (then the remaining critical point ω 1 (f ) belongs to A(f ), and no critical point of f belongs to

J(f )). A hyperbolic component U of F is of IA-capture type if U contains an IA-capture polynomial.
Hyperbolic components of IA-capture type will also be called IA-capture components.

Similarly to Remark 2.2, we emphasize that IA-capture polynomials have 0 as their attracting fixed point. Evidently, both critical points ω 1 (f ), ω 2 (f ) are well-defined for an IA-capture polynomial f . Observe also that, similarly to the above, the fact that polynomials f b,λ = z 3 + bz 2 + λz with |λ| = 1 are not hyperbolic implies that any hyperbolic component U of F of IA-capture type is contained in F at . Thus, the principal hyperbolic component P • of F and the hyperbolic components of F of IA-capture type are subsets of F at .

We also need the concepts of rational lamination and full lamination. Denote by D the open unit disk in the complex plane centered at the origin and by S the unit circle which is the boundary of D. We will identify R/Z with S via x → e 2πix .

Let f be a monic polynomial of degree greater than 1 and connected Julia set. In this case all external rays with rational arguments land. Given two rational angles α, β ∈ R/Z, we declare α ∼ r β iff the landing points of the corresponding external rays coincide. This defines an equivalence relation on Q/Z. The equivalence classes are finite (see Theorems 3.5 and 3.6 with references). We then consider the collection L r f of all edges of the convex hulls (in D) of all equivalence classes and call it the rational lamination of f .

If the Julia set J(f ) is locally connected, then all external rays land. Given any two angles α, β ∈ R/Z we declare that α ∼ β iff the landing points coincide. This defines an equivalence relation on R/Z, and in this case too the equivalence classes are finite (see Theorems 3.5 and 3.6 and Theorem 1.1 of [START_REF] Kiwi | Wandering orbit portraits[END_REF]). The collection of all edges of the convex hulls of all classes is denoted L f and is called the (full) lamination of f . We will refer to the elements of L f as leaves.

We include in each lamination the singletons {e 2πiα } and call them degenerate leaves, with α ∈ Q/Z for L r f , resp. α ∈ R/Z for L f . The set C of all possible chords of the unit disk and singletons in the unit circle is equipped with a natural topology that associates to a chord ab of S with endpoints a, b ∈ S the pair {a, b} of the symmetric product S × S/(a, b) ∼ (b, a).

Clearly, in the case when J(f ) is locally connected we have L r f ⊂ L f and, since L f is closed (see Section 3), we have L r f ⊂ L f . Contrary to what one may expect, it is not always true that L r f = L f . A typical example is the case of a quadratic polynomial Q with invariant Siegel domain and locally connected Julia set. Then L r Q consists only of degenerate leaves and, therefore, coincides with the rational lamination of z 2 (abusing the language we will call such a lamination the empty lamination). For IS-capture polynomials, we relate rational and full laminations in Subsection 3. Recall that a polynomial with connected Julia set that belongs to a hyperbolic component has a locally connected Julia set and, hence, a well-defined lamination.

thmA Theorem A. If f ∈ F is an IS-capture polynomial, then there is a unique bounded hyperbolic component U in F, whose boundary contains f . Moreover, U ⊂ F at , for all P ∈ U we have L r f = L r P , L P = L r f , and there are two possibilities:

(1) the Julia set of f contains no periodic cutpoints, then U = P • ;

(2) the Julia set of f has a repelling periodic cutpoint, then U is of IA-capture type.

A polynomial is said to be J-stable with respect to a family of polynomials if its Julia set admits an equivariant holomorphic motion over some neighborhood of the map in the given family [START_REF] Lyubich | Some typical properties of the dynamics of rational mappings[END_REF][START_REF] Mañé | On the dynamics of rational maps[END_REF]. Say that f ∈ F λ is λ-stable if it is J-stable with respect to F λ with λ = f (0), otherwise f is called λ-unstable. A component of the set of λ-stable polynomials in F λ is called an IS-capture component if some (equivalently, all) polynomials from this component are IScapture polynomials. Thus IS-capture components are complex onedimensional analytic disks in the two-dimensional space F. Every such disk is contained in a slice F λ represented as a straight (complex) line in coordinates (λ, b) of F.

In [Zak99, Theorem 5.3], Zakeri proved that every IS-capture polynomial belongs to some IS-capture component. From Theorem A it follows that every IS-capture component is contained in the boundary of a unique hyperbolic component U of F. Moreover, U = P • or U is of IA-capture type. Conversely: thmB Theorem B. Let U be either an IA-capture component or P • . Then the boundary of U contains uncountably many IS-capture components lying in F λ , where λ = e 2πiθ , and θ runs through all Brjuno numbers in R/Z.

A more precise formulation of Theorem B is contained in Theorem 6.5. We will apply Theorem A to the study of P, the closure of P • in F. The following are some properties of polynomials in P.

t:prophd Theorem 2.4 ([BOPT14a]). If f = f λ,b ∈ P, then |λ| 1, the Julia set J(f ) is connected, f has no repelling periodic cutpoints in J(f )
, and all its non-repelling periodic points, except possibly 0, have multiplier 1.

These properties extend almost verbatim to the higher degree case [START_REF] Blokh | The main cubioid[END_REF]. Theorem 2.4 motivates Definition 2.5. [START_REF] Blokh | The main cubioid[END_REF]). Let CU be the family of cubic polynomials f ∈ |λ| 1 F λ such that J(f ) is connected, f has no repelling periodic cutpoints in J(f ), and all its non-repelling periodic points, except possibly 0, have multiplier 1. The family CU is called the Main Cubioid of F.

d:cubio Definition 2.5 ([
Note that P • and CU are subsets of F that play a similar role to the principal hyperbolic component PHD 3 and the main cubioid CU in the (unmarked) moduli space of cubic polynomials. However, the difference is that, when defining P • and CU, we take into account the special role of the marked fixed point 0 for polynomials in F. As a consequence, the sets P • and CU are not stable under arbitrary affine conjugacies. By Theorem 2.4, Definition 2.5 immediately implies that P ⊂ CU.

corC1 Corollary C. IS-capture polynomials do not belong to CU \ P.

We prove Corollary C at the end of Section 5. For a compact set X ⊂ C, define the topological hull TH(X) of X as the union of X with all bounded components of C \ X. We write P λ for the λ-slice of P, i.e., for the set P ∩ F λ .

corC2 Corollary D. If W is a component of TH(P λ ) \ P λ and f ∈ W, then the following holds.

(1) Any such polynomial f is λ-stable.

(2) Critical points of f are distinct and belong to J(f ).

(3) The Julia set J(f ) has positive Lebesgue measure and carries an invariant line field.

We prove it in Section 7. Part of it follows from [Zak99, Theorem 3.4].

In Section 7, we aslo obtain interesting corollaries of Theorem B that help distinguish between Siegel and Cremer fixed points of a given multiplier.

In the end of this section we include a glossary of non-standard terms and notation used throughout the paper. We appreciate the suggestion made by one of the referees as it is certain to make reading easier.

GLOSSARY OF IMPORTANT TERMS AND NOTATION

PHD 3 : the Principal Hyperbolic Domain in M 3 consisting of classes of hyperbolic cubic polynomials with a fixed (super)attracting point and the Jordan curve Julia set. IS-capture polynomial: a cubic polynomial with invariant Siegel domain and a critical point that eventually maps to that domain. F: the space of polynomials f λ,b given by the formula

f λ,b (z) = λz + bz 2 + z 3 , λ ∈ C, b ∈ C.
F λ : the λ-slice of the space F consisting of all polynomials g ∈ F with g (0) = λ. F at : the union of all F λ with |λ| < 1. A(g): the immediate basin of attraction of 0 for g ∈ F at . F nr : the union of all F λ with |λ| 1. ∆(f ): if f ∈ F has a Siegel disk around 0, then we denote this disk ∆(f ). P • : the set of all hyperbolic polynomials f ∈ F such that f ∈ F at and J(f ) is a Jordan curve. P: the closure of P • in F. IA-capture polynomial: a hyperbolic polynomial f ∈ F at such that a critical point of f is eventually mapped to A(f ) but does not lie in A(f ). component of IA-capture type: a hyperbolic component U of F that contains an IA-capture polynomial. re(f ): a recurrent critical point of an IS-polynomial f . ca(f ): a non-recurrent critical point of an IS-polynomial f ; it eventually maps to ∆(f ). L r f : the rational lamination of a polynomial f . L f : the full lamination of a polynomial f , defined if J(f ) is locally connected. CU: the Main Cubioid of F (see Definition2.5). TH(Z): the topological hull of a set Z.

Rays and laminations ss:pcuts

We will make use of the concepts of the full/rational lamination associated to a polynomial with connected Julia set. These concepts are due to Thurston [START_REF] Thurston | The combinatorics of iterated rational maps[END_REF] and Kiwi [Kiw97,[START_REF] Kiwi | Rational laminations of complex polynomials[END_REF][START_REF] Kiwi | Real laminations and the topological dynamics of complex polynomials[END_REF]. In fact, in [START_REF] Thurston | The combinatorics of iterated rational maps[END_REF] full laminations are defined independently of polynomials as a combinatorial concept and are often studied in that setting (see, e.g., [START_REF] Blokh | Laminations in the language of leaves[END_REF]). Laminations are important tools of combinatorial complex polynomial dynamics. Some of these tools are applicable to polynomials of arbitrary degree, including those with non-locally connected Julia sets. However, for the sake of brevity in this paper we avoid unnecessary generality and define full lamination only in the case when P has a locally connected Julia set.

3.1. Rays. Studying periodic external rays of polynomials is a powerful tool in complex dynamics. Given a polynomial f with connected Julia set we denote by R f (α) the external ray of f with argument α. (According to our convention, arguments of external rays are elements of R/Z rather than R/2πZ.) The arguments of external rays depend on the choice of a Böttcher coordinate near infinity. For an arbitrary cubic polynomial, such coordinate is defined up to a sign, i.e., up to the involution z → -z. However, for f ∈ F, we can distinguish a linearizing coordinate asymptotic to the identity. We assume that, whenever f ∈ F, the linearizing coordinate near infinity is chosen in this way.

l:land-ratio Lemma 3.1 (see, e.g., [START_REF] Milnor | Holomorphic dynamics[END_REF], Section 18). Let f be a polynomial. All external rays of f with rational arguments land. The landing points eventually map to periodic parabolic or repelling points. If J(f ) is connected then all rays landing at points that are eventually mapped to parabolic or repelling periodic points have rational arguments.

Call an external ray smooth if it does not contain an escaping (pre)critical point. The next lemma can be found in [START_REF] Goldberg | Fixed points of polynomial maps. Part II. Fixed point portraits[END_REF] (Lemma B.1) or [START_REF] Douady | Étude dynamique des polynômes complexes I & II Publ[END_REF] (Lecture VIII, Section II, Proposition 3). l:rep Lemma 3.2. Let f be a polynomial, and z be a repelling periodic point of f . If a smooth periodic ray R f (θ) lands at z, then, for every polynomial g sufficiently close to f , the ray R g (θ) lands at a repelling periodic point w close to z, and w depends holomorphically on g.

By a periodic argument we mean an element of R/Z periodic under the d-tupling map θ → dθ. c:converge Corollary 3.3 (Lemma 4.7 [START_REF] Blokh | Complementary components to the cubic principal domain[END_REF]). Suppose that h n → h is an infinite sequence of polynomials of degree d with connected Julia sets, and {α, β} is a pair of periodic arguments such that the external rays R hn (α), R hn (β) land at the same repelling periodic point x n of h n . If the external rays R h (α), R h (β) do not land at the same periodic point of h, then one of these two rays must land at a parabolic point of h.

Lemma 4.7 of [START_REF] Blokh | Complementary components to the cubic principal domain[END_REF] is more general and includes (with provisions) the case when Julia sets of polynomials h n are disconnected.

The following result is purely topological and is based on local behavior of polynomials at points of the plane. Given a polynomial f with connected Julia set J(f ) and a point z ∈ J(f ), denote by A z the set of arguments of rays landing at z. It is known [START_REF] Hubbard | Local connectivity of Julia sets and bifurcation loci: three theorems of Yoccoz[END_REF] that A z is finite. Given a finite set X ⊂ S, the points a, b, c ∈ X are said to be consecutive if the positively oriented arcs (a, b) and (b, c) are disjoint from X (observe that the order of points in this definition is essential).

t:a1 Theorem 3.4 (cf Lemma 18.1 [Mil06]). Let f be a polynomial of de- gree d > 1 whose Julia set J(f ) is connected. (We do not assume that J(f ) is locally connected.) Let z ∈ J(f ) be a point such that A z = ∅. Then σ d | Az is a k-to-1 map between A z and A f (z)
, and, if z is noncritical, then k = 1. Moreover, there are two possibilities.

(1) The set The next result is classical and has a proof using the Schwarz-Pick metric in [START_REF] Douady | Étude dynamique des polynômes complexes I & II Publ[END_REF]. Recall that the (pre)periodic external rays are exactly those whose arguments are rational. t:a2 Theorem 3.5 (Proposition 2, Section II, Lecture VIII [START_REF] Douady | Étude dynamique des polynômes complexes I & II Publ[END_REF]). Let f be a polynomial of degree d > 1 with connected Julia set. Then all rational external rays for f land, and their landing points are (pre)periodic points eventually mapped to repelling or parabolic periodic points.

σ d (A z ) = A f (z) is a singleton. ( 2
Theorem 3.6, due to A. Douady, is a form of converse of Theorem 3.5.

t:a3 Theorem 3.6 (Theorem I.A [START_REF] Hubbard | Local connectivity of Julia sets and bifurcation loci: three theorems of Yoccoz[END_REF]). Let f be a polynomial of degree d > 1 whose Julia set J(f ) is connected. Let z ∈ J(f ) be a repelling or parabolic periodic point. Then:

i. The point z ∈ J(f ) is the landing point of at least one periodic external ray. ii. Every external ray landing at z is periodic. iii. All periodic external rays landing at z have the same period. iv. There are finitely many external rays landing at z.

Once one proves the first claim, the others follow from it, Theorem 3.4 and properties of the d-tupling map. In [START_REF] Hubbard | Local connectivity of Julia sets and bifurcation loci: three theorems of Yoccoz[END_REF] there is another proof, using the Yoccoz inequality.

The following nice theorem will not be used in its full strength; we add it for the sake of completeness. A wandering point in J(f ) is a point whose orbit is infinite: this is the opposite of being (pre)periodic. If f is a polynomial of degree d and J(f ) is locally connected, one defines an equivalence relation ∼ f on S by declaring α, β ∈ S equivalent if R f (α) and R f (β) land at the same point. Then J(f ) is homeomorphic to S/ ∼ f . By Theorems 3.6 and 3.7, any ∼ f -class is finite. It is wellknown that the graph of ∼ f is a closed subset of S × S.

Definition 3.8 is based upon ∼ f but is not related to polynomials. By an edge of a ∼-class we mean an edge of its convex hull.

d:si-inv-lam Definition 3.9 (Laminational equivalences and dynamics). A laminational equivalence relation ∼ is (σ d -)invariant if: (D1) ∼ is forward invariant: for a class g, the set σ d (g) is a class too; (D2) for any ∼-class g, the map τ = σ d | g extends to S as an orientation preserving covering map τ such that g is the full preimage of τ (g) under the covering map τ .

To each laminational equivalence relation ∼ we associate the corresponding geodesic lamination L ∼ defined as the collection of all edges of convex hulls of ∼-classes together with all points of S. Call the lamination all of whose leaves are singletons in S the empty lamination.

With every ∼-class G , we associate its convex hull G = CH(G ). The geodesic lamination L ∼ is the set of all edges of all such polygons G together with all singletons in S. Elements of L ∼ are leaves. A leaf is degenerate if it coincides with a point in S; otherwise it is non-degenerate. If = ab is a leaf, then, by Theorem 3.4, the chord

σ d (a)σ d (b) is a (possibly degenerate) leaf denoted σ d ( ). A critical leaf is a leaf that is a critical chord. A gap of L f is the closure of a compo- nent of D \ L f . For any gap G of L f , define σ d (G) as CH(σ d (G ∩ S)). A gap G is invariant if σ d (G) = G. If L f has a gap G such that G ∩ S is infinite, then the interior of G is disjoint from the convex hulls of all ∼ f -classes. Finally, the geodesic lamination L ∼ f = L f is called the (full) geodesic lamination associated with f .
3.3. Invariant gaps of cubic laminations. Let L ∼ be a cubic lamination. The degree of a gap G of L ∼ is the maximal number of disjoint critical chords that fit in G and are not edges of G, plus 1, except for the case when G is a triangle with critical edges in which case the degree of G is 3 (since chords include their endpoints, disjoint critical chords have distinct endpoints). Degree 2 (respectively, 3) gaps are said to be quadratic (respectively, cubic).

By [START_REF] Blokh | The main cubioid[END_REF], a quadratic σ 3 -invariant gap G has a unique longest edge M (G) called the major (of G). The major M (G) is critical (then G is of regular critical type) or periodic (then G is of periodic type). For every edge = ab of G, let H (G) be the arc of S with endpoints a and b and no points of G in H (G). Let us normalize the length of S to 1; then the major M (G) is singled out by the fact that the length of H M (G) (G) is greater than or equal to 1/3. Proof. Since f is hyperbolic, J(f ) is locally connected so that Theorem 3.10 applies to f . Evidently, neither critical point of f belongs to J(f ). Hence case (1) of Theorem 3.10 does not apply to f while case (2) does apply. The cutpoint cannot be parabolic for otherwise f would not be hyperbolic. This proves all claims of the corollary except for the last one. To see that f ∈ F λ \ P it remains to apply Lemma 3.2 which implies that small perturbations of f will have a periodic cutpoint in their Julia sets and, therefore, cannot belong to P • .

3.4. Rational lamination. Rational laminations L r f are introduced by Kiwi (see [START_REF] Kiwi | Rational rays and critical portraits of complex polynomials[END_REF][START_REF] Kiwi | Rational laminations of complex polynomials[END_REF][START_REF] Kiwi | Real laminations and the topological dynamics of complex polynomials[END_REF]) and are based upon the work of Goldberg and Milnor [START_REF] Goldberg | Fixed points of polynomial maps. Part II. Fixed point portraits[END_REF].

l:a5 Lemma 3.12. Let f be a polynomial of degree d 2 with connected Julia set. If a chord is a limit of leaves i ∈ L r f and one of its endpoints is periodic, then its other endpoint is periodic of the same period.

This lemma follows from Lemma 3.16 of [START_REF] Blokh | Laminations from the Main Cubioid[END_REF] since L r f is generated by a laminational equivalence relation. (2) for each ∈ L there exists 1 ∈ L so that σ d ( 1 ) = .

3

(3) for each ∈ L so that σ d ( ) is a non-degenerate leaf, there exist d disjoint leaves 1 , . . . , d in L so that = 1 and σ d ( i ) = σ d ( ) for all i = 1, . . . , d.

l:ratio-sibl Lemma 3.14 ([Kiw97, Kiw01]). For a polynomial f with connected Julia set the rational lamination L r f is sibling invariant. We are ready to prove the next lemma. l:ratiofull Lemma 3.15. If f is a polynomial of degree d 2 with locally connected Julia set and there is no bounded Fatou domain of f whose boundary contains a critical point with infinite orbit, then L r f = L f . Proof. Recall that always L r f ⊂ L f . Suppose that L r f L f . By Lemma 3.14, the collection L r f is sibling invariant. Moreover, let x and y be rational arguments. By Theorems 3.5 and 3.6, if x ∼ y and x is periodic for σ 3 , then y is periodic of the same period. By Lemma 3.12 it follows that there are no critical leaves in L r f with a periodic endpoint. Moreover, it follows also that if x ∈ S is periodic and xy = xz are leaves of L r f , then σ d (xy) = σ d (xz). Sibling invariant collections of leaves with these properties are called proper ; such collections as well as their closures are studied in [START_REF] Blokh | Laminations in the language of leaves[END_REF]. In particular, it follows from Theorem 4.9 of [BMOV13] that L r f is a lamination associated with an equivalence relation, say, ≈, on the unit circle. This means that L r f is formed by the edges of the convex hulls of all ≈-classes. Recall that L f is generated by a specific equivalence relation on S denoted by ∼ f . Now, by the assumption L r f L f . This implies that there is a gap G of L r f that contains leaves of L f inside (so that only the endpoints of these leaves belong to the boundary of G). The gap G cannot be finite because then all its vertices must be ∼ f -equivalent, and leaves of L f cannot intersect the interior of G. Suppose that G is infinite. We claim that there are no infinite gaps H of L f properly contained in G. Indeed, suppose otherwise. Then an edge of H must be contained in the interior of G (except for its endpoints). Observe that any edge of an infinite gap of any lamination is either (pre)critical or (pre)periodic (cf. [BOPT17a, Lemma 4.5]). Since ∈ L f \ L r f , this implies that is (pre)critical with infinite orbit, a contradiction with the assumption of the lemma. Thus, all gaps of L f in G are finite.

By [Kiw02, Theorem 1.1], all infinite gaps are (pre)periodic. Hence for some n the infinite gap G = σ n d ( G) is periodic. By the previous paragraph all gaps of L f in G are finite. Then the quotient space (G ∩ S)/ ∼ f is a so-called dendrite, which carries a self-map induced by σ p d where p is the minimal period of G. Theorem 7.2.7 from [START_REF] Blokh | Fixed point theorems for plane continua with applications[END_REF] implies that there are infinitely many periodic cutpoints in this dendrite, hence G contains leaves of L r f , a contradiction.

Preliminaries to Theorem A s:prel

In this section, we list various preliminary results. Some of them are well known and therefore given without proof.

ss:pl 4.1. A perturbation lemma. Consider a sequence λ n ∈ D converging to λ ∈ S. We say that λ n converges to λ non-tangentially if all λ n belong to a cone with the following properties. The vertex of the cone is λ. The axis of symmetry of the cone is the radius (radial line) through λ. The angle between the edges of the cone and its axis of symmetry is less than π/2. For an open set U ⊂ C and a holomorphic map g : U → C with attracting fixed point 0, let A(g) be the immediate basin of attraction of 0 with respect to g. Recall a part of Corollary 2 from [START_REF] Buff | On the size of linearization domains[END_REF], based on ideas of [Yoc95, Proposition 1, page 66]: l:perturb1 Lemma 4.1 (Corollary 2 of [START_REF] Buff | On the size of linearization domains[END_REF]). Suppose that λ n ∈ D converge non-tangentially to λ ∈ S. Let U ⊂ C be an open set, and f : U → C be a holomorphic map with f (0) = 0 and f (0) = λ. Assume that f has a Siegel disk ∆ around 0. If the sequence f n : U → C satisfies f n (0) = 0, f n (0) = λ n , and for every compact subset

K ⊂ ∆ max z∈K |f n (z) -f (z)| = O(|λ -λ n |), n → ∞, then any compact set K ⊂ ∆ is contained in A(f n ) for n large enough.
We now go back to our family F. Below, we define some special perturbations of polynomials in F nr . Let f (z) = f λ,b (z) = λz + bz 2 + z 3 ∈ F nr so that |λ| 1. Then denote by f ε the polynomial eq:feps eq:feps (4.1.1)

f (1-ε)λ,b (z) = (1 -ε)λz + bz 2 + z 3 ∈ F at ,
where ε > 0. The following is an easy corollary of Lemma 4.1.

c:perturb2 Corollary 4.2. If f = f λ,b has a Siegel disk ∆(f ) around 0, then, for every compact set K ⊂ ∆(f ), there exists δ( K) > 0 such that every polynomial f ε has the property K ⊂ A(f ε ) for any 0 < ε < δ( K).

Proof. Assume the contrary. Then there exists a sequence ε n → 0 with K ⊂ A(f εn ). Set λ n = (1-ε n )λ; then λ n converge to λ non-tangentially.

To use Lemma 4.1, observe that for a compact set K ⊂ ∆(f )

max z∈K |f εn (z) -f (z)| = O(|λ -λ n |), n → ∞
because the left-hand side equals ε n max z∈K |z| while |λ-λ n | = ε n . This yields a contradiction with Lemma 4.1 and proves the corollary.

ss:bp 4.2. Blaschke products. Here we deal with the dynamics of Blaschke products. As we do not need Blaschke products of higher degrees and for the sake of simplicity we only consider quadratic Blaschke products with fixed point 0. For a complex number a, we let a denote the complex conjugate of a.

d:bp Definition 4.3 (Blaschke products). Let b and s be complex numbers such that 0 < |b| < 1 and |s| = 1. Then the formula eq:bp eq:bp (4.2.1) B b,s (z) = sz b -z 1 -bz defines a quadratic Blaschke product with fixed point 0. It is not hard to see that the Blaschke product (4.2.1) is conjugate by a rotation to a so-called normalized quadratic Blaschke product Q a of the form eq:bp1 eq:bp1 (4.2.2)

Q a (z) = z a -z 1 -az ;
for some complex number a with |a| < 1.

Our normalized Blaschke product Q a differs by a sign from the traditional one in which the numerator is z -a, not a -z. It is well known that Q a is a quadratic rational function that preserves D, its complement C \ D, and the unit circle S. Moreover, eq:bpder eq:bpder (4.2.3)

Q a (z) = az 2 -2z + a (1 -az) 2 ,
which implies that Q a (0) = a; an easy computation shows that the multiplier of the fixed point at ∞ is a. Thus, both 0 and infinity are attracting fixed points of Q a . Set D r = {|z| < r}; then, by the Schwarz Lemma (or directly), we have

Q a (D r ) ⊂ D r . Similarly, |Q a (z)| > |z| if |z| > 1.
Hence the Julia set of Q a is S. In fact, Q a is expanding on S, see [START_REF] Tischler | Blaschke products and expanding maps of the circle[END_REF]. It is easy to see, that eq:bpcrpt eq:bpcrpt (4.2.4)

c a = 1 -1 -|a| 2 a = a 1 -1 -|a| 2 |a| 2 = a 1 + 1 -|a| 2
is the unique critical point of Q a that belongs to D. Also, by (4.2.4) a and c a belong to the same radial segment of D so that c a is located between 0 and a. Observe that if a → s ∈ S, then c a → s too. To describe the limit behavior of the entire orbit of c a as a → s ∈ S, we need Lemma 4.4. For a complex number w, set R w (z) = wz.

l:limit-onk Lemma 4.4. Suppose that s ∈ S and K ⊂ C \ {s} is a compact set.

Then the maps Q a converge to R s uniformly on K as a → s.

Proof. Since |s| = 1, we have ss = 1. Therefore s -z = s -ssz = s(1 -sz). Dividing on both sides by 1 -sz, we see that s-z 1-sz = s for all z = 1 s = s. Since K ⊂ C \ {s} is a compact set, standard continuity arguments imply the conclusions of the lemma. This does not yet yield the limit behavior of the orbit of c a as a → s ∈ S as then c a → s too, and Lemma 4.4 does not apply. l:bpcror Lemma 4.5. Suppose that s = e 2πiθ , where θ is irrational. Let ε be a positive real number and m be a positive integer. Then there exists δ > 0 such that for any a ∈ D with |s -a| < δ we have |Q i a (c a )| > 1 -ε for all i = 0, 1, . . . , m.

In other words, if a = Q a (0) is close to s, then the orbit of c a stays close to the unit circle for any given period of time. The conclusions of the lemma are sensitive with respect to the point whose trajectory we consider. For example, Q a (a) = 0 so that the orbit of a under Q a is (a, 0, 0, . . . ) and, thus, the limit behavior of the orbits of a and of c a are very different even though both a and c a converge to s = e 2πiθ .

Proof. We will use the following notation and terminology. Given a small arc T ⊂ S of length |T | with endpoints of arguments α and β, denote by U T a "polar rectangle" built upon T with vertices (in polar coordinates) given by (1

-|T |, α), (1 + |T |, α), (1 + |T |, β), (1 -|T |, β).
Simple computations show that eq:bpcrim eq:bpcrim (4.2.5)

Q a (c a ) = (1 -1 -|a| 2 ) 2 a 2 = c 2 a
Since θ is irrational, there exists a closed arc I ⊂ S symmetric with respect to s such that I, R s (I), R 2 s (I), . . . , R m s (I) are pairwise disjoint circle arcs. By Lemma 4.4, we can choose a small arc T ⊂ R s (I) centered at s 2 such that for all a sufficiently close to s we have that

Q i a (U T ) ⊂ U R i+1 s (I)
for all i = 0, . . . , m -1. We can then choose a small neighborhood W of s so that ζ 2 ⊂ U T provided that ζ ∈ W ; by (4.2.4) and (4.2.5) this implies that for any a sufficiently close to s we have c a ∈ W and Q j a (c a ) ∈ U R j s (I) for every j = 1, . . . , m.

C \ K is connected) continuum K ⊂ U .
If K is not a singleton and U = C, then we will call U \ K nondegenerate. It is well known [START_REF] Ahlfors | Complex analysis[END_REF] that any non-degenerate annulus is conformally equivalent to a non-degenerate round annulus and that two round annuli A(r, R) and A(r , R ) are conformally equivalent if and only if R r = R r [START_REF] Schottky | Über konforme Abbildung von mehrfach zusammenhängenden Fläche[END_REF]. Given a topological annulus A that is conformally equivalent to the round annulus A = A(r, R), we define its modulus m( A) as ln(R)-ln(r) 2π . By the above results the modulus of an annulus is well defined and invariant under conformal equivalence. We will use Theorem 4.6 in the proof of Lemma 5.1; below ρ(X, Y ) denotes the infimum of the distance between points x ∈ X and y ∈ Y for sets X, Y ⊂ C. [f ] top of rational functions topologically conjugate to f coincides with the set of rational functions qc-conjugate to f and is connected.

Suppose now that f and g are hyperbolic polynomials in F with connected Julia sets. Recall that then J(f ), J(g) are locally connected. A critical orbit relation for f is a constraint of the form f n (c) = f m (c), m = n, where c and c are critical points of f , not necessarily different. As in Section 3, we can associate geodesic laminations L f and L g with f and g, respectively.

l:topconj Lemma 4.8. Let f and g be two degree d > 1 hyperbolic polynomials with connected Julia sets such that L f = L g . If f and g have no critical orbit relations, then f and g are topologically conjugate.

See [START_REF] Mcmullen | Quasiconformal homeomorphisms and dynamics III: The Teichmüller space of a holomorphic dynamical system[END_REF] for very similar statements. The same methods prove Lemma 4.8. It follows that g ∈ [f ] top . Note however that, in the cubic case, the intersection of [f ] top with F may be disconnected. c:samecomp Corollary 4.9. If polynomials f and g belong to the same bounded hyperbolic component of F, then L f = L g . On the other hand, suppose that f , g ∈ F at are hyperbolic polynomials with connected Julia sets such that L f = L g = L. If f and g have no attracting fixed points except 0, then f , g belong to the same hyperbolic component of F.

Proof. The first claim is a variation of a well-known property of hyperbolic components; it is left to the reader. To prove the rest, we may assume that neither f nor g has critical orbit relations. Indeed, otherwise we can slightly perturb f and g within their hyperbolic components of F so that the perturbed maps have no critical orbit relations. Then f and g are topologically conjugate by Lemma 4.8. Suppose that

f = f λ f ,b f = z 3 + b f z 2 + λ f z and g = g λg,bg = z 3 + b g z 2 + λ g z.
By Lemma 4.7, there is a continuous family f t , t ∈ [0, 1] of cubic rational functions qc-conjugate to f such that f 0 = f and f 1 = g. Indeed, a qc-conjugacy between f and g takes the standard complex structure on the dynamical plane of g to some invariant qc-structure on the dynamical plane of f . The latter is represented by a Beltrami differential ν. Considering the family of Beltrami differentials ν t = tν and using the Ahlfors-Bers theorem, we obtain a family f t with the desired properties. Observe that all rational functions f t are hyperbolic. Let M t be a complex affine transformation such that

h t = M t • f t • M -1 t ∈ F. Since [0, 1]
is simply connected, we may choose M t to depend continuously on t and so that M 0 = id. Let U be the hyperbolic component of F containing f . Then h t ∈ U for all t by continuity; in particular, h 1 ∈ U. On the other hand,

h 1 = M 1 • g • M -1 1 ∈ F
and g are affinely conjugate. This implies that either h 1 = g or h 1 = z 3 -b g z 2 + λ g z. In the former case, we are done. In the latter case, observe that h 1 and g have the same linearizing coordinate near infinity (this follows from the fact that z → z 3 commutes with the involution z → -z) while the orbits of g are obtained from the orbits of h 1 by z → -z. Therefore, the geodesic lamination of g differs from the geodesic lamination of h 1 by a half-turn.

On the other hand, by our construction L coincides with the geodesic lamination of h 1 . Thus, L is invariant with respect to the rotation by 180 degrees about the center of the unit disk. Then, by [START_REF] Blokh | Laminations from the Main Cubioid[END_REF], the major of an invariant quadratic gap G in L corresponding to the basin of immediate attraction of 0 (of either f or g) is 0 1 2 . This implies that there are two invariant attracting domains of g (or f ), corresponding to G and the 180-degree rotation of G with respect to the center of the unit disk. A contradiction with the assumption that g (and f ) has only one attracting fixed point. The statement now follows.

Proofs of Theorem A and Corollary C s:mt

Let f be an IS-capture polynomial. We refer to the glossary in the end of Section 2. Let m f > 0 be the smallest positive integer for which we have f m f (ca(f )) ∈ ∆(f ). Observe that, given sufficiently small ε > 0, for all polynomials g close enough to f , there exist a unique critical point re(g) of g that is ε-close to re(f ) and a unique critical point ca(g) of g that is ε-close to ca(f ). Notice that the functions re(g) and ca(g) are holomorphic functions of the coefficients of g. However re(g) is not necessarily recurrent, and g may not have a Siegel invariant domain.

Lemma 5.1 is based on special perturbations (4.1.1).

l:zeta-out Lemma 5.1. Suppose that f is an IS-capture polynomial. Then, for sufficiently small ε > 0, we have re(f ε ) ∈ A(f ε ). In particular, if

f ε / ∈ P • , then ca(f ε ) / ∈ A(f ε ). Proof. Set f = f λ,b . Then λ = e 2πiθ
, where θ is irrational. Take a closed Jordan disk K and an open Jordan disk U such that

0 ∈ K ⊂ U ⊂ U ⊂ ∆(f ).
We may assume that f m f (ca(f )) lies in the interior of K.

Observe that if f ε ∈ P • then re(f ε ) ∈ A(f ε ) as desired. In particular, if for sufficiently small ε > 0 we have that f ε ∈ P • , then we are done. Thus we need to consider the case when there are positive values of ε arbitrarily close to 0 and such that f ε / ∈ P • . We need to show that re(f ε ) ∈ A(f ε ) for all these values of ε. Observe that in any case at least one critical point must belong to A(f ε ) for all ε > 0. Hence, if ca(f ε ) / ∈ A(f ε ) for some ε > 0, then re(f ε ) ∈ A(f ε ) for this ε as desired. Thus, to prove the lemma it would suffice to prove the following claim.

Claim. For sufficiently small ε > 0, if f ε / ∈ P • then ca(f ε ) / ∈ A(f ε ).
Proof of the Claim. Suppose that there are positive values of ε arbitrarily close to 0 and such that f ε / ∈ P • . Moreover, suppose by way of contradiction that the Claim fails. Then there exists a sequence ε n → 0 with f εn / ∈ P • and ca(f εn ) ∈ A(f εn ). Since f εn / ∈ P • , then ca(f εn ) is the only critical point in A(f εn ). A Riemann map ϕ : A(f εn ) → D with ϕ(0) = 0 conjugates f εn | A(fε n ) with a normalized quadratic Blaschke product Q an , where a n ∈ D. Then ϕ(ca(f εn )) = c an is the unique critical point of Q an in D. This yields the following contradiction.

(i) By Lemma 4.5, the point Q

m f
an (c an ) approaches the unit circle as ε n → 0. (ii) By Corollary 4.2 and by continuity, the point Q m f an (c an ) is bounded away from the unit circle as ε n → 0. A more detailed proof follows. (i) Clearly, the multiplier (1 -ε n )λ of f εn at 0 converges to λ = e 2πiθ . It follows that the multiplier of Q aε n at 0 also converges to λ. By Lemma 4.5, the point Q m f an (c an ) approaches the unit circle as ε n → 0. (ii) On the other hand, take a polynomial f ε with small ε > 0. By Corollary 4.2, we have U ⊂ A(f ε ) for all sufficiently small ε > 0. By continuity, f

m f ε (ca(f ε )) ∈ K if ε > 0 is sufficiently small. Thus, the point f m f εn (ca(f εn )
) is separated from Bd(A(f εn )) by the annulus U \ K of a definite positive modulus. It follows, by the conformal invariance of the modulus, that the point Q m f aε n (c an ) must also be separated from S by an annulus of a definite positive modulus. However, this contradicts Theorem 4.6 and the conclusions of (i) above.

Recall (Definition 2.3) that for an IA-capture polynomial f we denote by ω 1 (f ) its critical point that belongs to A(f ) and by ω 2 (f ) its critical point that does not belong to A(f ) but eventually (after one or more iterations) maps into A(f ). Observe that our notation for critical points ω 1 (f ) and ω 2 (f ) is consistent with Definition 2.3. Finally, recall that by potentially renormalizable polynomials we mean polynomials in F that do not belong to P = P • . c:cri-id Corollary 5.2. Suppose that f is an IS-capture polynomial. If f is potentially renormalizable, then ω 1 (f ) = re(f ) and ω 2 (f ) = ca(f ).

Proof. Since f is potentially renormalizable, all maps f ε of f are outside

P • if ε is small. By definition and Lemma 5.1, re(f ) = ω 1 (f ) and ca(f ) = ω 1 (f ).
Observe that, if W is a hyperbolic component non-disjoint from F at such that polynomials in W have a critical point which maps into a cycle of attracting Fatou domains but does not belong to it, then W ⊂ F at is an IA-capture component consisting of polynomials f with an invariant attracting Fatou domain A(f ) 0, a well-defined critical point ω 1 (f ) ∈ A(f ) and a well-defined critical point

ω 2 (f ) = ca(f ) / ∈ A(f ) such that for some minimal m f > 0 we have f m f (ω 2 (f )) ∈ A(f ). t:1hypcomb Theorem 5.3. If f ∈ F nr is an IS-capture polynomial, then f belongs
to the boundary of exactly one bounded hyperbolic component W in F at . Every polynomial g ∈ W has a locally connected Julia set so that L g = L r g , and W is either P • , or an IA-capture component. Proof. First we consider maps f ε . By Lemma 5.1, for some δ > 0 and any ε > 0 with ε < δ, we have re(f ) ∈ A(f ε ). By Corollary 4.2 and continuity, f m f (ca(f ε )) ∈ A(f ε ). Thus, f ε is hyperbolic, and there is a unique hyperbolic component U of F containing all polynomials f ε with ε < δ. Clearly, U is either P • , or an IA-capture component.

By way of contradiction, assume now that U and V are different bounded hyperbolic components in F at whose boundaries contain f . All polynomials in U have locally connected Julia sets, are conjugate on their Julia sets, and give rise to the same cubic invariant lamination L U ; similarly, all polynomials in V give rise to the same cubic lamination L V (cf. Corollary 4.9). Since, for a hyperbolic polynomial, the iterated forward images of a critical point cannot lie on the boundary of a Fatou component, then, by Lemma 3.15, we have

L U = L r U and L V = L r V
where L r U and L r V are the corresponding rational laminations. Consider a leaf ∈ L r f . It corresponds to a (pre)periodic point in J(f ). Since all periodic points in J(f ) are repelling, then, by Lemma 3.2, we have ∈ L U and ∈ L V . Since this holds for any ∈ L r f , we conclude that L r f ⊂ L r U and L r f ⊂ L r V . Now consider a leaf αβ ∈ L r U . Then R g (α), R g (β) land at the same (pre)periodic point x g , for every g ∈ U. The periodic cycle, into which the point x g eventually maps, is repelling. Consider a sequence g n ∈ U converging to f . By Corollary 3.3 applied to this sequence, we have αβ ∈ L r f . Since αβ is an arbitrary leaf of L r U , we conclude that L r U ⊂ L r f . Similarly, L r V ⊂ L r f . Together with the opposite inclusions proved earlier, this implies that L r U = L r V = L r f . By the first paragraph, it follows that L U = L V . Finally, by Corollary 4.9, we have

U = V = W.
Proof of Theorem A. Let f ∈ F λ be an IS-capture polynomial. By Theorem 5.3, there is a unique bounded hyperbolic component U in F at with f ∈ Bd(U). A priori, there could exist a different hyperbolic component V outside of F at with f ∈ Bd(V). Since for g ∈ V the fixed point 0 is repelling, there is a periodic angle θ such that R g (θ) lands at 0 for all g ∈ V. Consider a sequence g n ∈ V converging to f . By Lemma 3.1, the ray R f (θ) lands at a periodic point y = 0 (recall that 0 is a Siegel point). By Lemma 3.2, the point y is parabolic. However, an IS-capture has no parabolic periodic points, a contradiction. Thus, U is the only bounded hyperbolic component in F containing f in its boundary. It remains to observe that, if U is an IA-capture, then, by Corollary 3.11, the polynomial f has a repelling periodic cutpoint in its Julia set.

Proof of Corollary C. Suppose that f ∈ F λ with |λ| = 1 is a cubic IScapture polynomial. By way of contradiction, assume that f ∈ CU \ P. By Theorem 5.3, all polynomials f ε (see Equation (4.1.1)) for small ε > 0 belong to some IA-capture component U (since f / ∈ P, we have f ε / ∈ P o for small ε). On the other hand, then, by Theorem A, the map f contains a repelling periodic cutpoint in its Julia set, a contradiction with f ∈ CU.

Existence of IS-capture components

In this section, we find IS-capture components on the boundary of P • as well as on the boundaries of IA-capture components. Thus we will prove Theorem B.

Let U be an IA-capture component in F. Then, for every f ∈ U, we write A(f ) for the immediate attracting basin of 0. There is a unique critical point ω 2 (f ) not in A(f ), and we have f m f (ω 2 (f )) ∈ A(f ) for some positive integer m f . We may assume that m f is the smallest positive integer with this property. Observe that m f does not depend on f ; it depends only on U. We call this integer the preperiod of U. l:ag Lemma 6.1. Let U be a hyperbolic component in F that is either P • or an IA-capture component. In the latter case, let m be the preperiod of U; in the former case, set m = 2. For every Brjuno θ ∈ R/Z and every n m, there exists a map f ∈ Bd(U) ∩ F λ , where λ = e 2πiθ and f n (c) = 0 for some critical point c of f . Additionally, it can be arranged that f k (c) = 0 for k < n.

Let X n be the set of all polynomials f ∈ F such that f n (c) = 0 for some critical point c of f , and n is the smallest non-negative integer with this property. It is clear that X n is a complex algebraic curve in F = C 2 . Define a function µ on X n as µ(f ) = f (0). l:center Lemma 6.2. Let U be an IA-capture component. Consider a slice F λ with λ = 0 such that F λ ∩U = ∅; then clearly |λ| < 1. Take any integer n m, where m is preperiod of U. There is a polynomial

f ! ∈ F λ ∩ U such that f n ! (c ! ) = 0 for some critical point c ! of f ! , and f k ! (c ! ) = 0 for k < n.
Proof. The proof is a standard qc-deformation argument, cf. [START_REF] Branner | Quasiconformal surgery in holomorphic dynamics[END_REF]. Take any f ∈ F λ ∩ U. Then there is a critical point c of f with

f m (c) ∈ A(f ). The point v = f (c) is contained in a strictly preperiodic Fatou component V of f such that f m-1 (V ) = A(f ). Consider a C 1 - homeomorphism h : C → C that
coincides with the identity outside of some compact subset of V . Taking iterated h • f -pullbacks of the standard complex structure in iterated pullbacks of V , we obtain an h•f -invariant complex structure on C that coincides with the standard one outside of iterated pullbacks of V . By the Measurable Riemann Mapping theorem, h • f is conjugate to a rational function f h by a qcconjugacy fixing ∞. Since ∞ is a fixed critical point of f h of multiplicity 2, we conclude that f h is a polynomial. We may also arrange that f h ∈ F by an affine change of variables. In a small neighborhood of 0, we have h • f = f , and f is conformally conjugate to f h . Therefore, f and f h have the same multiplier at 0, and f h ∈ F λ . Note that f h depends continuously on h, and f h = f for h = id. Thus any connected set of homeomorphisms h gives rise to a connected subset of F λ lying entirely in U.

We now consider a connected set H of homeomorphisms as above (i.e., all h ∈ H equal the identity outside of some compact subset of V ). Let D be the corresponding set of maps f h , where h runs through H. Clearly, D is connected. For g = f h ∈ D, define v g as the image of h(v) under the conjugacy between h • f and f h . Then v g is a critical value of g. We can choose a homeomorphism h ! so that f n-1 (h ! (v)) = 0 and that f k-1 (h ! (v)) = 0 for k < n. Moreover, we can arrange that f m-1 (h ! (v)) is any given f n-m -preimage of 0 in A(f ). This chosen homeomorphism h ! can be included into a connected set H of homeomorphisms. The corresponding polynomial f ! = f h ! has a critical point c ! corresponding to the critical point c of h ! • f . Set v ! = f ! (c ! ) to be the corresponding critical value; clearly, it corresponds to the critical value h ! (v) of h ! • f . We have f n ! (c ! ) = 0 and f k ! (c ! ) = 0 for k < n. On the other hand, f ! belongs to a connected set D of hyperbolic polynomials; therefore,

f ! ∈ F λ ∩ U.
The component P • has been extensively studied in [START_REF] Petersen | Analytic coordinates recording cubic dynamics[END_REF]. In particular, the following is an immediate corollary of the parameterization of P • obtained in [START_REF] Petersen | Analytic coordinates recording cubic dynamics[END_REF]: l:center0 Lemma 6.3. Let λ be any complex number with |λ| < 1, and n be any integer that is at least 2. Then P • ∩ F λ contains a polynomial f ! with the following properties: f n ! (c ! ) = 0 for some critical point c ! of f ! , and

f k ! (c ! ) = 0 for k < n.
Thus, both in the case U = P • and in the case where U is an IAcapture component, we found a certain map f ! ∈ U.

Proof of Lemma 6.1. Recall that the function µ : X n → C was defined by the formula µ(f ) = f (0). We claim that µ(X n ∩ U) coincides with D, possibly with finitely many punctures. In the case U = P • , this follows from Lemma 6.3. Thus it suffices to assume that U is an IAcapture component. The inclusion µ(X n ∩ U) ⊂ D is obvious. It now suffices to show that µ(X n ∩ U) is open and closed in D. It is open by the Open Mapping Theorem and since µ is a non-constant holomorphic map. Suppose now that λ belongs to the boundary of µ(X n ∩ U) in D but not to µ(X n ∩ U). Then there is a polynomial f ∈ F λ ∩ X n ∩ U. In other words, there is a sequence f i ∈ X n ∩ U with f i → f ∈ F λ as i → ∞. For every i, there is a critical point c i of f i with f n i (c i ) = 0. Passing to a subsequence, we may assume that c i → c as i → ∞, where c is a critical point of f , and f n (c) = 0. On the other hand, |λ| < 1, hence f is hyperbolic. A hyperbolic polynomial belongs to the closure of a hyperbolic component U only if it belongs to U. Therefore, f ∈ U, but then by definition we have f ∈ X n ∩ U unless f is a puncture of X n (which means that f k (c) = 0 for some k < n). The latter case is ruled out for the following reason. There is δ > 0 such that f is injective on the δ-disk D δ around 0, and f (D δ ) D δ . Then, by continuity, D δ ⊂ A(f i ) for all large i. This implies that f k (c) = 0 for k < n. It follows that µ(f ) as f runs through X n takes all values in S, in particular, all values of the form e 2πiθ , where θ is Brjuno.

Choose a point f ∈ X n ∩ U with µ(f ) = e 2πiθ , where θ is Brjuno. It is clear that f is on the boundary of U. We will now prove that f is IS-capture. Indeed, f (0) = λ = e 2πiθ and θ is Brjuno, hence f has a Siegel disk ∆ around 0 (we distinguish between the function µ and its particular value λ). On the other hand, since f ∈ X n , there is a critical point c of f such that f n (c) = 0. We have in fact f ∈ X n (and f k (c) = 0 for k < n) for the same reason as above. By definition, this means that f is an IS-capture polynomial.

The following statement is proved as Theorem 5.3 in [START_REF] Zakeri | Dynamics of Cubic Siegel Polynomials[END_REF] for a different parameterization of basically the same slices. The only difference with [START_REF] Zakeri | Dynamics of Cubic Siegel Polynomials[END_REF] is that Zakeri considers critically marked cubic polynomials.

l:no-isol Lemma 6.4. Suppose that f ∈ F λ , where |λ| = 1, and f has a Siegel disk ∆ around 0. If f n (c) ∈ ∆ for some critical point c of f , then there is an IS-capture component in F λ containing f .

Proof. The proof is based on the same qc-deformation argument as the proof of Lemma 6.2. We will use the notation introduced in Lemma 6.2, in particular, v, H, D and f h . Then D = {f h |h ∈ H} a connected subset of F λ consisting of IS-capture polynomials. Recall that v = f (c) is a critical value of f . We choose the set H of homeomorphisms so that D = {h(v) | h ∈ H} is open. For every g ∈ D, we let ∆ g be the Siegel disk of g around 0. We let V g denote the Fatou component of g containing a critical value and such that g n-1 (V g ) = ∆ g . These properties define V g in a unique way. We will also write v g for the critical value of g contained in V g . Note that, if g = f h , then v g is the image of h(v) under the conjugacy between h • f and f h . Consider the Riemann map φ g : ∆ g → D such that φ(0) = 0 and φ (0) ∈ R >0 . The map g → φ g (g n-1 (v g )) takes D to the open set φ f (f n-1 (D)). Indeed, the image of f h under this map is φ f (f n-1 (h(v))). Thus, an analytic map takes D to some open set. It follows that D contains an open subset of F λ . Since D consists of IScapture polynomials, it is contained in some IS-capture component.

Finally, we can prove the main theorem of this section.

t:IS-BdP Theorem 6.5. Let U be a hyperbolic component of F that is either P • or an IA-capture component. In the latter case, set m to be the preperiod of U; in the former case set m = 2. For every Bjuno θ ∈ R/Z and every n m, there exists an IS-capture component D in Bd(U)∩F λ with λ = e 2πiθ such that, for all g ∈ D, we have g n (c g ) ∈ ∆(g) for some critical point c g of g.

Recall that ∆(g) is the Siegel disk of g around 0.

Proof. By Lemma 6.1, for any Brjuno θ ∈ R/Z and any n m, there is a cubic polynomial f with the following properties:

(1) we have f ∈ F λ , where λ = e 2πiθ ;

(2) there is a critical point c of f with f n (c) = 0;

(3) we have f k (c) = 0 for k < n. By Lemma 6.4, there is an IS-capture component D in F λ containing f . By Theorem A, the component D belongs to the boundary of a unique hyperbolic component V of F. Moreover, by Theorem 5.3, the polynomial f lies on the boundary of a unique hyperbolic component. But f is in the boundary of U. It follows that V = U, hence D is contained in the boundary of U.

Theorem 6.5 establishes the existence of many analytic disks on the boundary of the cubic connectedness locus. Observe that Lemma 6.4 and Theorem 6.5 imply Theorem B.

We conclude this section with a remark which relates our results concerning IA-capture components and laminations. A cubic invariant lamination L is said to be an IA-capture lamination if the following assumptions hold:

(1) there is an invariant Fatou gap A such that σ 3 | A∩S is two-to-one;

(2) there is a Fatou gap V = A such that σ 3 | V ∩S is two-to-one;

(3) we have σ m L 3 (V ) = A, where m L = m is the minimal integer with this property. The number m is called the preperiod of L. It is well-known (and easy to see) that any IA-lamination is the closure of its restriction upon all the rational angles (i.e., the closure of the corresponding rational lamination).

It follows from the appendix to [START_REF] Milnor | Hyperbolic components. With an appendix by A. Poirier[END_REF] written by Poirier that, for each IA-capture lamination L, there exists a unique IA-capture component U L ⊂ F with the following property. No matter which f ∈ U L we take, the lamination generated by f coincides with L. The result of [START_REF] Milnor | Hyperbolic components. With an appendix by A. Poirier[END_REF] is stated in the language of Hubbard trees and so-called reduced mapping schemes, however, a straightforward translation of this result into the language of laminations yields the claim stated above. Similarly, if L is the empty lamination, then we set U L = P • . Evidently, Theorem 6.5 can be restated to emphasize the role of IA-capture laminations, e.g., as follows.

Theorem 6.5 . Let L be the empty lamination or an IA-capture lamination. In the latter case, set m to be the preperiod of L; in the former case set m = 2. For every Brjuno θ ∈ R/Z and every n m, the hyperbolic component U L contains an IS-capture component D in Bd(U L )∩F λ with λ = e 2πiθ such that, for all g ∈ D, we have g •n (c g ) ∈ ∆(g) for some critical point c g of g, and n is the least such integer.

The main cubioid of F s:cu

In this section, we prove Corollary D and obtain corollaries related to the problem of distinguishing between Siegel and Cremer fixed points. Recall that the Main Cubioid CU was introduced in Definition 2.5.

Let W be a component of TH(P λ ) \ P λ , where |λ| 1. It is called a queer domain (or is said to be of queer type) if there exists a polynomial f ∈ W so that all of its critical points are in J(f ). Polynomials from such W are also said to be of queer type. Observe that IS-polynomials and polynomials of queer type have connected Julia sets. If f is an IS-polynomial, then ca(f ) is a critical point of f that does not belong to J(f ), hence f is not a polynomial of queer type.

The following theorem relies on [Zak99, Theorem 3.4], where the most difficult case is worked out.

t:nosiegel Theorem 7.1 ( [START_REF] Blokh | Complementary components to the cubic principal domain[END_REF]). Let W be a component of TH(P λ ) \ P λ of queer type. Then, for any polynomial f ∈ W, the Julia set J(f ) has positive Lebesgue measure and carries an invariant line field.

Properties of polynomials from P listed in Theorem 2.4 are inherited by polynomials from the topological hulls TH(P λ ). Moreover, all components of the set TH(P λ )\P λ , where |λ| 1, consist of λ-stable polynomials.

In [START_REF] Blokh | Complementary components to the cubic principal domain[END_REF], we consider components of the set TH(P λ )\P λ , where |λ| 1. Let us describe some results of [START_REF] Blokh | Complementary components to the cubic principal domain[END_REF][START_REF] Blokh | Quadratic-like dynamics of cubic polynomials[END_REF]. A cubic polynomial f ∈ F λ \ P = F λ \ P λ with |λ| 1 is said to be potentially renormalizable. A critical point c of a potentially renormalizable polynomial f is said to be principal if there is a neighborhood U of f in F and a holomorphic function ω 1 : U → C defined on U such that c = ω 1 (f ), and, for every g ∈ U ∩ F at , the point ω 1 (g) is the critical point of g contained in A(g). By Theorem 7.3, if f ∈ F nr is potentially renormalizable, then the point ω 1 (f ) is well-defined; let the other critical point of f be ω 2 (f ). It is easy to see that ω 1 (f ) ∈ K(f ). It immediately follows from [START_REF] Blokh | Laminations from the Main Cubioid[END_REF] that an IA-capture polynomial g has a repelling periodic cutpoint of the Julia set J(g). Hence an IA-capture polynomial g is not in CU, thus not in P, i.e., it is potentially renormalizable, and the notation for its critical points ω 1 (g), ω 2 (g), introduced in Definition 2.3, is consistent with the just introduced notation for all potentially renormalizable polynomials.

Recall that, by Theorem 7.2, all polynomials in a component W of TH(P λ ) \ P λ are conjugate on their Julia set. Moreover, if some polynomial in W is an IS-capture, then it is easy to see that so are all polynomials in W. This inspires the following definition. Let W be a component of λ-stable polynomials, where |λ| 1. Then W is said to be of IS-capture type if any f ∈ W is an IS-capture polynomial. We also say in this case that W is an IS-capture component. It is easy to construct examples of IS-captures in F λ \ TH(P λ ). By Theorem B the first possibility listed in Theorem 7.4 is impossible.

Corollary D now follows from Theorem 7.4. For the sake of completeness we also prove the next lemma.

l:nohyp Lemma 7.5. The only hyperbolic component of F intersecting CU is P • .

Proof. Assume, to the contrary, that there exists a hyperbolic component V = P • intersecting CU. Set V λ = V ∩ F λ and CU λ = CU ∩ F λ . Choose λ with V λ ∩ CU λ = ∅. We must have |λ| 1 since otherwise CU λ = ∅. From V λ = ∅, it follows that V ∩ F at = ∅. But then V ⊂ F at and |λ| < 1. Note also that, since polynomials in CU have connected Julia sets, all polynomials in V have connected Julia sets, i.e., the component V is bounded. Take g ∈ V λ ∩ CU λ . Then J(g) is locally connected; let L be the corresponding geodesic lamination. There is a gap G of L corresponding to A(g). By Theorem 3.10, the major M of G is either critical or periodic. The former implies that a critical point of g belongs to Bd(A(g)), a contradiction. Therefore, M = αβ is periodic. The rays R g (α), R g (β) land at the same periodic point x of g. Since g is hyperbolic, x must be repelling. Thus g has a repelling periodic cutpoint of J(g), a contradiction with g ∈ CU.

A question as to whether a fixed irrationally indifferent point of a polynomial is Cremer or Siegel depending on the multiplier at this point is addressed in a conjecture by A. Douady. Let us now state a related corollary based upon results of Perez-Marco.

Below we verify this for cubic polynomials f λ,b = λz +bz 2 +z 3 except for polynomials that belong to the set P λ . An important ingredient of our arguments is a result of R. Perez-Marco [START_REF] Perez-Marco | Total convergence or general divergence in small divisors[END_REF]; again for brevity we state only a relevant corollary of Perez-Marco's theorem reduced to our spaces of polynomials (the actual results of [START_REF] Perez-Marco | Total convergence or general divergence in small divisors[END_REF] are much stronger and much more general). c:pm Corollary 7.6 (Corollary 1 [START_REF] Perez-Marco | Total convergence or general divergence in small divisors[END_REF]). Suppose that λ = e 2πiθ and θ is irrational. Then the set of parameters b for which f λ,b has 0 as a Siegel fixed point is either the entire F λ , or, otherwise, has Hausdorff dimension 0 (in particular, it has empty interior).

Combining these results with our tools we prove Corollary 7.7.

c:nop Corollary 7.7. If θ / ∈ B is not a Brjuno number and λ = e 2πiθ , then the fact that f ∈ F λ \ P λ implies that 0 is a Cremer fixed point of f . Proof. Suppose first that f = f λ,b / ∈ TH(P λ ). Then, by [START_REF] Blokh | Quadratic-like dynamics of cubic polynomials[END_REF], the map is immediately renormalizable; moreover, 0 belongs to the filled quadratic-like Julia set K * ⊂ K(f ) of f . By Theorem 1.3, this implies that 0 is a Cremer point of f . By Corollary 7.6, it follows then that the set of parameters b for which f λ,b has 0 as a Siegel point has empty interior. Since, by [START_REF] Blokh | Quadratic-like dynamics of cubic polynomials[END_REF], in each component of TH(P λ ) \ P λ the polynomials are conjugate, then polynomials in those bounded domains cannot have 0 as their fixed Siegel point. This completes the proof.

  ) Given any three consecutive points a, b, c in A z , the points σ d (a), σ d (b) and σ d (c) form a triple of consecutive points in A f (z) .

  t:a4 Theorem 3.7 ([START_REF] Kiwi | Wandering orbit portraits[END_REF]). Let f be a polynomial of degree d > 1 with locally connected Julia set J(f ). Then there exists an integer k = k(d) independent of f , such that every wandering point z ∈ J(f ) can be the landing point of at most k external rays. 3.2. Full lamination. For a (finite or infinite) set A ⊂ S, denote by CH(A) its (closed Euclidian) convex hull. A chord ab between any two points a, b ∈ S is CH({a, b}) and contains the endpoints a and b. If b = a the chord is called degenerate. Consider a closed set A ⊂ S and its convex hull CH(A). An edge of CH(A) is a closed straight segment I connecting two points of S such that I ⊂ Bd(CH(A)). Define the map σ d : S → S by σ d (s) = s d ; here we assume S ⊂ C. Then the (σ d -)image of a chord ab is by definition the chord σ d (a)σ d (b). A (σ d -)critical chord is a non-degenerate chord whose endpoints have the same σ d -image.

  t:major Theorem 3.10 ([START_REF] Blokh | Laminations from the Main Cubioid[END_REF]). Consider a polynomial f ∈ F at \ P • with locally connected Julia set J(f ). Then the geodesic lamination L f has a quadratic invariant gap G, and there are two possibilities.(1) The major M (G) of G is critical, the corresponding critical point of f belongs to Bd(A(f )), and periodic cutpoints of J(f ) do not exist. (2) The major M (G) of G is periodic, and the corresponding point of J(f ) is a repelling or parabolic periodic cutpoint of J(f ).Corollary 3.11 easily follows.c:attracapt Corollary 3.11. If f ∈ F λ , |λ| < 1, is an IA-capture polynomial, then J(f ) is locally connected, the geodesic lamination L f has a quadratic invariant gap G with periodic major M (G), the Julia set J(f ) contains a periodic repelling cutpoint associated to M (G), and f ∈ F λ \ P.

  d:siblinv Definition 3.13 ([BMOV13]). A collection of chords L is sibling σ dinvariant provided that: (1) for each ∈ L, we have σ d ( ) ∈ L, 2

  Modulus. A round annulus A(r, R) ⊂ C is an open annulus formed by two concentric circles of radii r < R. A topological annulus U \ K is formed by a simply connected domain U ⊂ C and a non-separating (i.e., such that

  t:annuprop Theorem 4.6. Suppose that A ⊂ A are two annuli such that A is not null-homotopic in A . Then m(A) m(A ). Moreover, there exists a function ψ :R >0 → R >0 such that ρ(K, S) ψ(m(D \ K)) for any non-separating continuum K ⊂ D.The first part of Theorem 4.6 is well known and can be found in various textbooks; the second part easily follows, e.g., from [McM94, Theorem 2.4] or from [Ahl06, Problem I of Section A, Chapter III]. 4.4. Hyperbolic components. We will make use of the following result [McS98, Corollary 2.10]: l:McS Lemma 4.7. Let f be a hyperbolic rational function. Then the set

  t:extendclo Theorem 7.2 ([BOPT14a]). Suppose that |λ| 1. We have TH(P λ ) ⊂ CU.

  t:princ Theorem 7.3 ([BOPT14b]). A potentially renormalizable polynomial has a unique principal critical point.

  t:sie-quee Theorem 7.4 ([START_REF] Blokh | Complementary components to the cubic principal domain[END_REF]). Let W be a component of TH(P λ ) \ P λ , where |λ| 1. Then W is either of IS-capture type or of queer type.

The set F at was denoted by A in[START_REF] Blokh | Complementary components to the cubic principal domain[END_REF][START_REF] Blokh | Quadratic-like dynamics of cubic polynomials[END_REF]. We adopt a more consistent notation in this paper.

Acknowledgements

The first author was partially supported by NSF grant DMS-1201450. The third named author was partially supported by NSF grant DMS-1807558. The fourth named author was partially funded within the framework of the HSE University Basic Research Program and by the Russian Academic Excellence Project '5-100'. Finally, we would like to thank the referees for valuable and thoughtful suggestions.