Immediate renormalization of complex polynomials
aa r X i v : . [ m a t h . D S ] F e b IMMEDIATE RENORMALIZATION OF COMPLEXPOLYNOMIALS
ALEXANDER BLOKH, LEX OVERSTEEGEN, AND VLADLEN TIMORIN
Abstract.
A cubic polynomial P with a non-repelling fixed point b is said to be immediately renormalizable if there exists a (con-nected) quadratic-like invariant filled Julia set K ∗ such that b ∈ K ∗ . In that case exactly one critical point of P does not belongto K ∗ . We show that if, in addition, the Julia set of P has no(pre)periodic cutpoints then this critical point is recurrent. Introduction
In the introduction we assume knowledge of basics of complex dy-namics.We study polynomials P with connected Julia sets J ( P ). An (ex-ternal) ray with a rational argument always lands at a point that iseventually mapped to a repelling or parabolic periodic point. If twoexternal rays like that land at a point x ∈ J ( P ), then such rays aresaid to form a rational cut (at x ) . The family of all rational cuts ofa polynomial P may be empty (then one says that the rational lami-nation of P is empty); if it is non-empty it provides a combinatorialtool allowing one to study properties of P even in the presence of suchcomplicated irrational phenomena as Cremer or Siegel periodic points.Consider quadratic polynomials with connected Julia set. It is knownthat any quadratic polynomials Q / ∈ PHD has rational cuts (PHD is the Principal Hyperbolic Domain of the Mandelbrot set). Thus,any arc from PHD towards the rest of the Mandelbrot set consists ofpolynomials with rational cuts.The purpose of this paper is to investigate a similar phenomenon inthe cubic case. Then there is a “gray area” G in-between the set ofcubic polynomials with rational cuts, and the set PHD , the PrincipalHyperbolic Domain of the cubic connectedness locus. We conjecture
Date : February 20, 2021.2010
Mathematics Subject Classification.
Primary 37F20; Secondary 37C25,37F10, 37F50.
Key words and phrases.
Complex dynamics; Julia set; Mandelbrot set.The study has been funded by the Russian Academic Excellence Project ‘5-100’. that G is empty. Thus, the set G is the true object of our study eventhough, paradoxically, in the end we want to establish that it is empty.As a step in this direction we prove that a polynomial from G musthave specific properties. Our approach is not unusual: according tothe nature of the contrapositive argument one studies a phenomenonin great detail only to discover that an elaborate list of its propertiesleads to contradictions thus disproving its existence. d:ql1 Definition 1.1 ([DH85]) . A polynomial-like map is a proper holomor-phic map f : U → f ( U ) of degree k >
1, where U , f ( U ) ⊂ C areopen Jordan disks and U ⊂ f ( U ). The filled Julia set K ( f ) of f is theset of points in U that never leave U under iteration of f . The Juliaset J ( f ) of f is the boundary of K ( f ). Call U a basic neighborhood of K ( f ) and assume that if f is given, then its basic neighborhood isfixed. If k = 2, then the corresponding polynomial-like maps are saidto be quadratic-like .We can now state our main result. t:recur Theorem 1.2.
Let f be a cubic polynomial with empty rational lami-nation that has a quadratic-like restriction with a connected quadratic-like filled Julia set K ∗ ( f ) = K ∗ . Then the critical point of f that doesnot belong to K ∗ is recurrent. In the situation of Theorem 1.2 we will always denote a connectedquadratic-like filled Julia set by K ∗ ; also, we will fix its neighborhood U ∗ on which f is quadratic-like and denote f | U ∗ by f ∗ .2. Preliminaries: polynomial-like maps and cubicpolynomials with empty rational lamination s:prelim By classes of polynomials, we mean affine conjugacy classes. Fora polynomial f , let [ f ] be its class, let K ( f ) be its filled Julia set,and let J ( f ) be its Julia set. The connectedness locus M d of degree d is the set of classes of degree d polynomials whose critical points donot escape (i.e., have bounded orbits). Equivalently, M d is the set ofclasses of degree d polynomials f whose Julia set J ( f ) is connected. Theconnectedness locus M of degree 2 is called the Mandelbrot set ; theconnectedness locus M of degree 3 is called the cubic connectednesslocus . The principal hyperbolic domain PHD of M is defined asthe set of classes of hyperbolic cubic polynomials whose Julia sets areJordan curves. Equivalently, [ f ] ∈ PHD if both critical points of f are in the immediate basin attraction of the same (super-)attractingfixed point. A polynomial is hyperbolic if the orbits of all critical pointsconverge to (super-)attracting cycles. MMEDIATE RENORMALIZATION 3 ss:pl
Polynomial-like maps. d:ql2
Definition 2.1 ([DH85]) . Two polynomial-like maps f : U → f ( U )and g : V → g ( V ) of degree k are said to be hybrid equivalent ifthere is a quasi-conformal map ϕ from a neighborhood of K ( f ) to aneighborhood of K ( g ) conjugating f to g in the sense that g ◦ ϕ = ϕ ◦ f wherever both sides are defined and such that ∂ϕ = 0 almosteverywhere on K ( f ).The terminology is explained by the following fundamental result. Straightening Theorem ([DH85]) . Let f : U → f ( U ) be a polynomial-like map. Then f is hybrid equivalent to a polynomial P of the samedegree. Moreover, if K ( f ) is connected, then P is unique up to ( aglobal ) conjugation by an affine map. We will need the next definition. d:plrays
Definition 2.2.
Let f be a polynomial, and for some Jordan disk U ∗ the map f ∗ = f | U ∗ be polynomial-like. Let g be a monic polyno-mial hybrid equivalent to f ∗ . Then the corresponding filled Julia set K ( f ∗ ) = K ∗ of f ∗ is called the polynomial-like filled invariant Juliaset . The curves in C \ K ( f ∗ ) = K ∗ corresponding (through the hybridequivalence) to dynamic rays of g are called polynomial-like rays of f ∗ .If the degree of f ∗ is two, then we will talk about quadratic-like rays .Denote polynomial-like rays R ∗ ( β ), where β is the argument of the ex-ternal ray of g corresponding to R ∗ ( β ). We will also call them K ∗ -rays to distinguish them from rays external to K ( f ) = K called K -rays .Here K ∗ -rays are defined in a bounded neighborhood of K ∗ while K -rays are defined on the entire plane. By Straightening Theoremcombined with well known results from complex dynamics the com-position ψ ∗ : C \ K ∗ → C \ D of the hybrid conjugacy for K ∗ (seeStraightening Theorem) and the inverse Riemann map for C \ K withreal derivative at infinity transfers the dynamics of f to the plane onwhich, loosely, the role of K ∗ is played by the unit circle while the restof the plane (i.e., the set C \ K ∗ ) remains “the same”. Thus, the restof K (i.e., the set K \ K ∗ ) is transferred to C \ D and looks like acollection of pieces “growing” out of D . In terms of dynamics the map P is transferred by the map ψ ∗ to a Jordan annulus c U ∗ = S ∪ ψ ∗ ( U ∗ )to produce the map z s : c U ∗ → S ∪ ψ ∗ ( f ( K ∗ )) ( S ⊂ C is the unit circlecentered at the origin) with the appropriate choice of s .Evidently, if f is a polynomial of degree d and T $ J ( f ) is a properpolynomial-like invariant Julia set then the degree of f | T is less than d . A. BLOKH, L. OVERSTEEGEN, AND V. TIMORIN
In particular, if f is a cubic polynomial and K ∗ ⊂ K ( f ) is a polynomial-like filled invariant Julia set, then the polynomial-like map f | K ∗ isquadratic-like. The following lemma is proven in [BOT16] (it is basedupon Theorem 5.11 from McMullen’s book [McM94]). l:7.2 Lemma 2.3 (Lemma 6.1 [BOT16]) . Let f be a complex cubic polyno-mial with a non-repelling fixed point a . Then the quadratic-like filledinvariant Julia set K ∗ with a ∈ K ∗ (if any) is unique. ss:emp-lam Polynomials with empty rational lamination.
As was saidin the Introduction, we want to study cubic polynomials f ∈ M with-out rational cuts (equivalently, with empty rational lamination). Thisreduces the family of polynomials of interest to us. l:gray-1 Lemma 2.4.
Suppose that a cubic polynomial f has empty rationallamination. Then f must have exactly one fixed non-repelling pointand all other periodic points of f are repelling.Proof. By Theorem 7.5.2 [BFMOT12] if all fixed points of f are re-pelling then at one of them the combinatorial rotation number is not 0and hence several rays must land, a contradiction. Also, if f has a fixednon-repelling point and a distinct periodic non-repelling point, by Kiwi[Kiw00] the rational lamination of f is non-empty, a contradiction. (cid:3) We will need the following corollary. c:7.2
Corollary 2.5.
A cubic polynomial f with empty rational laminationcontains, in its filled Julia set K ( f ) , at most one set K ∗ ; this set mustcontain a unique non-repelling fixed point of f .Proof. By Lemma 2.4, the map f has a unique non-repelling fixed point,say, a , and all other periodic points of f are repelling. Thus, if K ∗ doesnot contain a , then all its periodic points are repelling. In particular,by Theorem 7.5.2 [BFMOT12] the map f ∗ has a fixed point b suchthat K -rays landing at b rotate under f m , a contradiction. Thus, K ∗ contains a ; by Lemma 2.3 it is unique. (cid:3) Preliminaries: full continua and their decorations s:fucode
In this section we consider certain ordered by inclusion pairs of fullcontinua on the plane (a compact set X ⊂ C is full if C \ X is con-nected). This is a natural situation occurring in complex dynamics,both when studying polynomials and their parameter spaces. Indeed,let a cubic polynomial f have a connected filled Julia set K ( f ) = K .Suppose that K ∗ and U ∗ exist; Then the situation is exactly like theone described above because K ∗ ⊂ K . Another example is when one MMEDIATE RENORMALIZATION 5 takes the filled Main Cardioid of the Mandlebrot set M . It is easy togive other dynamical or parametric examples.Let X ⊂ Y be two full planar continua. We would like to represent Y as the union of X and decorations . d:dec Definition 3.1.
Components of Y \ X are called decorations (of Y relative to X ) , or just decorations (if X and Y are fixed).Decorations are connected but not closed; thus, decorations maybehave differently from what common intuition suggests. In Lemma3.2 we discuss topological properties of decorations. Given sets A and B , say that A accumulates in B if A \ A ⊂ B . l:triv Lemma 3.2.
Any decoration D of Y rel. X accumulates in X . Theset D \ D = D ∩ X is a continuum. The sets D and D ∪ X = D ∪ X are full continua.Proof. Suppose, by way of contradiction, that there exists x ∈ D \ ( D ∪ X ). Then we have D ⊂ A = D ∪ { x } ⊂ D while A ∩ X = ∅ . Since D is connected, and since D ⊂ A ⊂ D , then A is connected too. Hence D is not a component of K \ X , a contradiction.The continuum D is full as otherwise its complementary domainswould be complementary to Y .Now, by the first paragraph, D \ D = D ∩ X is compact. Supposethat D ∩ X is disconnected. Then there exists a bounded component U of C \ ( X ∪ D ) that at least partially accumulates to X and partiallyto D . Since Y is full, then U ⊂ Y ; hence U is a subset of a decorationthat accumulates (partially) to points of D . By the first paragraph thisimplies that U ⊂ D , a contradiction. Thus, D ∩ X is connected; then D ∩ X is a full continuum as both X and D are full. (cid:3) We will use the inverse Riemann map ψ : C \ X → C \ D with realderivative at infinity. Loosely, one can say that under the map ψ thecontinuum X is replaced by the closed unit disk D while the rest of theplane is conformally deformed. Thus, under ψ the decorations becomesubsets of C \ D . l:triv2 Corollary 3.3.
Let D be a decoration of Y rel. X . Then ψ ( D ) \ D isa (perhaps, degenerate) continuum (arc or unit circle) I D ⊂ S .Proof. Follows from Lemma 3.2. (cid:3)
Observe that the set I D can, indeed, coincide with the entire unitcircle (e.g., D can spiral onto D ). The arcs I D = S are also possible as ψ ( D ) may approach an arc I D by imitating the behavior of the functionsin(1 /x ) as x → + . Moreover, two distinct decorations D and T may A. BLOKH, L. OVERSTEEGEN, AND V. TIMORIN well have equal arcs I D and I T , or it may be so that, say, I D $ I T , or I D and I T can have a non-trivial intersection not coinciding with eitherarc (all these examples can be constructed by varying the behaviorof components similar to the behavior function sin(1 /x ) as x → + ).However there are some cases in which one can guarantee that eachdecoration has a degenerate arc I D . Ray Assumption on X and Y . Suppose that there is a dense set
A ⊂ S and a family of curves R x landing in A and disjoint from ψ ( Y ) . Suppose that Ray Assumption holds for X and Y . Moreover, sup-pose that there is a neighborhood U of Y and a homeomorphism ϕ : U → W ⊂ C . Then Ray Assumption holds for ϕ ( X ) ⊂ ϕ ( Y )too. l:triv3 Lemma 3.4.
Suppose that Ray Assumption holds for X ⊂ Y . Thenfor every decoration D the arc I D is degenerate.Proof. Suppose that I D is a non-degenerate arc. Choose three points x, y, z ∈ A ∩ I D . It follows that ψ ( D ) is contained in one of the twodisjoint open strips formed by the curves R x , R y and R z . Howeverthen ψ ( D ) can accumulate to only one of the circle arcs formed by thepoints x, y and z , a contradiction. (cid:3) Definition 3.5.
Under Ray Assumption and in the above notation,the argument of a decoration D is the angle α such that I D = { α } .Since we study decorations in the complex dynamical setting, makingthe ray assumption is not overly restrictive because, as we will now see,it holds in important for us dynamical cases. l:raya1 Lemma 3.6.
Suppose that K ∗ is a connected filled invariant polynomial-like Julia set contained in a connected filled Julia set K of a polynomial P . Then K ∗ ⊂ K satisfy the ray assumption.Proof. Choose a periodic repelling point x ∈ K ∗ and a K -ray R landingat x . Under ψ ∗ this ray becomes a curve ψ ∗ ( R ) landing at the appro-priate point of S , and these points are dense in S . The remark afterwe define Ray Assumption now shows, that K ∗ ⊂ K satisfy it. (cid:3) Cubic parameter slices s:para-sli
Let F be the space of polynomials f λ,b ( z ) = λz + bz + z , λ ∈ C , b ∈ C . An affine change of variables reduces any cubic polynomial to the form f λ,b . Clearly, 0 is a fixed point for every polynomial in F . Define the MMEDIATE RENORMALIZATION 7 λ -slice F λ of F as the space of all polynomials g ∈ F with g ′ (0) = λ ,i.e. polynomials f ( z ) = λz + bz + z with fixed λ ∈ C . We write P λ for the set of polynomials f ∈ F λ for which there are polynomials g ∈ F arbitrarily close to f with | g ′ (0) | < f ] of f then belongs to PHD . Also,denote by F nr the space of polynomials f λ,b with | λ | n on- r epelling”). For a fixed λ with | λ | λ -connectedness locus C λ , of the λ -slice of the cubic connectedness locus is defined as the setof all f ∈ F λ such that K ( f ) is connected. This is a full continuum[BrHu88, Z99]. We study sets C λ ⊂ F λ as we want to see to whatextent results concerning the quadratic Mandelbrot set hold for C λ . ss:imre-vs-hyp Immediately renormalizable polynomials vs the closure ofthe principal hyperbolic component.
Let us describe what hap-pens to quadratic-like invariant filled Julia sets of f ∈ F nr that contain0 when f is slightly perturbed (assuming such a set exists for a given f ). In the rest of the paper by the “quadratic counterpart” of f ∗ wemean a quadratic polynomial hybrid conjugate to f ∗ by StraighteningTheorem. l:cnct Lemma 4.1.
Let f ∈ F nr be a polynomial, K ∗ be a quadratic-like filledinvariant Julia set containing . Then K ∗ is connected. Every cubicpolynomial g ∈ F nr sufficiently close to f has a quadratic-like Julia set B ∗ containing ; the set B ∗ here is also connected. Moreover, if is anattracting fixed point for g then g has a quadratic-like Julia set whichis a Jordan curve; in particular, g / ∈ PHD .Proof. Since 0 is non-repelling, then, by the Fatou-Shishikura inequal-ity, the critical point of the quadratic counterpart of f ∗ cannot escape.Hence K ∗ is connected. Let f ∗ : U ∗ → V ∗ , f ∗ = f | U ∗ be the associatedquadratic-like map. If g is very close to f then 0 ∈ U ∗ and, moreover,we can arrange for a new Jordan disk W ∗ with g ( W ∗ ) = V ∗ . By theabove, the associated quadratic-like Julia set B ∗ is connected. Finally,if f i → f are polynomials with 0 as an attracting fixed point then, bythe above, for large i the polynomial f i has a quadratic-like filled Juliaset coinciding with the closure of the basing of immediate attraction of0. Therefore, [ f i ] / ∈ PHD for large i as desired. (cid:3) Call a cubic polynomial f ∈ F nr immediately renormalizable if thereare Jordan domains U ∗ and V ∗ such that 0 ∈ U ∗ , and f ∗ = f : U ∗ → V ∗ is a quadratic-like map; denote by K ∗ the filled quadratic-like Julia setof f ∗ (in the future we always use the notation U ∗ , V ∗ , f ∗ and K ∗ when talking about immediately renormalizable maps). Denote theset of all im mediately r enormalizable polynomials by ImR, and let A. BLOKH, L. OVERSTEEGEN, AND V. TIMORIN
ImR λ = F λ ∩ ImR. Let P be the set of polynomials f ∈ F nr withthe following property: there are polynomials g ∈ F arbitrarily closeto f with | g ′ (0) | < g ] ∈ PHD . Then clearly [ f ] ∈ PHD (observe, that there may be polynomials outside of P whose classesare also in PHD ). Also, set P λ = P ∩ F λ . Corollary 4.2 follows fromLemma 4.1. c:cnct Corollary 4.2. If f ∈ ImR , then K ∗ is connected. The set ImR isopen in F nr . The set ImR λ is open in F λ for any λ, | λ | . The sets ImR and P are disjoint. We want to study the sets ImR and P ; Corollary 4.2 shows thatthey are disjoint, so this investigation may be done in parallel. In facta lot is known about the sets ImR λ and P λ in the case when | λ | < t:roesch Theorem 4.3 ([Roe06]) . P λ is a Jordan disk for any λ with | λ | < . We want to combine this result with [BOPT16a] where sufficientconditions on polynomials for being immediately renormalizable aregiven. t:when-imr
Theorem 4.4 ([BOPT16a]) . If f ∈ F λ , | λ | belongs to the un-bounded complementary domain of P λ in F λ , then f is immediatelyrenormalizable. Combining these theorems and Lemma 4.1, we get Corollary 4.5. c:<1
Corollary 4.5. If | λ | < then ImR λ = C \ P λ . We study the neutral case | λ | = 1; to obtain more general results, asmuch as possible we study neutral slices without using specifics of thenumber λ . l:lambda<0 Lemma 4.6. If | λ | < then [ z + λz ] ∈ PHD .Proof. We claim that J ( f ) is a Jordan curve. Let U be the basin ofimmediate attraction of 0 (which is an attracting fixed point of f ).Since f n ( − z ) = − f n ( z ) for every n then U is centrally symmetric withrespect to 0. Since there exists a critical point c ∈ U and f ′ ( − z ) = f ′ ( z )for any z , then − c is critical. The central symmetry of U with respectto 0 now implies that − c ∈ U . Since both critical points of f belongto U , the claim follows. (cid:3) Theorem 4.7 develops Lemma 4.6. l:0in
Theorem 4.7.
For any λ, | λ | < , we have ∈ Int( P λ ) . For any λ, | λ | , we have ∈ P λ , and P λ is a continuum. MMEDIATE RENORMALIZATION 9
Proof.
The first claim is proven in Lemma 4.6. To prove the rest,observe that if | λ | = 1 then P λ = lim sup P τ where τ → λ, | τ | < ∈ Int( P τ ) for all these numbers τ , then 0 ∈ P λ , and P λ , beingthe lim sup of the continua P τ which all share a common point 0, isalso a continuum as claimed. (cid:3) ss:stru-fla The structure of the slice F λ . Following [BOT16], define theset CU λ , | λ | f ∈ F λ with connectedJulia sets and such that the following holds:(1) f has no repelling periodic cutpoints in J ( f );(2) f at most one non-repelling cycle not equal to 0, and all itspoints have multiplier 1.The set CU λ is a centerpiece, literally and figuratively, of the λ -slice C λ of the cubic connectedness locus. A big role in studying polynomialsfrom C λ is played by studying properties of the quadratic polynomial z + λz whose fixed point 0 has multiplier λ . Aiming at most generalresults, we consider the general case of λ with irrational argumentwithout any extra-conditions. To state some theorems proven earlierwe need new notions. For a closed subset A ⊂ S of at least 3 points,call its convex hull CH( A ) a gap . Given a chord ℓ = ab of the unit circlewith endpoints a and b , set σ ( ℓ ) = σ ( a ) σ ( b ) (we abuse the notationand identify the angle-tripling map σ : R / Z with the map z : S → S ;similarly we treat the map σ ). For a closed set A ⊂ S , call eachcomplementary arc of A a hole of A . Given a compactum A ⊂ C let the topological hull Th( A ) be the complement to the unboundedcomplementary domain of A . sss:family Family of invariant quadratic gaps.
Let us discuss properties of quadratic σ -invariant gaps [BOPT16]. For our purposes it suffices toconsider gaps G such that G ∩ S has no isolated points. “Invariant”means that an edge of a gap G maps to an edge of G , or to a pointin G ∩ S ; “quadratic” means that after collapsing holes of G the map σ | Bd( G ) induces a locally strictly monotone two-to-one map of the unitcircle to itself that preserves orientation and has no critical points.For convenience, normalize the length of the circle so that it equals 1.Let V be a quadratic σ -invariant gap with no isolated points. Thenthere is a unique arc I V (called the major hole of V ) complementaryto V ∩ S whose length is greater than or equal to 1 /
3; the length ofthis arc is at most 1 /
2. The edge M V of V connecting the endpointsof I V is called the major of V . If M V is critical then itself and V aresaid to be of regular critical type ; if M V is periodic then itself and V are said to be of periodic type . Collapsing edges of V to points, we construct a monotone map τ : V → S that semiconjugates σ | Bd( V ) and σ : S → S .The map τ is uniquely defined by the fact that it is monotone andsemiconjugates σ | Bd( V ) and σ : S → S . Indeed, if there had beenanother map τ ′ like that then there would have existed a non-trivialorientation preserving homeomorphism of the circle to itself conjugat-ing σ with itself. However it is easy to see that the only such map isthe identity (recall, that σ has a unique fixed point).The family of all invariant quadratic gaps can be parameterized (see[BOPT16]). Namely, by Lemmas 3.22 and 3.23 of [BOPT16], thereexists a Cantor set Q ⊂ S such that if we collapse every hole of Q toa point, we obtain a topological circle whose points are in one-to-onecorrespondence with all quadratic invariant gaps U such that U ∩ S isa Cantor set. Moreover, the following holds:(1) for each point θ ∈ S of Q that is not an endpoint of a hole of Q ,the critical chord ( θ + 1 / θ + 2 /
3) is the major of a quadraticinvariant gap U such that U ∩ S is a Cantor set;(2) for each hole ( θ , θ ) of Q the chord ( θ + 1 / θ + 2 /
3) is theperiodic major of a quadratic invariant gap U such that U ∩ S is a Cantor set.Thus, to choose the tag of a quadratic invariant gap V we first takeits major M V and then choose the edge or vertex ℓ of V distinct from M V but with the same image as M V . Evidently, (1) ℓ is an edge of V if M V is not critical (and is, therefore, of periodic type), (2) ℓ is a vertexof V if M V is critical (and is, therefore, of regular critical type).The convex hull Q of Q in the plane is called the Principal QuadraticParameter Gap (see Figure 1). The set Q plays a somewhat similar roleto that of the following set appearing in quadratic dynamics: the set ofarguments of all parameter rays (rays to the Mandelbrot set) landingat points of the main cardioid. Holes of Q will play an important role.The period of a hole ( θ , θ ) of Q is defined as the period of θ + 1 / θ + 2 / Q are (1 / , /
3) and (2 / , / sss:la-slice Properties of the λ -slice. For every polynomial f ∈ F λ and everyangle α ∈ R / Z , we will define the dynamic ray R f ( α ). Also, for everyangle θ , in the parameter plane of F λ we define the parameter ray R λ ( θ ). We use rays to show that the picture in F λ resembles thepicture in the parameter plane of quadratic polynomials. MMEDIATE RENORMALIZATION 11
13 1623 56 1112 23245121124 1241121324 712
Figure 1.
The Principal Quadratic Parameter Gap fig:Qft:main-bot16
Theorem 4.8 (Main Theorem of [BOT16]) . Fix λ with | λ | . Theset CU λ is a full continuum. The set C λ is the union of CU λ and acountable family of limbs LI H of C λ parameterized by holes H of Q .The union is disjoint. For a hole H = ( θ , θ ) of Q , the following holds. (1) The parameter rays R λ ( θ ) and R λ ( θ ) land at the same point f root ( H ) . (2) Let W λ ( H ) be the component of C \ R λ ( θ ) ∪ R λ ( θ ) containingthe parameter rays with arguments from H . Then, for every f ∈ W λ ( H ) , the dynamic rays R f ( θ + 1 / , R f ( θ + 2 / landat the same point, either a periodic and repelling point for all f ∈ W λ ( H ) , or the point for all f ∈ W λ ( H ) . Moreover, LI H = W λ ( H ) ∩ C λ . (3) The dynamic rays R f root ( H ) ( θ + 1 / , R f root ( H ) ( θ + 2 / land atthe same parabolic periodic point, and f root ( H ) belongs to CU λ . Figure 2 shows the parameter slice F e πi/ in which several parameterrays and several wakes are shown.Given a compact set A ⊂ C , let the topological hull of A be theunbounded complementary domain of A . t:cubio Theorem 4.9 ([BOT16]) . We have that
Th( P λ ) ⊂ CU λ . The set CU λ is a full continuum. In this paper we study the set CU λ \ Th( P λ ). By Theorems 4.8 and4.9, except for vertices f root ( H ) of parameter wakes W λ ( H ), no points of CU λ belong to those parameter wakes. The only places at which pointsof CU λ \ Th( P λ ) may be located can be associated with parameter rays Figure 2.
Parameter slice F e πi/ with some parameter rays fig:Fc1d3 R λ ( θ ) where θ ∈ Q is a parameter, associated with a regular criticalquadratic invariant gap U with regular critical major M U .5. Decorations
The notation below will be used in what follows. Namely, we assumethat f is an immediately renormalizable cubic polynomial; recall thatby Lemma 2.3 its filled Julia set K ∗ is unique. Let f : U ∗ → V ∗ be aquadratic-like map where V ∗ is very tight around K ∗ . Let ω be thecritical point of f belonging to K ∗ ; let ω be the other critical pointof f (this notation will be used in what follows) . In order to indicatethe dependence on f , we may write K ∗ ( f ), ω ( f ), etc. Observe thatin this section we put no restrictions on the rational lamination of f .A pullback of a connected set D under f is defined as a connectedcomponent of f − ( D ). l:pull Lemma 5.1.
A pullback of a connected set A under f maps onto A .In particular, there exists a pullback e K ∗ of K ∗ disjoint from K ∗ andmapped by f onto K ∗ in the one-to-one fashion.Proof. The first claim of the lemma is proven in Lemma 4.1 of [LT90].Moreover, since K ∗ is a quadratic-like Julia set, it has a pullback e K ∗ disjoint from itself. By the first claim, e K ∗ is a continuum that maps onto K ∗ . Moreover, f | e K ∗ is one-to-one. Indeed, otherwise there are MMEDIATE RENORMALIZATION 13 points x = y ∈ e K ∗ with f ( x ) = f ( y ) = z ∈ K ∗ . It follows that z hasoverall four preimages (two in K ∗ and two in e K ∗ ), a contradiction. (cid:3) The notation e K ∗ will be used from now on. Also, from now on bydecorations we mean those of K rel. K ∗ . Definition 5.2.
A decoration is said to be critical if it contains e K ∗ .Thus there is only one critical decoration denoted D c . All other deco-rations are said to be non-critical .Let v = f ( ω ) be the critical value associated with the point ω . l:v-nin-k Lemma 5.3.
Neither ω nor v belong to K ∗ .Proof. We have ω / ∈ K ∗ since ω ∈ K ∗ and K ∗ contains at most onecritical point. If v ∈ K ∗ , then there are two preimages of v in K ∗ andtwo preimages of v outside of K ∗ (both numbers take multiplicitiesinto account). This contradicts f being three-to-one. (cid:3) For x / ∈ K ∗ let D ( x ) be the decoration containing x ; set D v = D ( v )and call it critical value decoration . Initial dynamical properties ofdecorations are listed in Theorem 5.4. Set L to be { ω } (if ω ∈ J ( f )),or the closure of the Fatou domain of f containing ω (if any). InTheorem 5.4 we use the following E-construction , and we use the samenotation whenever we implement it.
The E-construction.
Draw a K -ray E landing at a periodic repellingpoint x = f ( ω ) , x ∈ K ∗ . Construct the two pullback K -rays E ′ and E ′′ of E landing at distinct points x ′ , x ′′ ∈ K ∗ where x ′ = x ′′ by thechoice of x . The set C \ ( K ∗ ∪ E ′ ∪ E ′′ ) consists of components Z and Z . Assume that Z contains all K -rays with arguments from anopen arc I of length 1 / Z contains all K -rays with argumentsfrom the open arc I of length 2 / I . Notice that since x ′ = x ′′ then some periodic repelling points of K ∗ belong to Z and areaccessible by K -rays from within Z (the same claim holds for Z ). t:decor Theorem 5.4.
The critical decoration D c maps onto the entire K ( f ) while any other decoration maps onto a decoration in the one-to-onefashion. Any decoration D = D v has three homeomorphic pullbacks: D such that D \ D ⊂ e K ∗ and D , D that are decorations. D i it-self is a decoration for i = 2 , . Decoration D v has a homeomorphicpullbacks D ′ v which is itself a decoration, and a pullback T that mapsonto D v in the two-to-one fashion, contains ω , is contained in D c , andaccumulates to both K ∗ and e K ∗ .Proof. We prove Theorem 5.4 step by step.
Step 1. If D is a decoration of f , then every pullback of D is a subsetof some decoration of f . Moreover, if D ′ and D ′′ are decorations and f ( D ′ ) ∩ D ′′ = ∅ , then f ( D ′ ) ⊃ D ′′ .Proof of Step 1. Let Γ be a pullback of D . Clearly, Γ ⊂ K \ K ∗ . SinceΓ is connected, it must lie in some decoration. Now, if D ′ and D ′′ aredecorations and f ( D ′ ) ∩ D ′′ = ∅ then we can choose a pullback D ′′′ of D ′′ which is non-disjoint from D ′ . By the above D ′′′ ⊂ D ′ ; by Lemma5.1, f ( D ′′′ ) = D ′′ . Hence f ( D ′ ) ⊃ D ′′ as desired. Step 2. If D is a non-critical decoration then f ( D ) is a decoration. Proof of Step 2. The set f ( D ) is connected and disjoint from K ∗ (bydefinition of a non-critical decoration). Hence it is contained in onedecoration. By Step 1, the set f ( D ) coincides with this decoration. Step 3. ω ∈ D c .Proof of Step 3. Let ω / ∈ D c . Since by Lemma 3.2, A = D c ∪ K ∗ iscompact, a neighborhood of ω is disjoint from A . It follows that theset L is contained in a decoration D = D c . Choose a neighborhood U of f ( L ) so that a pullback W of U with ω ∈ W is disjoint from A .Then W is a neighborhood of L that maps two-to-one onto U .Choose a K -ray R that enters U and denote the first (coming frominfinity) point of intersection of R and Bd( U ) by x . Denote the unionof the segment of R from infinity to x and a curve I ′ ⊂ U from x to v by R ′ . The pullback R ′′ of R ′ containing ω consists of two segments oftwo K -rays each of which maps to the segment of R from infinity to x ,and an arc I ′′ that double-covers I ′ . Clearly, R ′′ partitions the plane intwo half-planes on one of which f acts in the one-to-one fashion whileon the other one it acts in the two-to-one fashion. However by theconstruction R ′′ is disjoint from A and points of K ∗ ⊂ A have threepreimages in A , a contradiction. Thus, ω ∈ D c . Step 4.
We have ω ∈ Z , and, hence, D c ⊂ Z . The map f is ahomeomorphism on Z . If D is a decoration then Z contains a uniquehomeomorphic pullback D ′ of D which is itself a non-critical decoration.No decoration has a unique pullback. Proof of Step 4. Both Z and Z contain points from K \ K ∗ ; in fact,their union contains the entire K \ K ∗ . Hence, all the decorations arecontained in Z ∪ Z . Notice though, that the K -rays approachingpoints of K \ K ∗ have arguments from the disjoint open arcs of anglesat infinity, namely, from I and I , respectively. Since all K -rays in Z have arguments from the arc I , then they all have distinct images.We claim that ω / ∈ Z . Indeed, suppose that ω ∈ Z . By Step 3 then D c ⊂ Z and so e K ∗ ⊂ Z . Choose a repelling periodic point y ∈ K ∗ accessible by a K -ray R to K ∗ from within Z . Then choose the first MMEDIATE RENORMALIZATION 15 preimage y ′ ∈ e K ∗ of f ( y ). Clearly, y ′ is also accessible by a K -ray R ′ which itself is a pullback of f ( R ). However both rays must havearguments from I , a contradiction (recall that I is an arc of length1 / ω ∈ Z and, hence, D c ⊂ Z .Let us show that f | Z is a homeomorphism. It is a homeomorphismon the union of all K -rays with arguments from I . Suppose thatpoints x, y ∈ Z are such that f ( x ) = f ( y ) = z . We may assume that z / ∈ E ∪ { v } . Since e K ∗ ⊂ D c ⊂ Z , then z / ∈ K ∗ . Hence we canconstruct a ray H from z to infinity bypassing K ∗ ∪ v ∪ E . Considerpullbacks H x and H y of H such that x ∈ H x and y ∈ H y . By thechoices we made, H x ⊂ Z , H y ⊂ Z , and H x ∩ H y = ∅ . It follows thatthere exist distinct points x ′ ∈ H x and y ′ ∈ H y with the same imageand such that both x ′ and y ′ belong to C \ K . However then x ′ and y ′ must belong to K -rays with arguments from I , a contradiction. Hence f | Z is one-to-one. By Brouwer’s Invariance of Domain Theorem, f | Z is a homeomorphism onto f ( Z ).Let D be a decoration. Choose y ∈ D ∩ J ( f ) and a sequence y i ∈ C \ K of points that converge to y and belong to K -rays R i = E .Choose K -rays R ′ i ∈ Z with f ( R ′ i ) = R i and then points y ′ i ∈ R ′ i suchthat f ( y ′ i ) = y i . Then y ′ i → y ′ where f ( y ′ ) = y , and hence y ′ ∈ D ′ where D ′ ⊂ Z is a decoration. The rest of the claim follows. Step 5. If D = D v is a decoration then it has three pullbacks each ofwhich maps onto D homeomorphically. Two of the pullbacks accumu-late into K ∗ ; one pullback accumulates into e K ∗ . Proof of Step 5. By Lemma 3.2 the set K ∗ ∪ D is a full continuum;clearly, v / ∈ K ∗ ∪ D . Apply the E-construction to D . Then draw a ray R from v to infinity so that R ∩ ( K ∗ ∪ D ∪ E ) = ∅ . Construct thepullback C of R passing through ω ; the set C cuts the plane in twohalf-planes, X and Y , such that X maps onto C \ R in the one-to-onefashion while Y maps onto C \ R in the two-to-one fashion. The cut C is disjoint from K ∗ ∪ e K ∗ as well as from the pullbacks of D . Hence C separates K ∗ and e K ∗ ; it is easy to see that e K ∗ ⊂ X and K ∗ ⊂ Y .It follows that there exists a unique pullback D of D contained in X . Since it has to accumulate to points mapped to K ∗ , the set D accumulates into e K ∗ . Hence D ⊂ D c . Since D c ⊂ Z , then D ⊂ Z .In addition to D , by Lemma 5.1 there may be either one pullbackof D mapped onto D in the two-to-one fashion, or two pullbacks of D mapped onto D in the one-to-one fashion. By Step 4, D has ahomeomorphic pullback D ⊂ Z which is a non-critical decoration.Clearly, D = D . Hence D has three homeomorphic pullbacks ofwhich D accumulates to e K ∗ and D accumulates to K ∗ . Let D be the remaining pullback of D . If it accumulated to e K ∗ it would followthat f | e K ∗ is not one-to-one, a contradiction. Hence S accumulates to K ∗ . Step 6.
The decoration D v has two pullbacks. One of them, say, T ,maps onto D v in the two-to-one fashion, contains ω , is contained in D c , and accumulates to both K ∗ and e K ∗ ; the other one is the homeo-morphic pullback D ′ v defined in Step 4. Proof of Step 6. Clearly, D v cannot have three pullbacks as otherwisethe point v will have four preimages (counted with multiplicity), acontradiction. By Step 4, D v has a homeomorphic pullback D ′ v ⊂ Z .Thus, we only need to study the remaining two-to-one pullback T of D v . Clearly, ω ∈ T which implies that T ⊂ D c . To prove that T accumulates in both K ∗ and e K ∗ , notice that there are points of T closeto e K ∗ . Indeed, a neighborhood of e K ∗ maps homeomorphically ontoa neighborhood of K ∗ and therefore contains points of the preimageof D v . These points cannot belong to D ′ v because D ′ v \ D ′ v ⊂ K ∗ byLemma 3.2. Hence they belong to T as claimed. Thus, T accumulatesinto e K ∗ . To show that T accumulates into K ∗ too, choose a sequence ofpoints y i ∈ D v such that y i → y ∈ K ∗ . For each i , let y ′ i , y ′′ i ∈ T be twodistinct preimages of y i in T . We may assume that they converge to y ′ , y ′′ respectively. If y ′ , y ′′ / ∈ K ∗ then y ′ , y ′′ ∈ e K ∗ which implies that inany neighborhood of e K ∗ there are pairs of points with the same image,a contradiction. Hence T accumulates into both K ∗ and e K ∗ . (cid:3) Quadratic arguments s:quadarg
Consider K ∗ -rays R ∗ ( α ). Clearly, f ( R ∗ ( α )) ⊃ R ∗ (2 α ) (the curve f ( R ∗ ( α )) extends the ray R ∗ (2 α ) into the annulus between the basicneighborhood of K ∗ and its image). A crosscut (of K ∗ ) is a closed arc I with endpoints x, y ∈ K ∗ such that [ I \ { a, b } ] ⊂ C \ K ∗ . If a n isa crosscut then the shadow Shad( a n ) of a crosscut a n is the boundedcomplementary component of a n ∪ K ∗ . A sequence of crosscuts a n , n =1 , , . . . is fundamental if a n +1 ⊂ Shad( a n ) for every n and the diameterof a n converges to 0 as n → ∞ . Two fundamental sequences of crosscutsare equivalent if crosscuts of one sequence are eventually contained inthe shadows of crosscuts of the other one, and vice versa. This isan equivalence relation whose classes are called prime ends . In whatfollows the set of endpoints of a closed arc I is denoted by end( I ).By the Carath´eodory theory, every quadratic-like ray R ∗ ( α ) to K ∗ corresponds to a certain prime end E ∗ ( α ) represented by a fundamentalsequence of crosscuts { a n } . Moreover, for every a n a tail of R ∗ ( α ) iscontained in Shad( a n ) (a tail of R ∗ ( α ) is defined by a point x ∈ R ∗ ( α ) MMEDIATE RENORMALIZATION 17 and coincides with the component of R ∗ ( α ) \ { x } that accumulates into K ∗ ). It is convenient to consider also the associated picture in C \ D ;the picture on the K ∗ -plane is transferred to that in C \ D by meansof the map ψ ∗ introduced earlier.Namely, for every n , the set ψ ∗ ( a n \ end( a n )) is an arc I n ⊂ C \ D without endpoints such that I is a closed ark with endpoints x n , y n ∈ S . One can choose the circle arc I ′ n positively oriented from x ′ n to y ′ n such that α ∈ I ′ n and consider the Jordan curve Q n = I n ∪ I ′ n ; thenthe radial ray R α with the initial point at z α ∈ S where z α ∈ S isthe point of the circle with argument α , intersected with the simplyconnected domain U a n with boundary Q n , contains a small subsegmentof R α with an endpoint z α . Observe that ( ψ ∗ ) − ( U a n ) is the shadowof the crosscut a n . The impression of E ∗ ( α ) is the intersection of theclosures of Shad( a n ). We say that a prime end E ∗ ( α ) is disjoint from aset S ⊂ C \ K ∗ if Shad( a n ) ∩ S = ∅ for all sufficiently large n . Riemannmap of C \ K ∗ . In what follows, when talking about crosscuts, we willuse this notation. l:cross-primends Lemma 6.1.
Suppose that X ⊂ C \ K ∗ is a connected set, and ψ ∗ ( X ) accumulates on exactly one point z α ∈ S with argument α . Then X isnon-disjoint from E ∗ ( α ) and disjoint from any other prime end.Proof. Let a be a crosscut associated with E ∗ ( α ). Consider the set U a .Since ψ ∗ ( X ) accumulates on z α , then ψ ∗ ( X ) is non-disjoint from U , andhence X is non-disjoint from Shad( a ). By definition, X is non-disjointfrom E ∗ ( α ). Also, for any point t = e πβi = z α we can find a crosscut b associated with β and so small that U b is disjoint from ψ ∗ ( X ). Then X is disjoint from Shad( b ) and hence X is disjoint from E ∗ ( β ). (cid:3) Recall that a set A accumulates in B if A \ A ⊂ B . l:accum Lemma 6.2.
Suppose that a connected subset b D ⊂ C \ D accumulateson a non-degenerate arc A $ S . Let b R be an arc in C \ D landing atan interior point y of A . Then b R crosses b D .Proof. By way of contradiction assume that b R ∩ b D = ∅ . Extend b R toinfinity still avoiding b D . Since A = S , then there exists a ray b R froma point x ∈ S \ A to infinity that is disjoint from b D . It follows that b D is contained in a component of C \ ( b R ∪ b R ∪ I ) where I is one ofthe two circle arcs with endpoints x and y . However then b D can onlyaccumulate on the part of A that is contained in I , a contradiction. (cid:3) Proposition 6.3 uses Lemma 6.2. p:pends
Proposition 6.3.
Every decoration D is disjoint from all prime endsof K ∗ except exactly one.Proof. The (connected) set D ′ = ψ ∗ ( D ) accumulates to a closed arc A ⊂ S . Suppose that A is non-degenerate, and bring this to a con-tradiction. First, for every repelling periodic point x ∈ K there existsan K -ray landing at x . Also, K -rays are disjoint from D . Choose twodistinct repelling periodic points in K ∗ , draw K -rays landing at them,and then map all this by ψ ∗ to C \ D . This will result in two curveslanding at two distinct points of S and disjoint from D ′ . Thus, D ′ does not accumulate onto the entire S .Now, choose a periodic K -ray R that lands at a repelling periodicpoint w ∈ K ∗ . Under the map ψ ∗ it is associated to a curve ψ ∗ ( R ) thatlands at a certain point x ∈ S and is otherwise disjoint from D . Pullingback R under f following backward orbit of w in K ∗ corresponds topulling back ψ ∗ ( R ) under z . Since A is non-degenerate, there exists anumber N such that some N -th pullback of R lands at a point w ′ ∈ K ∗ while the corresponding N -th pullback of ψ ∗ ( R ) is a curve Q in C \ D that lands at an interior point of A . Since Q is disjoint from D ′ , wesee by Lemma 6.2 that A = { α } is degenerate. By Lemma 6.1, D isdisjoint from all prime ends of K ∗ except for E ∗ ( α ). (cid:3) Suppose that E ∗ ( α ) is the only prime end non-disjoint from D . Then α is called the quadratic argument of D and is denoted by α ( D ). ByProposition 6.3 each decoration has only one quadratic argument (andso quadratic arguments are well defined) while different decorationsmay a priori have the same quadratic argument. A useful interpretationof these concepts is as follows. Using the map ψ ∗ for C \ K ∗ we cantransfer all decorations to the set C \ D ; then for any decoration D the set ψ ∗ ( D ) accumulates to the point e πiα ( D ) of the unit circle withargument α ( D ) (i.e., ψ ∗ ( D ) \ ψ ∗ ( D ) consists of one point from the unitcircle with argument α ( D )). Moreover, if U ∗ is a basic neighborhood of K ∗ then ψ ∗ conjugates z restricted onto ψ ∗ ( U ∗ \ K ∗ ) and f restrictedonto U ∗ \ K ∗ .The dynamics on the uniformization plane immediately implies thenext lemma stated here without proof. l:z2 Lemma 6.4. If D = D c is a decoration then α ( f ( D )) = σ ( α ( D )) . Onthe other hand, in the notation of Theorem . , we have that α ( D v ) = σ ( α ( D c )) . In what follows we use Riemann maps and their inverses for eitherthe entire filled Julia set K of f , or a quadratic-like Julia set K ∗ . In theformer case we talk about K -plane and z -plane (which are associated MMEDIATE RENORMALIZATION 19 to each other under the appropriate Riemann map), in the latter case,similarly, we talk about K ∗ -plane and z -plane.Corollary 6.5 follows from Proposition 6.3. c:2pbdec Corollary 6.5.
Consider a decoration D with quadratic argument α .Then both α/ and ( α + 1) / are quadratic arguments of decorationscontaining pullbacks of D . These two decorations are different. Cubic arguments s:cubarg
So far we have been working on establishing general facts concerningthe situation when a cubic polynomial f has a connected quadratic-like filled Julia set K ∗ (clearly, K ∗ ⊂ K ( f )). In this section we beginlooking into more specific cases; as the first step we describe someresults and concepts, mostly taken from [BOT16].Consider an immediately renormalizable polynomial f ∈ F λ with | λ |
1, and define an invariant quadratic gap U ( f ) associated with 0;as before, by f ∗ : U ∗ → V ∗ , we denote the corresponding quadratic-likemap. When J ( f ) is disconnected, gaps analogous to U ( f ) were studiedin [BCLOS16] where tools developed in [LP96] were used; howeverthis approach is based upon the fact that J ( f ) is disconnected in anessential way and, hence, is very different from that used in [BOT16]and here. Once we introduce U ( f ), we shall see that it coincides withthe similar gap in the disconnected case. Recall that by Lemma 2.3 if f is a cubic polynomial with a non-repelling fixed point a , then thereexists at most one quadratic-like filled invariant Julia set K ∗ containing a ; by Corollary 2.5 if f has empty rational lamination then it has aunique non-repelling fixed point a and at most one quadratic-like filledinvariant Julia set that, if it exists, must contain a .To define the gap U ( f ) associated with K ∗ , we use (pre)periodicpoints of f . Since K ∗ is a quadratic-like filled Julia set, then K ∗ is acomponent of f − ( K ∗ ). d:uf Definition 7.1.
Let b X ( f ) = b X be the set of all σ -(pre)periodic points α ∈ S such that R f ( α ) lands in K ∗ . Let X ( f ) = X be the closureof b X . Let U ( f ) be the convex hull of X . Let e K ∗ be the componentof f − ( K ∗ ) different from K ∗ (such a component of f − ( K ∗ ) existsbecause f | K ∗ is two-to-one). Let Y ( f ) = Y be the closure of the set ofall preperiodic points α ∈ S with R f ( α ) landing in e K ∗ . Observe, that e K ∗ is disjoint from U ∗ (otherwise points of f ( e K ∗ ∩ U ∗ ) must belong to K ∗ , a contradiction with dynamics of points of e K ∗ ∩ U ∗ ). From now on we fix an immediately renormalizable polynomial f ∈F nr and do not refer to f in our notation (we write U instead of U ( f )etc). Lemma 7.2 summarizes some results of Section 7 of [BOT16]. l:inv-quad Lemma 7.2.
The set e K ∗ is disjoint from U ∗ . The gap U is an invari-ant quadratic gap of regular critical or periodic type. The map σ | b X istwo-to-one, and Y lies in the closure of the major hole of X . This shows that the results of [BOPT16] and [BOT16] apply to U (recall that these results are described in Subsection 4.2). E.g., con-sider the map τ defined there for any quadratic invariant gap of σ ( τ collapses edges of U and semiconjugates σ | Bd( U ) and σ ). The map ψ ∗ maps K -rays to their counterparts on the z -plane. On the otherhand, the Riemann map defined by K sends the radial rays with ra-tional arguments in C \ D , to K -rays, including K -rays landing in K ∗ .Composing these two maps we obtain a map η that associates radialrays from the z -plane with arguments from b X to ψ ∗ -images of K -rayson the z -plane landing in S . Thus, if R ( β ) is a radial ray on z -planeand β ∈ b X , then the ray η ( R ( β )) is contained in z -plane and landsat a point of S ; this defines a map τ ′ : b X → S . By construction wesee that τ ′ semiconjugates σ | b X and σ ; the uniqueness of the map τ shows that τ ′ is the restriction of τ onto b X .Observe that K -rays with arguments from Bd( U ) do not necessarilyhave principal sets contained in K ∗ . Nevertheless the map τ allows usto relate decorations of K ∗ and their quadratic arguments with edgesand vertices of the gap U . l:cridi3 Lemma 7.3.
The quadratic argument of D c is τ ( M U ) .Proof. If the quadratic argument of D c is not τ ( M U ), then there is anedge/vertex v of U such that τ ( v ) is the quadratic argument of D c andwe can find angles α, β ∈ b X such that the arc I = ( α, β ) contains v but does not contain the endpoints of M U . For K -rays R ( α ) , R ( β ) witharguments α, β , consider the component W of C \ [ R ( α ) ∪ R ( β ) ∪ K ∗ ]containing K -rays with arguments from I . By Lemma 7.2 Y lies inthe closure of the major hole of X . Hence e K ( f ∗ ) is disjoint from W despite the fact that e K ( f ∗ ) ⊂ D c ⊂ W . (cid:3) In the end of this section we study the issue of landing at points of K ∗ for periodic and preperiodic angles from U . l:who-land Lemma 7.4.
Let α ∈ U be a (pre)periodic angle that never maps toan endpoint of the major M U of U . Suppose that the K -ray R ( α ) withargument α lands at a point x . Then x ∈ K ∗ . MMEDIATE RENORMALIZATION 21
In our notation the claim of the lemma simply means that α ∈ b X . Proof.
Let α = 0 ∈ U , yet x / ∈ K ∗ . Let y ∈ K ∗ be the fixed point atwhich R ∗ (0) lands. Then the only K -ray that can land at y is R ( )which implies that M U = 0 , a contradiction with the assumptions ofthe lemma. Similarly, if α = , then x ∈ K ∗ .We claim that the lemma holds for a (pre)periodic angle α ∈ U ifit holds for β = σ ( α ). By way of contradiction assume that x / ∈ K ∗ ; then x ∈ e K ∗ , α ∈ Y and, by Lemma 7.2, Y is contained in theclosure of the major hole of U . It follows that α is an endpoint of M ( U ), a contradiction with the assumptions of the lemma. By thefirst paragraph we conclude that the lemma holds if α eventually mapsto a σ -fixed angle.We claim that if β ∈ U ∩ b X and σ ( α ) = β then α ∈ b X . Indeed, byLemma 7.2 σ | b X is two-to-one. If α / ∈ b X then all three σ -preimages of β belong to U . However this is only possible if two of them are endpointsof M U . It follows that α is an endpoint of M U , again a contradiction.This and the first paragraph of the proof imply the claim of the lemmaif α eventually maps to a σ -fixed angle.Assume now that α never maps to a σ -fixed angle. The angle τ ( α )is (pre)periodic under σ and never maps to 0 under iterations of σ .The K ∗ -ray R ∗ ( τ ( α )) with quadratic argument τ ( α ) lands at a point x ′ ∈ K ∗ . Let a K -ray R ( γ ) land at x ′ . We claim that α = γ . Indeed, α ∈ U by the assumptions, and γ ∈ U since R ( γ ) lands at x ′ ∈ K ∗ .Moreover, neither α nor γ ever map to 0 or . It now follows from theconstruction that both angles (recall that α and γ belong to U ) behavein the same fashion with respect to the partition of U in two arcs by the σ point that belongs to U , and its preimage in U (or, in the appropriateexceptional cases, by the major M ( U ) = 0 and its preimage-edge in U ). Together with the assumptions of the lemma that α never mapsto an endpoint of the major M U this implies that α = γ and hence thelanding point x of R ( α ) belongs to K ∗ as desired. (cid:3) Sectors
Suppose that f is an immediately renormalizable cubic polynomial.Define a few objects depending on f and denote them with f as the sub-script; yet, if f is fixed, we may omit it from notation. Consider a pair ofexternal rays R ( α ), R ( β ) landing in K ∗ . The set Σ f = K ∗ ∪ R ( α ) ∪ R ( β )divides the plane into two components, one of which contains all exter-nal rays with arguments in ( α, β ) and the other contains all externalrays with arguments in ( β, α ). To formally justify this claim, collapse K ∗ to a point (i.e., consider the equivalence relation ∼ on C , whoseclasses are K ∗ and single points in C \ K ∗ ). By Moore’s theorem, thequotient space C / ∼ is homeomorphic to the sphere. The image ofΣ f ( α, β ) under the quotient map, together with the image of the pointat infinity, form a Jordan curve. The statement now follows from theJordan curve theorem. Let S ◦ ( α, β ) f be the component of C \ Σ f ( α, β )containing all external rays with arguments in ( α, β ). Observe that S ◦ ( α, β ) is defined only if the rays R ( α ), R ( β ) both land in K ∗ . Thesets S ◦ ( α, β ) will be called open sectors , and the sets Σ( α, β ) will becalled cuts . Images of sectors contain K ∗ iff sectors contain e K ∗ .An open sector S ◦ ( α, β ) is associated with its argument arc ( α, β ) ⊂ R / Z that consists of arguments of all rays included in S ◦ ( α, β ). Note,that this sector does not have to coincide with the union of those raysas open sectors may contain decorations. More generally, consider asubset T ⊂ C . We call the set T f ( f ) -radial if any ray intersecting T lies in T . For a radial set T we can define the argument set arg( T ) of T as the set of all γ ∈ R / Z with R ( γ ) ⊂ T . Every open sector is aradial set, whose argument set is an open arc. It is clear that, for anyradial set T , we havearg( f ( T )) = σ (arg( T )) , arg( f − ( T )) = σ − (arg( T )) . The following properties of open sectors are almost immediate. l:pb-opsec
Lemma 8.1.
Let S ◦ be an open sector and T ◦ an f -pullback of S ◦ .Then arg( T ◦ ) is the union of one or several components of σ − (arg( S ◦ )) .The number of critical points in T ◦ equals the number of componentsminus one. If ω / ∈ T ◦ and the closure of T ◦ intersects K ∗ , then T ◦ is an open sector mapping 1-1 onto S ◦ . Any pullback of S ◦ is disjointfrom K ∗ .Proof. The first claim (arg( T ◦ ) is a union components of σ − (arg( S ◦ )))is immediate. Note that f : T ◦ → S ◦ is proper. Therefore, this maphas a well-defined degree. The degree is clearly equal to the number ofcomponents in arg( T ◦ ). On the other hand, by the Riemann–Hurwitzformula, the degree is equal to the number of critical points in T ◦ plusone. Thus the second claim follows.Let us prove the third claim of the lemma. The only critical pointthat can lie in T ◦ is ω . Since we assume that ω / ∈ T ◦ , then arg( T ◦ )has only one component. Let arg( T ◦ ) = ( α, β ). Then T ◦ is boundedby R ( α ) ∪ R ( β ) and a part of K ∗ ∪ e K ∗ . If both R ( α ) and R ( β ) landin K ∗ , then, by definition, T ◦ coincides with the open sector S ◦ ( α, β ).If both R ( α ) and R ( β ) land in e K ∗ , then T ◦ is disjoint from K ∗ as itsimage S ◦ does not contain K ∗ ; this implies that T ◦ is disjoint with MMEDIATE RENORMALIZATION 23 K ∗ , a contradiction with our assumptions. Finally, if one of the tworays R ( α ), R ( β ) lands in K ∗ and the other lands in e K ∗ , then we get acontradiction too. Suppose, say, that R ( α ) lands in K ∗ and R ( β ) landsin e K ∗ . The sets K ∗ ∪ R ( α ) and e K ∗ ∪ R ( β ) are closed disjoint non-separating sets, whose union cannot separate the plane. Thus the onlypossibility is that T ◦ = S ◦ ( α, β ) and f : T ◦ → S ◦ is a homeomorphism.The last claim of the lemma now easily follows. (cid:3) l:img-sec Lemma 8.2.
Consider an open sector S ◦ ( α, β ) , whose argument arc ismapped one-to-one under σ . Then f ( S ◦ ( α, β )) = S ◦ (3 α, β ) . More-over, S ◦ ( α, β ) maps one-to-one onto S ◦ (3 α, β ) .Proof. Let T ◦ be the f -pullback of S ◦ (3 α, β ) that includes rays witharguments in ( α, β ). Clearly, the rays R ( α ), R ( β ) are on the boundaryof T ◦ . Since these rays land in K ∗ and T ◦ ∩ K ∗ = ∅ , we must have T ◦ ⊂ S ◦ ( α, β ). On the other hand, if T ◦ = S ◦ ( α, β ), then, by Lemma8.1, the arguments of rays in T ◦ form two intervals of R / Z rather thanone. A contradiction. Therefore, T ◦ = S ◦ ( α, β ) as desired. (cid:3) Definition 8.3 (minimal sectors) . Let x be a point outside of K ∗ . The minimal sector S ( x ) of x is defined as the intersection of all S ◦ ( α, β )such that x ∈ S ◦ ( α, β ).Note that, by definition, a minimal sector is always a closed set.It is clear from the definition that a minimal sector is bounded byat most two external rays and a piece of K ∗ . It may coincide withthe union of a single ray and its impression. The next lemma showsthat minimal sectors are related to decorations and immediately followsfrom the definitions. Recall that if x / ∈ K ∗ then D ( x ) is the decorationcontaining x . Recall also that the map τ : U → S collapses to pointsall edges of the gap U and semiconjugates σ | Bd( U ) and σ . l:dec-sec Lemma 8.4.
Let x / ∈ K ∗ . Consider the edge (possibly degenerate) ab = τ − ( α ( D ( x ))) of U . Then S ( x ) \ K ∗ is the union of all decorationswith quadratic argument α ( D ( x )) and all K -rays with arguments fromthe complementary arc of Bd( U ) in S with endpoints a and b . Define the critical sector as the minimal sector S ( ω ). l:noncr-sec Lemma 8.5.
Suppose that ω / ∈ S ( x ) . Then f ( S ( x )) = S ( f ( x )) .Proof. We first prove that f ( S ( x )) ⊂ S ( f ( x )). Indeed, S ( x ) is theintersection of all S ◦ ( α, β ) with x ∈ S ◦ ( α, β ). The f -image of the in-tersection lies in the intersection of images. Taking only those S ◦ ( α, β ),for which | β − α | < /
3, we see by Lemma 8.2 that f ( S ( x )) lies in the intersection of S ◦ (3 α, β ) ∋ f ( x ). The latter set obviously coincideswith S ( f ( x )).We now prove that S ( f ( x )) ⊂ f ( S ( x )), i.e., every point z in S ( f ( x ))has the form f ( y ) for some y ∈ S ( x ). Take an open sector S ◦ ( α, β ) ∋ x that contains only one f -preimage of z ; call this preimage y . We mayalso assume that ω / ∈ S ◦ ( α, β ) and that ( α, β ) maps one-to-one under σ . Then, by Lemma 8.2, we have f ( S ◦ ( α, β )) = S ◦ (3 α, β ). Let S ◦ beany open sector in S ◦ (3 α, β ) containing z . Then the pullback T ◦ of S ◦ in S ◦ ( α, β ) must be an open sector, and it must contain a preimageof z . The only option is that y ∈ T ◦ . Since y is contained in all such T ◦ , we have y ∈ S ( x ). (cid:3) l:bdminsec Lemma 8.6.
For any x ∈ C \ K ∗ , the rays on the boundary of S ( x ) map onto the rays on the boundary of S ( f ( x )) under f .Proof. Consider an open sector S ◦ around x . Its argument arc cov-ers one or several components of σ − (arg( S ( f ( x )))). Moreover, S ◦ can be chosen so that the endpoints of arg( S ◦ ) are arbitrarily close to σ − (arg( S ( f ( x )))). The lemma follows. (cid:3) Backward stability
In this section we study backward stability of decorations. The aim,to begin with, is to show that under certain circumstances decorationsshrink as we pull them back. Our arguments are based upon the fol-lowing theorem of Ma˜n´e. t:mane
Theorem 9.1 ([Man93]) . If f : C → C is a rational map and z ∈ C a point that does not belong to the limit set of any recurrent criticalpoint, then for some Jordan disk W around z , some C > and some < q < the spherical diameter of any component of f − n ( W ) is lessthan Cq n . Neighborhoods satisfying Theorem 9.1 are called
Ma˜ne neighbor-hoods . l:sn Lemma 9.2.
Fix q ∈ (0 , and b > . Consider a sequence of positivenumbers s n such that either s n +1 = qs n or s n +1 qs n + b . In theformer case call n the good index, and in the latter case call n the bad index. Suppose that the distance between adjacent bad indices tends toinfinity. Then s n → .Proof. It suffices to show that s n → n runs through all bad indices n < n < . . . ; fix ε > N such that q N < / q N − b < ε .Then, for i large, we have n i +1 − n i > N , and s n i +1 = q n i +1 − n i − (2 qs n i + b ) = q n i +1 − n i (2 s n i + q − b ) s n i ε. MMEDIATE RENORMALIZATION 25
Since the map h ( x ) = x/ ε has a unique attracting point 4 ε/ R , then s n i becomes eventually less than 4 ε . Since ε > s n → i → ∞ , as desired. (cid:3) Recall that we consider immediately renormalizable polynomials f and use for them the notation introduced above. Consider two rays R = R ( α ) and L = R ( β ) landing in K ∗ . Also, take any equipotential b E of P . Let ∆ = ∆( R, L, b E ) be the bounded complementary componentof K ∗ ∪ R ∪ L ∪ b E such that the external rays that penetrate into∆ have arguments that belong to the positively oriented arc from α to β . Evidently, ∆ is the intersection of S ◦ ( α, β ) with the Jordandisk enclosed by b E . Hence results of the previous section dealing withsectors apply to ∆ and similar sets. Let ∆ ′ be a pullback of ∆ suchthat ∆ ′ ∩ K ∗ = ∅ ; then say that ∆ ′ is a pullback of ∆ adjacent to K ∗ . If ∆ ′ is a P n -pullback of ∆ such that ∆ ′ ∩ K ∗ = ∅ , we call ∆ ′ an iterated pullback of ∆ adjacent to K ∗ . Let ∆ = ∆ and for every n theset ∆ n be a pullback of ∆ n − adjacent to K ∗ . Then the sequence ofsets ∆ n , n = 0 , , . . . is called a backward pullback orbit of ∆ adjacentto K ∗ . Set e U ∗ to be the pullback of U ∗ containing e K ∗ . l:Del Lemma 9.3.
Suppose that { ∆ n } is a backward pullback orbit of ∆ ad-jacent to K ∗ and n < n < . . . are all positive integers n such that ω ∈ ∆ n and ω is non-recurrent. If n i +1 − n i → ∞ , then ∆ n ⊂ U ∗ forlarge n so that the distance between ∆ n and K ∗ tends to .Proof. Consider a finite covering U of ∆ \ U ∗ by Ma˜ne neighborhoods.After adjustments we may assume that elements of U are subsets of∆. Similarly, fix a finite covering V of e U ∗ by Ma˜ne neighborhoodssuch that S V = e U ∗ . Set U = U and define U n inductively as follows.Assuming by induction that ∆ n \ U ∗ ⊂ S U n ⊂ ∆ n ∪ U ∗ , define U n +1 as the set of all open sets U satisfying one of the following:(1) there is U ′ ∈ U n such that U ⊂ ∆ n +1 is a pullback of U ′ ;(2) the point ω is in ∆ n +1 , and U ∈ V .Neighborhoods in U n +1 as in item (1) (resp., (2)) are called type (1)(resp., type (2)) neighborhoods.Any U ∈ U n is either obtained as a P k -pullback of some type (1)neighborhood in U n − k with k being maximal with this property, orcomes from V but only at the moments when ω ∈ ∆ n . In the formercase set s ( U ) = Cq k ; by the Ma˜ne theorem, diam( U ) s ( U ). In thelatter case set s ( U ) = diam( U ). Define s n = s n (∆) as the sum of s ( U )over all U ∈ U n . By the triangle inequality diam(∆ n \ U ∗ ) is bounded above by s n . Thus it suffices to show that s n → n → ∞ . Nowconsider two cases of transition from n to n + 1.(1) Assume that ω / ∈ ∆ n +1 . Then ∆ n +1 maps one-to-one to ∆ n ;moreover, no point of ∆ n +1 \ U ∗ maps into U ∗ . It follows that allneighborhoods in U n +1 are type (1). In this case we have s n +1 = qs n .(2) Assume that ω ∈ ∆ n +1 . Then there are at most twice as manytype (1) neighborhoods in U n +1 as neighborhoods in U n . Also, U n +1 includes V . Thus we have s n +1 qs n + diam( e U ∗ ).Thus, s n satisfies Lemma 9.2 with b = diam( e U ∗ ); in particular, s n → n → ∞ , and so the diameter of ∆ n \ U ∗ tends to 0. Replacing U ∗ with a smaller neighborhood of K ∗ and repeating the same argumentyields ∆ n ⊂ U ∗ for large n . Evidently, this completes the proof. (cid:3) If, in the setting of Lemma 9.3, the sequence n i +1 − n i does not tendto infinity, then it takes the same value infinitely many times. We nowconsider this case. l:bnd Lemma 9.4.
Suppose that n i +1 − n i takes the same value N infinitelymany times. Then the quadratic argument of ω is σ N -fixed and theassociated pullbacks [ α n , β n ] of [ α, β ] are shrinking segments aroundcorresponding points of the orbit of the quadratic argument of ω .Proof. Let R n and L n be the rays with quadratic arguments α n and β n landing in K ∗ and bounding ∆ n near K ∗ . Since ∆ n +1 is a P -pullbackof ∆ n , then α n = 2 α n +1 (mod 1) and β n = 2 β n +1 (mod 1), and theinterval [ α n +1 , β n +1 ] is twice shorter than [ α n , β n ]. By the assumption, ω ∈ ∆ n ∩ ∆ n + N for arbitrarily large n . Hence the intervals [ α n , β n ]contain the quadratic argument of ω . Passing to the limit, we see thatthis quadratic argument is σ N -fixed. The last claim now follows. (cid:3) Standard arguments now yield the next lemma. l:all-n
Lemma 9.5. If ω is non-recurrent and a decoration D has a quadraticargument α ( D ) which does not belong to the orbit of a periodic quadraticargument α ( D c ) , then there exists N such that any P N -pullback of D adjacent to K ∗ is contained in U ∗ .Proof. If D n is a P n -pullback of D adjacent to K ∗ with D n U ∗ , then f i ( D n ) U ∗ for every i, i n . By Lemma 9.4 any backward pull-back orbit of D adjacent to K ∗ satisfies conditions of Lemma 9.3; thus,for any such orbit D = D , D , . . . there exists the minimal n such that D n ⊂ U ∗ . By way of contradiction suppose that for any N there existsa P N -pullback of D adjacent to K ∗ and not contained in U ∗ . Thenthere are infinitely many such pullbacks. Now take the P -pullbacks of D adjacent to K ∗ (there are finitely many of them). Choose among MMEDIATE RENORMALIZATION 27 them one pullback that has P N -pullbacks adjacent to K ∗ and not con-tained in U ∗ for any N , and then apply the same construction to it.Evidently, in the end we will construct a backward pullback orbit of D adjacent to K ∗ , a contradiction with Lemma 9.3. (cid:3) All this implies Proposition 9.6. p:alldec
Proposition 9.6. If f is immediately renormalizable, and ω is non-recurrent then every decoration is eventually mapped to D c .Proof. Suppose that the quadratic argument α ( D ) = γ of D does notbelong to a periodic orbit of α ( D c ). Then by Lemma 9.5 for any U ∗ the P n -pullbacks of D will be contained in U ∗ for any n > N ( U ∗ )where N ( U ∗ ) depends on U ∗ . Consider now the union K d of K ∗ andall decorations that are eventually mapped to D c . We claim that theset K d is backward invariant. Indeed, take any decoration D ⊂ K d .Then any f -pullback of D is either a decoration in K d or a subset of D c .It follows that f − ( K d ) ⊂ K d , as desired. Also, the set K d is compactas a union of K ∗ and a sequence of sets that are closed in C \ K ∗ andaccumulate to K ∗ . Observe that if α ( D c ) is periodic, the sets in K d are decorations with periodic arguments from the σ -orbit of α ( D c ), ordecorations with non-periodic arguments-preimages of α ( D c ).Now, J ( f ) is a minimal by inclusion compact backward invariantsubset of C . (Equivalently, the backward orbit of any point from J ( f )is dense in J ( f ).) Thus, J ( f ) ⊂ K d since K d contains points of J ( f ).On the other hand, consider a bounded Fatou component Ω. If Ω K ∗ then the boundary of Ω is a subset of K d . Therefore, it is a subset ofsome decoration D ⊂ K d . It follows that Ω itself is included into D . Weshowed that K = K d which completes the proof of the proposition. (cid:3) The next corollary follows from Proposition 9.6. c:uni-dec
Corollary 9.7. If f is immediately renormalizable and ω is non-re-current then for each quadratic angle γ there is at most one decorationwith quadratic argument γ .Proof. Assume that α ( D c ) is non-periodic. If there are two distinct dec-orations D ′ and D ′′ with the same quadratic argument γ , then thereexists a unique number n such that σ ( γ ) = α ( D c ). This is the onlychance for D ′ and D ′′ to be mapped to D c . It follows from Proposition9.6 that f ( D ′ ) = f ( D ′′ ) = D c . However by Lemma 8.2 this is impos-sible (for small open sectors with argument arc containing α ( D c ) themap f restricted on them is one-to-one by Lemma 8.2). Now suppose that α ( D c ) is periodic of period n . Then, similar tothe previous paragraph, we see that if there are two distinct decora-tions D ′ and D ′′ with the same quadratic argument γ then there mustexist a decoration D ′′′ = D c with quadratic argument α ( D c ) such that f n ( D ′′′ ) = D c . Transferring this to the z -plane we see that the map z is not one-to-one in a small neighborhood of the point of the unitcircle with argument α ( D c ), a contradiction completing the proof. (cid:3) The next lemma specifies properties of the gap U . l:perio-type Lemma 9.8. If ω is non-recurrent, then U is of periodic type.Proof. Consider the critical value decoration D v . By Proposition 9.6, D v eventually maps back to D c . The rays on the boundary of the min-imal sector S ( ω ) map to the rays on the boundary of S ( f ( ω )) under f . This follows from Lemma 8.6. The sector S ( f ( ω )) is eventuallymapped to S ( ω ) by Lemma 8.5. Therefore, U is of periodic type. (cid:3) Main theorem s:mt
To prove the main result we rely upon recent powerful results ob-tained in [DL21]. To state them, we need to state a property of qua-dratic polynomials from the Main Cardioid of the Mandelbrot set es-tablished in [Chi08]. t:chi
Theorem 10.1 ([Chi08]) . Let Q be a quadratic polynomial with a fixedpoint a such that P ′ ( a ) = e πiθ with irrational θ . Then the limit set ω ( c Q ) of the critical point c Q of Q is a continuum. In the situation of Theorem 10.1 the set Th( ω ( c Q )) is called the mother hedgehog of Q and is denoted by M Q . t:md Theorem 10.2 ([DL21]) . Let Q be a quadratic polynomial with a fixedpoint a such that Q ′ ( a ) = e πiθ with irrational θ . Then Q − ( M Q ) is theunion of M Q and a continuum M ′ Q such that M Q ∩ M ′ Q = { c } . Theorem 10.2 allows one to view Q similar to how Siegel quadraticpolynomials with locally connected Julia sets are viewed. By Theorem10.2 there are infinite concatenations of pullbacks of M Q analogous toexternal rays in that they partition J ( Q ) in pieces and, thus, enable fur-ther study of topology J ( Q ). Indeed, by Theorem 10.2 countably manypullbacks of M ′ Q are attached to M Q at preimages of c Q that belong to M Q . Each of them is eventually mapped onto M Q . Hence each of themhas countably many attached to it pullbacks of M Q , etc. The entiregrand orbit of M Q can be viewed as a “spiderweb” of concatenated invarious ways pullbacks of M Q . MMEDIATE RENORMALIZATION 29
Each pullback T of M Q can be characterized by a sequence of integersthr( T ) called the thread of T . It reflects the “journey” of T to M Q ,and, simultaneously, the concatenation (the chain ) CH T (of pullbacksof M Q ) that connects M Q and T . In what follows we denote chains byboldface capital letters (e.g., A or X ) and threads by small boldfaceletters (e.g., a and x ). To study chains of pullbacks of M Q we need thenext lemma. l:chains Lemma 10.3.
Distinct pullbacks S and T of M Q can only intersect ifthey meet at a point that is a preimage of c Q . Two chains share aninitial finite string and are otherwise disjoint.Proof. Observe that M Q is a full continuum. Therefore all pullbacksof M q (including M ′ Q ) are full continua too. It follows that if T is apullback of M Q and c Q / ∈ T , then P | T is a homeomorphism to theimage. Suppose that S = T are pullbacks of M Q and x ∈ S ∩ T . Letus apply Q to S and T step by step. Then there are two cases.(1) At some earliest moment n we have Q n ( S ) = M Q and Q n ( T ) = M Q . Then Q n ( T ) is a Q m -pullback of M ′ Q for some m >
0. It followsthat Q m + n ( x ) = c Q as desired.(2) At some earliest moment n we have Q n ( S ) = Q n ( T ) = A , foreach i < n, Q i ( S ) = Q i ( T ) and neither of these sets equals M Q . Then Q n ( S ) = M Q either because otherwise we must have Q n − ( S ) = M Q ,a contradiction. The set Q n − ( S ∪ T ) = B is a pullback of the set A .Together with the fact that Q | B is not a homeomorphism this impliesthat c Q ∈ Q n − ( S ) ∩ Q n − ( T ). Since neither of these two sets equals M Q and they are distinct, it follows that at most one of them equals M ′ Q while the other one (denote is by Z ) is a pullback of M ′ Q . Wehave that c Q ∈ Z ∩ M ′ Q and, for some k > Q k ( Z ) = M ′ Q while Q k ( M ′ Q ) = M Q . It follows that Q K ( c Q ) = c Q , a contradiction.We leave the second claim of the lemma to the reader. (cid:3) Threads are one-sided sequences, infinite on the left. For each pull-back S in CH T choose the least number m ( S ) such that Q m ( S ) ( S ) = M Q while M Q itself is associated with infinite on the left string of zeros de-noted by 0. Thus, if CH T = ( M Q , S , . . . , S k − , S k = T ) then thr( T ) =(0 , m ( S ) , . . . , m ( T )) so that thr( M Q ) = (0) , thr( M ′ Q ) = (0 ,
1) etc. Ob-serve that given a pullback T of M Q , its thread thr( T ) is a sequencewith 0 followed by a finite string of strictly growing positive integers.If thr( T ) = (0 , m , . . . , m k ) then S is the Q m -pullback of M Q at-tached to M Q . Under Q m the set S maps forward so that Q m ( S )is attached to M Q and is later, under the action of Q m − m , mappedto M Q . In general, each power Q m j moves all pullbacks S i ∈ CH T , j i k forward so that the chain Q m j ( M Q , S , . . . , S k − , S k ) equals( M Q , Q m j ( S j +1 ) , Q m j ( S j +2 ) , . . . , Q m j ( S k ))and the associated thread is (0 , m j +1 − m j , . . . , m k − m j ). Thus, theaction of Q on threads will be denoted by η and translates as η :(0 , m , . . . , m k ) (0 , m − , . . . , m k −
1) (notice that if m = 1 thenumber m − M Q )and characterize them by their threads , i.e. infinite in both directionssequences of integers (0 , m , m , . . . , ) where 0 < m < . . . (we will stilluse the same notation for infinite chains/threads as for finite ones).The corresponding infinite chain of pullbacks of M Q is ( M Q , S , . . . )that can be described as follows: the set S is the Q m -pullback of M Q ,attached to M Q , the set S is the Q m -pullback of M Q , attached to S ,etc. Clearly, the map η can be defined on infinite threads as follows: if x = (0 , m , m , . . . ), then η ( x ) = (0 , m − , m − , . . . ). This actioncorresponds to the action of Q on infinite chains so that if thr( X ) = x ,then thr( Q ( X )) = η ( x )) (i.e., thr ◦ Q = η ◦ thr). Clearly, the map η defined on infinite threads has periodic points. E.g., by definition a = (0 , m , m , . . . ) is η -fixed if m − , m − m , etc, i.e.the only η -fixed thread is (0 , , , , . . . ). Of course, there are otherperiodic infinite chains.Suppose that (0 , m , . . . ) is periodic of (minimal) period N (in whatfollows by “period” we always mean “minimal period”). It is easy tosee that this means the following: there exists k such that m k = N and,moreover, for any j = lk + r, l, r < k we have m j = lN + m r .Recall that the topological hull Th( A ) of a compact set A ⊂ C is thecomplement of the unbounded complementary domain of A . l:exponent Lemma 10.4.
Let Q be a quadratic polynomial with a fixed point a such that Q ′ ( a ) = e πiθ with irrational θ . Then claims (1) − (4) hold. (1) Let T be a pullback of M Q disjoint from M Q . Then there exists C > and < q < such that any Q N -pullback of T hasdiameter less than C q N for any N . (2) Distinct infinite chains converge to distinct points. (3)
Suppose that thr( X ) = x is periodic of period N . Then X converges to a periodic point of period N that does not belongto M Q . Any preperiodic chain converges to a preperiodic point. (4) The only periodic point in M Q is the fixed point a . All periodicpoints y = a of Q are limits of the corresponding chains ofpullbacks of M Q . MMEDIATE RENORMALIZATION 31
Proof. (1) The fact that M Q = Th( ω ( c Q )) implies that T is disjointfrom ω ( c Q ). By Ma˜n´e [Man93] (see Theorem 9.1) we can cover Th( T )with small Ma˜n´e disks U , . . . , U k so that their union is itself a Jordandisk V with V ∩ M Q = ∅ so that any Q n -pullback of each U i is ofdiameter less than Cq n for some C > < q < Q ). Since V ∩ Th( ω ( c Q )) = ∅ then any Q N -pullback of V isa topological disk V ′ that homeomorphically maps onto V under Q N (observe that since c Q is recurrent, the orbit of c Q is contained in ω ( c Q ).Hence each U i is represented in V ′ by exactly one pullback U ′ i whosediameter, by Theorem 9.1, is less than Cq N . Hence the diameter of V ′ is less than kCq N and it remains to set C = kC .(2) Now, if two distinct chains of pullbacks of M Q converge to thesame point, it would follow that some points of these pullbacks areblocked from infinity by the union of these chains and, therefore, cannotbelong to J ( Q ), a contradiction. Observe that we are not claiming thatall chains converge. However it two chains do converge,they cannotconverge to the same as was just proven.(3) If x = (0 , m , . . . ), then there exists k such that m k = N and,moreover, for any j = lk + r, l, r < k we have m j = lN + m r .Suppose that S is the j -th element of X (associated with m j ). It ispreceded in X by another pullback S ′ of M Q , associated with m j − ,and until S ′ maps to M Q under Q m j , the set S remains detached from M Q . Thus, Q m j − − ( S ) is a specific pullback of M Q detached from M Q .Evidently, there is a finite collection of such “last detached” pullbacks of M Q that serve all elements of X . By (1) we can choose numbers C > < q < S ) < Cq m j − .Evidently, this implies that X converges to a Q N -fixed point z . If theperiod of z were less than N , there would exist a chain distinct from X but converging to z , a contradiction to (2). Moreover, z cannot belongto M Q as otherwise some points of M Q would be blocked from infinity(similar to the argument in (2)). The last part of claim (3) now follows.(4) Claims (1) - (3) hold for some quadratic polynomial P sie withSiegel fixed point b and locally connected Julia set. In case of suchpolynomials it is easy to check that all periodic points of P sie exceptfor b can be obtained as limits of periodic chains. However chains of Q and chains of P are in one-to-one correspondence with correspondingperiodic threads. Hence the number of periodic points of any period n is the same for Q and for P sie and coincides with the number ofperiodic threads of that period. Recall that except for b there are no Q -periodic points that belong to M Q as M Q is a Q -invariant Jordan curve on which Q is conjugate with an irrational rotation. It followsfrom these counts of periodic points for Q and P sie that no periodicpoint of Q , except for a , can belong to M Q . (cid:3) Let us now combine Lemma 10.4 and tools from [BFMOT12]. t:allc
Theorem 10.5.
Let P be a polynomial of any degree with a fixed non-repelling point a ∈ B ∗ where B ∗ is an invariant filled quadratic-likeJulia set of P . Assume that any periodic neutral point x = a of P is parabolic. Then for a periodic point z ∈ B ∗ \ { a } and a periodicexternal ray R of P the fact that z belongs to the impression I of R implies that I= { z } (thus, J ( P ) is locally connected at z ). It is well-known that if the impression of a ray is degenerate thenthis rays lands at some point z and the continuum is ready. Proof.
First assume that a is either attracting or parabolic. Then P has no Cremer or Siegel points. Hence by Corollary 7.5.4 [BFMOT12]any periodic impression is degenerate.The forthcoming argument which involves chains, thread etc appliesto P | B ∗ as B ∗ is quadratic-like; we will, therefore, use the same nota-tion as before despite the fact that P itself is not quadratic. Supposenow that P ′ ( a ) = e πiθ where θ is irrational. We will need the fol-lowing construction. Let X be the chain that converges to z . Let(0 , m , . . . ) = thr( X ) = x be its thread. Choose a pullback S of themother hedgehog M P of P that belongs to X assuming that S = M P ,S | neM ′ P .. Evidently, on the plane there are pullbacks L, R of M P at-tached to S from either side of the part of X connecting S and z . Thechoice of L and R depends on θ , however, regardless of θ , such L and R exist. Then choose (pre)periodic chains CH l and CH r that extend L and R and converge to points y l , y r , and external rays Y l , Y r of P thatland at y l , y r and have arguments θ l , θ r , resp. Set Z = Y l ∪ { y l } ∪ L ∪ S ∪ R ∪ { y r } ∪ Y r and use it in the proof. Repeat this construction for a different pullback S ′ ∈ X of M P assuming that in X the set S ′ is closer to M P than S and construct a similar set Z ′ for S ′ so that Z separates Z ′ from z .The set Z divides C in two open subsets, W a ∋ a and W z ∋ z . Aray whose impression contains z must be contained in W z or coincidewith Y l or with Y r . Now, the set Z ′ divides C in two open subsets, W ′ a ∋ a and W ′ z ∋ z , and W ′ z ⊃ Z ∪ W z . It follows that the impressionof any angle that contains z cannot contain a . By Corollary 7.5.4 from[BFMOT12] this impression is degenerate and, hence, coincides with { z } . This implies that J ( P ) is locally connected at z as desired. (cid:3) MMEDIATE RENORMALIZATION 33
The next result is used in dealing with an easier case. t:cutpts
Theorem 10.6 ([BOT21]) . Let P be a polynomial of any degree withconnected filled Julia set K ( P ) . Let x ∈ K ( P ) be a repelling or para-bolic periodic point and all K ( P ) -rays to x form m wedges W i , where i m . Moreover, suppose that x ∈ Q is a cutpoint of order n of aninvariant continuum Q ⊂ J . Then n m , each wedge W i intersects Q over a connected (possibly empty) set, and every Q -ray to x is isotopicrel. Q to a K ( P ) -ray that lands at x . We are finally ready to prove Theorem 1.2.
Proof of Theorem 1.2.
Since ω is non-recurrent, then by Lemma 9.8 U is of periodic type. Let M = αβ be its major and I = ( α, β ) beits major hole. Then α and β are σ -periodic angles each of whichis approached from the outside of I by periodic angles α i → α and β i → β whose rays land in J ∗ . These angles belong to U and correspond(through the map τ collapsing edges of U ) to σ -periodic angles α ′ i and β ′ i . Moreover, “quadratic” angles α ′ i and β ′ j converge to the same“quadratic” periodic angle γ = τ ( M ). Then by Theorem 10.5 thelanding points of “quadratic” external rays with arguments α ′ i and β ′ i converge to the periodic landing point of the “quadratic” ray withargument γ . Evidently, the same will happen in K ∗ , i.e. the landingpoints of P -rays with arguments α i and β j converge to a periodic pointin K ∗ . This point belongs to the impressions of α and of β . Hence,by Theorem 10.5, they are degenerate and coincide with this point, acontradiction with the assumption that the rational lamination of P isempty. (cid:3) References [BCLOS16] A. Blokh, D. Childers, G. Levin, L. Oversteegen, D. Schleicher,
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Department of Mathematics, Uni-versity of Alabama at Birmingham, Birmingham, AL 35294-1170 (Vladlen Timorin)
Faculty of Mathematics, HSE University, 6 UsachevaSt., 119048 Moscow, Russia (Vladlen Timorin)
Independent University of Moscow, Bolshoy VlasyevskiyPereulok 11, 119002 Moscow, Russia
Email address , Alexander Blokh: [email protected]
Email address , Lex Oversteegen: [email protected]
Email address , Vladlen Timorin:, Vladlen Timorin: