A model of the cubic connectedness locus
aa r X i v : . [ m a t h . D S ] A ug A MODEL OF THE CUBIC CONNECTEDNESS LOCUS
ALEXANDER BLOKH, LEX OVERSTEEGEN, AND VLADLEN TIMORINA
BSTRACT . For the cubic connectedness locus, we define a combinato-rial upper semi-continuous partition. This can be regarded as a combi-natorial model of the cubic connectedness locus.
1. I
NTRODUCTION
The parameter space of complex degree d polynomials P is the spaceof affine conjugacy classes [ P ] of these polynomials. The connectednesslocus M d consists of classes of all degree d polynomials P , whose Juliasets J P are connected. The Mandelbrot set M has a complicated self-similar structure understood through the “pinched disk” model [8, 9, 14].In this paper, we find a combinatorially defined upper semi-continuous(USC) partition of M . A property of a polynomial is combinatorial ifit can be stated based only upon knowing which pairs of rational externalrays land at the same point and which pairs do not. A combinatorial USCpartition of M yields a continuous map of M to a quotient space of CrP ,the space of unordered cubic critical portraits. Let us describe our approach.Let a cubic polynomial P have a connected Julia set. A point x is ( P -)stable if its forward orbit is finite and contains no critical or non-repellingperiodic points. An unordered pair of rational angles { α, β } is ( P -)stable ifthe external rays with arguments α and β land at the same stable point of P .Write S P for the set of all P -stable pairs of angles; call S P the s-set of P .A cubic polynomial P ∈ M is visible if S P = ∅ and invisible otherwise.If P is visible, denote by C P the set of all critical portraits compatible with S P (i.e., no critical chord from a critical portrait in C P separates a P -stablepair of angles). The set C P is the combinatorial counterpart of P . We willshow that C P depends upper semi-continuously on P .We now want to define, for every [ P ] ∈ M , a closed subset A P of CrP called an alliance . The main properties of alliances are:(1) if P is visible, then C P ⊂ A P ; Mathematics Subject Classification.
Primary 37F20; Secondary 37F10, 37F50.
Key words and phrases.
Complex dynamics; laminations; Mandelbrot set; Julia set.The second named author was partially supported by NSF grant DMS–1807558.The third named author has been supported by the HSE University Basic ResearchProgram and Russian Academic Excellence Project ’5-100’. (2) distinct alliances are disjoint;(3) the alliances form an USC partition of
CrP .One special alliance is said to be prime . It contains C P for all visiblepolynomials P with a non-repelling fixed point and some other combina-torial counterparts. Also, we associate the prime alliance with all invisiblepolynomials P . All other alliances are called regular ; they are combina-torial counterparts of certain polynomials, and there are infinitely many ofthem. Main Theorem.
The sets A P form an USC partition {A P } of CrP . Themap P
7→ A P is continuous and maps M to the quotient space CrP / {A P } . We will show that the union of regular alliances is open and dense in
CrP .Even though the prime alliance is special (as it has to represent all invisiblepolynomials), it is small in the sense of Baire. A more detailed combinato-rial study of M will characterize combinatorial counterparts and the waysthey can intersect. Thurston [14] gave a similar description of M in termsof the Quadratic Minor Lamination QML . Combinatorial counterparts for M correspond to certain gaps or leaves of QML . We have to pass fromcombinatorial counterparts to alliances in order to have a partition of M .Alliances for M correspond to maximal baby Mandelbrot sets or singlenon-renormalizable quadratic polynomials, except for the central alliance,which may be called prime and which covers countable concatenations ofhyperbolic components growing from the main cardioid. Our Main Theo-rem gives a cubic analog of this partition. On the other hand, the centralpart of M is more complicated than that of M : it is not locally connected[13] and is highly nontrivial combinatorially [3, 6].2. C RITICAL PORTRAITS AND LAMINATIONS
We assume familiarity with complex polynomial dynamics, including Ju-lia sets, external rays, etc. All cubic polynomials in this paper are assumedto be monic , i.e., of the form z + a quadratic polynomial, and to have con-nected Julia sets. We can parameterize the external rays of a cubic poly-nomial P by angles , i.e., elements of R / Z . The external ray of argument θ ∈ R / Z is denoted by R P ( θ ) . Clearly, P maps R P ( θ ) to R P (3 θ ) . Lemma 2.1 ([9], cf. Lemma B.1 [10]) . Let g be a polynomial, and z be astable point of g . If an external ray R g ( θ ) with rational argument θ lands at z , then, for every polynomial ˜ g sufficiently close to g , the ray R ˜ g ( θ ) landsat a stable point ˜ z close to z . Moreover, ˜ z depends holomorphically on ˜ g . For a pair of sets A , B , let A ∨ B denote the set of all unordered pairs { a, b } with a ∈ A and b ∈ B . Thus, the s-set S P of P consists of all pairs MODEL OF THE CUBIC CONNECTEDNESS LOCUS 3 { α, β } ∈ ( Q / Z ) ∨ ( Q / Z ) such that R P ( α ) and R P ( β ) land at the samestable point of P .A chord ab is a closed segment connecting points a, b of the unit circle S = { z ∈ C | | z | = 1 } . If a = b , the chord is degenerate . Write σ d forthe self-map of S that takes z to z d . A chord ab is said to be ( σ d -) critical if σ d ( a ) = σ d ( b ) . Let CCh be the set of all σ -critical chords equipped withthe natural topology; CCh is homeomorphic to S . A critical portrait is anunordered pair { c, y } ∈ CCh ∨ CCh such that c and y do not intersect inthe unit disk D = { z ∈ C | | z | < } . The space of all critical portraits isdenoted by CrP . It is homeomorphic to the M ¨obius band, cf. [15].Let ∼ P be the equivalence relation on S defined as follows: e πiα ∼ P e πiβ if { α, β } ∈ S P or α = β . Let S lP be the set of all edges of the convexhulls in D of all ∼ P -classes. Thus S lP is a set of chords. The superscript l isfrom “laminational”. Two distinct chords of D cross if they intersect in D .Alternatively, crossing chords are said to be linked . Two sets of chords are compatible if no chord of one set crosses a chord of the other set. Proposition 2.2.
The dependence P
7→ C P is upper semi-continuous.Proof. We prove that if P i → P and we choose K i ∈ C P i with K i → K ,then K ∈ C P . Assume the contrary: K = { c, y } , where c crosses some ℓ = ab ∈ S lP . By Lemma 2.1, we have a ∼ P i b for large i , and K i containsa critical chord c i close to c , a contradiction, since then c i also crosses ℓ . (cid:3) In [14] Thurston defined invariant laminations as families of chords withcertain dynamical properties. We use a slightly different approach (see [2]).
Definition 2.3 (Laminations) . A lamination is a collection L of chordscalled leaves such that distinct leaves are unlinked, all degenerate chords(points of S ) are leaves, the set L + = S ℓ ∈L ℓ is compact. Without the lastcondition L is called a prelamination .From now on L denotes a lamination (unless we specify that we considerprelamination). Gaps of L are the closures of components of D \ L + . Agap G is countable ( finite, uncountable ) if G ∩ S is countable and infinite(finite, uncountable). Uncountable gaps are called Fatou gaps. For a closedconvex set H ⊂ C , straight segments in Bd( H ) are called edges of H .In what follows, convergence of prelaminations L i to a set of chords E isalways understood as Hausdorff convergence of leaves of L i to chords from E . Evidently, E is a lamination. Call L nonempty if it has nondegenerateleaves, otherwise it is empty (denoted L ∅ ). Say that L is countable if ithas countably many nondegenerate leaves and uncountable otherwise; L is perfect if it has no isolated leaves.If G ⊂ D is the convex hull of G ∩ S , define σ d ( G ) as the convex hullof σ d ( G ∩ S ) . Sibling ( σ d )-invariant laminations modify Thurston’s [14] A. BLOKH, L. OVERSTEEGEN, AND V. TIMORIN invariant geodesic laminations. A sibling of a leaf ℓ is a leaf ℓ ′ = ℓ with σ d ( ℓ ′ ) = σ d ( ℓ ) . Call a leaf ℓ ∗ such that σ d ( ℓ ∗ ) = ℓ a pullback of ℓ . Definition 2.4 ([2]) . A (pre)lamination L is sibling ( σ d )-invariant if(1) for each ℓ ∈ L , we have σ d ( ℓ ) ∈ L ,(2) for each ℓ ∈ L there exists ℓ ∗ ∈ L with σ d ( ℓ ∗ ) = ℓ ,(3) for each ℓ ∈ L such that σ d ( ℓ ) is a nondegenerate leaf, there exist d pairwise disjoint leaves ℓ , . . . , ℓ d in L such that ℓ = ℓ and σ d ( ℓ ) = · · · = σ d ( ℓ d ) .Collections of leaves from (3) above are full sibling collections . Theirleaves cannot intersect even on S . By cubic (resp., quadratic) laminations,we always mean sibling σ -(resp., σ -) invariant laminations. When dealingwith cubic laminations, we write σ instead of σ . From now on L (possiblywith sub- and superscripts) denotes a cubic sibling invariant lamination.These are properties of cubic sibling invariant laminations [2]: gap invariance: if G is a gap of L , then H = σ ( G ) is a leaf of L (possibly degenerate), or a gap of L , and in the latter case, the map σ : G ∩ S → H ∩ S extends to a map of the boundary of G onto theboundary of H so that the extended map is an orientation preservingcomposition of a monotone map and a covering map; compactness: if a sequence of sibling invariant prelaminations con-verges to a set of chords, this set of chords is a sibling invariantlamination.Gap invariance is a part of Thurston’s original definition [14].A chord ℓ is inside a gap G if ℓ is, except for the endpoints, in the interiorof G . A gap G of L is critical if either all edges of G are critical, or thereis a critical chord inside G . A critical set of L is a critical leaf or a criticalgap. We also define a lap of L as either a finite gap of L or a nondegenerateleaf of L not on the boundary of any gap. Lemma 2.5.
Suppose that L i → L are σ d -invariant laminations, and let G be a periodic lap of L . Then G is also a lap of L i for all sufficiently large i .Proof. Let G be a lap and ℓ an edge of G ; we write k for the minimal periodof ℓ . Then L i , for large i , must have a lap G i with G i → G . Choose an edge ℓ i of G i so that ℓ i → ℓ . Then ℓ i does not cross ℓ for large i as otherwise theleaves σ kd ( ℓ i ) and ℓ i would cross. Moreover, ℓ i is disjoint from the interiorof G for large i as otherwise σ kd ( ℓ i ) would intersect the interior of G i (notethat ℓ i maps farther away from ℓ under σ kd ). By way of contradiction assumethat L i do not contain G . Then G i % G and ℓ i = ℓ for at least one edge ℓ of G . It follows that σ kd ( G i ) % G i , a contradiction. (cid:3) MODEL OF THE CUBIC CONNECTEDNESS LOCUS 5
We now discuss some special classes of laminations. A lamination L is clean if any pair of non-disjoint leaves of L is on the boundary of a finitegap. Clean laminations give rise to equivalence relations: a ∼ L b if either a = b or a , b are in the same lap of L . By Lemma 3.16 of [4], any cleanlamination has the following period matching property : if one endpoint ofa leaf is periodic, then the other endpoint is also periodic with the sameminimal period. Limits of clean σ d -invariant laminations are called limitlaminations , cf. [5]. Chords are always considered in the Hausdorff metric. Definition 2.6 (Perfect laminations [7]) . A closed set of chords is perfect ifit has no isolated chords. The maximal perfect subset L p of L is called the perfect part of L . Clearly, a lamination L is perfect if L = L p .One can define the perfect part of L as the set of all leaves ℓ ∈ L such thatarbitrarily close to ℓ there are uncountably many leaves of L . By Lemma3.12 of [7], the set L p is an invariant lamination. Definition 2.7 (Oldest ancestors) . If L is nonempty, an oldest ancestor of L is defined as a minimal by inclusion nonempty sublamination of L .An oldest ancestor is perfect or countable: if L is uncountable, then L p ⊂L is nonempty. Given a chord ℓ = ab denote by | ℓ | the length of the smallerarc with endpoints a and b (computed with respect to the Lebesgue measureon S such that the total length of S is 1); call | ℓ | the length of ℓ . Lemma 2.8. If L is nonempty, then L contains an oldest ancestor.Proof. Let L α be a nested family. Definition 2.4 implies that then T L α is asibling invariant lamination too. Any nonempty lamination contains leavesof length > d +1 . Indeed, for a nondegenerate leaf ℓ either | σ d ( ℓ ) | = d | ℓ | or | ℓ | > d +1 . Thus, if all L α above are nonempty, then T L α is nonempty.Now the desired statement follows from the Zorn lemma. (cid:3) The next lemma follows from the definitions and the compactness prop-erty of invariant laminations.
Lemma 2.9.
Let L be an oldest ancestor. If ℓ ∈ L is a nondegenerate leaf,then the iterated pullbacks of the nondegenerate iterated images of ℓ aredense in L .
3. I
NVARIANT GAPS AND PRIME PORTRAITS An invariant gap is an invariant gap of a cubic L , not necessarily spec-ifying the latter. An infinite invariant gap is quadratic if it has degree 2.By Section 3 of [4], any quadratic invariant gap can be obtained as follows.A critical chord c gives rise to the complementary circle arc L ( c ) of length / with the same endpoints as c . The set Π( c ) of all points with orbits in A. BLOKH, L. OVERSTEEGEN, AND V. TIMORIN L ( c ) is nonempty, closed and forward invariant. The convex hull G ( c ) of Π( c ) is an invariant quadratic gap, and any invariant quadratic gap is of thisform. For any invariant gap G , finite or infinite, a major of G is an edge M = ab of G , for which there is a critical chord ax or by disjoint from theinterior of G . By Section 4.3 of [4], a degree 1 invariant gap has one or twomajors; every edge of G eventually maps to a major and if G is infinite, atleast one of its majors is critical. An invariant gap G is rotational if σ actson its vertices (i.e., G ∩ S ) as a combinatorial rotation. Recall that σ = σ .For brevity say that a chord is compatible with a finite collection of gaps ifit does not cross edges of these gaps.Define I ( c ) as the complement of L ( c ) , define I ( y ) similarly. Call K weak if the forward orbit of c is disjoint from I ( y ) , or the forward orbit of y is disjoint from I ( c ) ; otherwise call K strong . Lemma 3.1.
The set of strong critical portraits is open and dense in
CrP while the set of weak critical portraits is closed and nowhere dense in
CrP . Lemma 3.1 is left to the reader.
Lemma 3.2.
A critical portrait { c, y } is compatible with an invariant qua-dratic gap if and only if it is weak.Proof. Let T be the forward orbit of y . If T is disjoint from I ( c ) , then T ⊂ Π( c ) ; hence, { c, y } is compatible with G ( c ) . Assume now that { c, y } is compatible with an invariant quadratic gap U . Then U contains c or y ,say, y ⊂ U . Since U is quadratic, c is disjoint from the interior of U , thus I ( c ) ∩ U = ∅ . We conclude that T never visits I ( c ) . (cid:3) Observe that a strong critical portrait K is not compatible with an in-finite invariant gap G as otherwise, by [4], at least one chord from K isnon-disjoint from G . Thus, a critical portrait is weak if and only if it iscompatible with an infinite invariant gap and strong otherwise. Theorem 3.3.
Suppose that a nonempty cubic L has an infinite periodicgap U and either σ ( U ) = U or U shares an edge with a finite rotational lapof L . Then there is a weak critical portrait compatible with L .Proof. Choose a critical chord c in a gap from the orbit of U with an end-point of the same period as U . Choose a critical chord y = c compatiblewith L and not crossing c . Then { c, y } is compatible with a quadratic in-variant gap by Lemma 3.2. (cid:3) From now on, by an oldest ancestor, we mean an oldest ancestor of some limit lamination.
A gap G of a lamination is invariant if σ ( G ) = G (with“ = ” rather than “ ⊂ ”). Lemma 3.4.
Any oldest ancestor has an invariant lap or infinite gap.
MODEL OF THE CUBIC CONNECTEDNESS LOCUS 7
Proof.
By Lemma 3.7 of [10], any clean lamination has a lap or an infinitegap G such that σ ( G ) = G . Passing to the limit, we conclude that any limitlamination has the same property (even though the limit of finite invariantgaps may be infinite). An oldest ancestor of a limit lamination must thenalso have the above mentioned property. (cid:3) Definition 3.5 (Friends, prime critical portraits) . Critical portraits K , K ∈ CrP are friends (through an oldest ancestor L ) if K , K are compatible with L . A critical portrait K is prime if some friend of K has a weak friend.The next lemma is straightforward; we leave it to the reader. Lemma 3.6. If K , K are friends, then there is an oldest ancestor withwhom they are compatible. Also friendship is a closed relation: if K i → K and K ′ i → K ′ and K i and K ′ i are friends for all i , then so are K and K ′ . Given L and a nondegenerate leaf ℓ ∈ L , let G ( ℓ ) be the set of iteratedpullbacks of the nondegenerate iterated images of ℓ . Lemma 3.7.
Let L be a countable oldest ancestor. Then (1) for any nondegenerate leaf ℓ ∈ L , the set of all nondegenerateleaves in L coincides with G ( ℓ ) ; (2) all nondegenerate leaves of L are isolated; (3) at least one weak critical portrait is compatible with L .Proof. (1) Choose an isolated leaf ℓ ∈ L . If G ( ℓ ) does not coincide withthe set of all nondegenerate leaves of L , choose a nondegenerate leaf ℓ ∈L \ G ( ℓ ) . Then leaves of G ( ℓ ) cannot approximate ℓ or coincide with ℓ ,a contradiction with Lemma 2.9. Let now ℓ be any nondegenerate leaf of L ; we proved that ℓ ∈ G ( ℓ ) . Therefore, G ( ℓ ) = G ( ℓ ) is the set of allnondegenerate leaves of L , as desired.(2) All non-isolated leaves in L form a forward invariant closed family ofleaves. If ℓ is non-isolated, choose leaves ℓ i → ℓ , choose their pullbacks q i ,and choose a converging subsequence of these pullbacks; in the end we willfind a non-isolated leaf q with σ ( q ) = ℓ . Now, let ℓ be non-isolated and non-critical. Choose a sequence ℓ i → ℓ so that each ℓ i has exactly two siblingsin L (the only way a leaf ℓ can have more siblings is when there is a critical4-gon or 6-gon that maps onto σ ( ℓ ) ). We may assume that these siblingsare ℓ ′ i , ℓ ′′ i and that ℓ ′ i → ℓ ′ while ℓ ′′ i → ℓ ′′ . Clearly, σ ( ℓ ) = σ ( ℓ ′ ) = σ ( ℓ ′′ ) .We claim that ℓ , ℓ ′ and ℓ ′′ are pairwise disjoint. Indeed, if, say, ℓ = ab and ℓ ′ = bc , where σ ( c ) = σ ( a ) , then ℓ i and ℓ ′ i have distinct endpoints close to b and mapping to the same point; a contradiction. Hence by definition theset of all non-isolated leaves of L is itself a sibling-invariant lamination, acontradiction with L being an oldest ancestor. A. BLOKH, L. OVERSTEEGEN, AND V. TIMORIN (3) By Lemma 3.4, we can find an invariant lap or infinite gap G of L . If G is infinite, our claim follows from Theorem 3.3. Hence we may assumethat G is finite. Let ℓ be an edge of G ; it is isolated by (2). Let H be a gap of L attached to G along ℓ . If H is infinite, the desired statement follows fromTheorem 3.3. Assume that H is finite. Let n be the minimal period of edgesof G , and there are two cases: σ n ( H ) = H and σ n ( H ) = ℓ . The former casecontradicts (1), hence σ n ( H ) = ℓ , and we may assume that σ ( H ) = σ ( ℓ ) .Choose a critical chord y ∈ H that shares an endpoint with ℓ , and a criticalchord c in a critical gap or leaf of L disjoint from H . By Lemma 3.2, thecritical portrait { c, y } is compatible with G ( c ) as desired. (cid:3) An oldest ancestor is regular if all its critical portraits have only strongfriends.
Lemma 3.8.
A regular oldest ancestor L does not share a critical portraitwith another oldest ancestor.Proof. The lamination L is uncountable by Lemma 3.7; hence it is perfect.Let K = { c, y } be a critical portrait compatible with L and an oldest an-cestor L ′ = L . Since K is strong, invariant sets of L and L ′ are finite. Let G be an invariant lap of L . Let G ′ be an invariant leaf or gap L ′ located inthe same component of D \ S K as G . Since all friends of K are strong, L ′ is uncountable (and hence perfect) by Lemma 3.7. Moreover, by Theorem3.3, any gap of L ′ non-disjoint from G ′ is finite. There are no leaves of L intersecting the interior of G ′ since otherwise uncountably many leavesof L would intersect edges of G ′ . Since K is compatible with both L and L ′ , this is impossible by [7, Lemma 3.53]. Therefore, G ′ ⊂ G . A similarargument shows that G ⊂ G ′ , hence G = G ′ .If iterated images of c and y avoid G , then iterated L -pullbacks of G anditerated L ′ -pullbacks of G ′ are the same. Hence L = L ′ since the iteratedpullbacks of G are dense in both L and L ′ . Let for some minimal n > thepoint σ n ( c ) be a vertex of G . Let C , C ′ be the critical sets of L , resp., L ′ containing c . Infinite gaps of L and L ′ are disjoint from G by Theorem 3.3;so, n > and C , C ′ are finite. Since L and L ′ are compatible and perfect, C = C ′ by [7, Theorem 3.57]. Similarly, we see that either y never maps to G or the critical sets of L , L ′ containing y coincide. In both cases pullbacksof G in L are the same as pullbacks of G in L ′ , hence L = L ′ . (cid:3) A critical portrait is regular if it is compatible with a regular oldest an-cestor. Corollary 3.9 follows immediately from definitions and Lemma 3.8.
Corollary 3.9.
A friend of a regular critical portrait is regular. All its crit-ical portraits of a regular oldest ancestor form a closed subset of
CrP con-sisting of friends, and no other critical portrait can be their friend.
MODEL OF THE CUBIC CONNECTEDNESS LOCUS 9
One can define regular critical portraits through the concept of a friend.
Lemma 3.10.
A critical portrait K is regular if and only if all friends offriends of K are strong (i.e., if K is not prime).Proof. Let K be a regular critical portrait. Then there is a unique regularoldest ancestor L compatible with K . All critical portraits of L are strongand have only strong friends; by Lemma 3.8, none of them is compatiblewith an oldest ancestor L ′ = L . Hence all friends of K are compatible with L , i.e. are regular. Repeating this, we see that friends of friends of K arecompatible with L and, hence, strong. On the other hand, suppose that allfriends of friends of a critical portrait K are strong. Take an oldest ancestor L compatible with K . Then all its critical portraits have only strong friends.By definition L is regular which implies that K is regular as desired. (cid:3) Lemma 3.11.
A limit of prime critical portraits is prime.Proof. If K i are prime and K i → K , then, by definition, some friends K ′ i of K i have weak friends K ′′ i . Passing to a subsequence we can arrange that K ′ i → K ′ and K ′′ i → K ′′ . By Lemma 3.6, the portrait K ′ is a friend of K , and K ′′ is a friend of K ′ . By Lemma 3.1, the portrait K ′′ is weak. By definition, K is prime. (cid:3)
4. A
LLIANCES
The prime alliance A is the set of all prime critical portraits. Section 3implies that the prime alliance is closed topologically and under friendship.For a visible polynomial P such that S lP is not compatible with prime crit-ical portraits, define the regular alliance A P as the set of friends of criticalportraits from C P . If S lP is compatible with a prime critical portrait (e.g.,if S lP = ∅ ), then A P is defined as the prime alliance. This defines A P for any polynomial P with [ P ] ∈ M . The prime alliance is special as itserves all invisible polynomials, however diverse they are. It also serves allpolynomials with non-repelling fixed points and some other polynomials.For [ P ] ∈ M , define a clean lamination L sP (“s” from stable) as follows.First, define an equivalence relation ≈ P on S by declaring e πiα ≈ P e πiβ if R P ( α ) and R P ( β ) land at the same point eventually mapped to a stablepoint of P ; a ∼ P b implies a ≈ P b but not vice versa. Then L sP is definedas the set of all edges of the convex hulls of all ≈ P -classes and the limits ofthese edges. Clearly, L sP is a cubic lamination containing S P (cf [2]).Recall some results of [12]. A polynomial P with no neutral periodicpoint defines a clean lamination L R P called the real lamination of P . Here e πiα and e πiβ are vertices of a lap of L R P iff there are angles α = α, α ,. . . , α k = β such that the impressions of angles α i and α i +1 intersect. Aclean lamination without infinite degree 1 gaps has the form L R P for some P . For a periodic lap G of L R P , external rays corresponding to vertices of G land at the same stable point. Lemma 4.1.
A regular alliance A P has the form C P for some visible poly-nomial P , possibly different from P .Proof. Suppose that S lP is not compatible with a prime critical portrait.Then L sP cannot be compatible with a prime critical portrait either. Con-sider an oldest ancestor L of L sP ; it is compatible with some K ∈ C P . ByLemma 3.8, the lamination L is not compatible with any other oldest an-cestor. By [12], there is a polynomial P without neutral periodic pointssuch that L R P = L . We want to prove that A P = C P , that is, if K ∈ C P and K ′ is a friend of K , then K ′ ∈ C P . Indeed, K is compatible with L sP hence also with L . If K ′ is a friend of K , then there is an oldest ancestor L ′ compatible with both K ′ and K . Since L is not compatible with any otheroldest ancestor, L = L ′ . Hence K ′ ∈ C P as desired. (cid:3) A regular alliance is closed topologically (because C P is closed by defi-nition) and under friendship (by Corollary 3.9). Lemma 4.2.
For any visible P we have C P ⊂ A P .Proof. If there are no prime portraits in C P , then C P ⊂ A P by definition ofa regular alliance. If C P has a prime portrait, then A P is the prime alliance.In this case C P ⊂ A P since A P is closed under friendship. (cid:3) Lemma 4.3. If P has no neutral cycles, and G is a periodic lap of L sP ,then the external rays for P corresponding to vertices of G land at the samepoint.Proof. Indeed, since L R P ⊃ L sP , the set G is also a lap of L R P . The conclusionnow follows from [12]. (cid:3) Theorem 4.4 implies the Main Theorem.
Theorem 4.4.
The map P
7→ A P from M to the quotient space of CrP generated by alliances is continuous.Proof.
Consider a sequence P i → P of polynomials, and set A i = A P i .Suppose that K i → K , where K i ∈ A i ; we claim that K ∈ A P . Passing to asubsequence, assume that either all K i are prime, or all K i are regular. If K i are prime, then K is prime by Lemma 3.11. Assume that all K i are regular.By Proposition 2.2, we have K ∈ C P and hence K ∈ A P by Lemma 4.2 if P is visible. It remains to show that if S P = ∅ then K is prime.Assume that K is regular. Then K is compatible with a regular oldestancestor L ◦ ; in particular, L ◦ is perfect. By Lemma 3.21 of [7], the equiva-lence on S that collapses all laps and infinite gaps of L ◦ semiconjugates σ to MODEL OF THE CUBIC CONNECTEDNESS LOCUS 11 an induced map on a dendrite. This map satisfies assumptions of Theorem7.2.6 of [1], which implies that it has infinitely many periodic points. Since L ◦ has only finitely many infinite periodic gaps, lifting the self-map of thedendrite back to L ◦ and S , we obtain infinitely many periodic laps of L ◦ .Thus, there is a periodic lap G of L ◦ with the following property. Forevery vertex e πiα of G , the ray R P ( α ) lands at a stable periodic point of P .Suppose that, for two vertices e πiα , e πiβ of G , the rays R P ( α ) , R P ( β ) landat distinct points. By Lemma 2.1, the rays R P i ( α ) , R P i ( β ) land at distinctstable points of P i , for large i . However, by Lemma 2.5, the set G is a lap of L i for all large i , a contradiction with Lemma 4.3. The conclusion is that theexternal rays of P corresponding to the vertices of G land at the same stablepoint of P . A contradiction with the assumption that P is invisible. (cid:3) Finally, let us show that the prime alliance is a small subset of
CrP . Lemma 4.5.
An open and dense subset of
CrP consists of non-prime criti-cal portraits.Proof.
The union of all regular alliances is open. Let K = { c, y } be acritical portrait such that the orbits of σ ( c ) and σ ( y ) are dense in S . Suchportraits are dense in CrP . We prove that K is regular by proving that, forany friend K ′ = { c ′ , y ′ } of K , the orbits of σ ( c ′ ) and σ ( y ′ ) are dense in S .Let L be an oldest ancestor compatible with K and K ′ . Let C be the leaf c if c ∈ L or the critical gap of L containing c otherwise. Define Y similarly.Arrange that c ′ ⊂ C and y ′ ⊂ Y , possibly renaming c ′ and y ′ . We claimthat the orbit of σ ( c ′ ) is dense in S . Otherwise consider the nondegeneratechord q = xx ′ , where x = σ ( c ) and x ′ = σ ( c ′ ) ,. There is ε > and anarc I ⊂ S such that σ n ( x ′ ) is never ε -close to I . On the other hand, iteratedimages of x are dense in I ; the corresponding images of q have length > ε .Therefore, all leaves of L originating in I have length ε or more.Note that C and Y are not periodic, therefore, no σ -periodic point of S isan eventual image of x or x ′ . There is a positive integer N with σ N ( I ) = S .Since any σ -periodic point a of S has a σ N -preimage in I , we have ab ∈ L for some b = a . Thus, the horizontal diameter Di connecting the two σ -fixed points of S is a leaf of L . Consider a nondegenerate chord ℓ withendpoint i = e πi (1 / . Then σ n ( ℓ ) crosses c or y for some n > . Thus ℓ / ∈L , a contradiction. We conclude that the orbit of σ ( c ′ ) is dense. Similarly,the orbit of σ ( y ′ ) is dense. (cid:3) R EFERENCES [1] A. Blokh, R. Fokkink, J. Mayer, L. Oversteegen, E. Tymchatyn,
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Degree d invariant laminations , in: “What’s Next?: The Mathematical Legacy of William P.Thurston”, ed. by D. Thurston, AMS – 205, Princeton U. Press (2020).(Alexander Blokh and Lex Oversteegen) D EPARTMENT OF M ATHEMATICS , U
NIVER - SITY OF A LABAMA AT B IRMINGHAM , B
IRMINGHAM , AL 35294(Vladlen Timorin) F
ACULTY OF M ATHEMATICS , HSE U
NIVERSITY , 6 U
SACHEVASTR ., M
OSCOW , R
USSIA , 119048
E-mail address , Alexander Blokh: [email protected]
E-mail address , Lex Oversteegen: [email protected]
E-mail address , Vladlen Timorin:, Vladlen Timorin: