COMBINATORIAL MODELS FOR SPACES OF CUBIC POLYNOMIALS

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1 COMBINATORIAL MODELS FOR SPACES OF CUBIC POLYNOMIALS ALEXANDER BLOKH, LEX OVERSTEEGEN, ROSS PTACEK, AND VLADLEN TIMORIN ABSTRACT. Thurston constructed a combinatorial model for the Mandelbrot set. No combinatorial model is known for the analogous spaces M d of (affine conjugacy classes of) polynomials of degree d 3. To address this problem, we define linked geolaminations L 1 and L 2 (with critical sets divided into groups of specifically linked sets). An accordion is defined as the union of a leaf l of L 1 and leaves of L 2 crossing l. We show that any accordion behaves like a gap of one lamination and use this to prove that the maximal perfect (i.e., without isolated leaves) sublaminations of L 1 and L 2 coincide. In the cubic case let D 3 M 3 be the set of all cubic dendritic (i.e., having only repelling cycles) polynomials. Let MD 3 be the space of all critically marked polynomials (P, c, w), where P D 3 and c, w are critical points of P (perhaps, c = w). Let c be the co-critical point of c, i.e., the point c c with P (c ) = P (c) (if c w) or the point c = c (if c = w). By Kiwi, to P D 3 one associates its lamination P so that each x J(P ) corresponds to a convex polygon G x with vertices in S. We associate to (P, c, w) MD 3 its mixed tag Tag(P, c, w) defined as G c G P (w) D D and show that mixed tags of distinct marked polynomials from MD 3 are either disjoint or coincide. Let Tag(MD 3 ) + = D 3 Tag(P, c, w) D D. The sets Tag(P, c, w) form a partition of Tag(MD 3 ) + and generate the corresponding quotient space of Tag(MD 3 ) + denoted by CML. We prove that Tag : MD 3 CML is continuous and thus CML can serve as a combinatorial model space for MD INTRODUCTION We assume basic knowledge of complex dynamics and use standard notation (for precise definitions see Section 2). Let P be a polynomial of degree Date: May 28, Mathematics Subject Classification. Primary 37F20; Secondary 37F10, 37F50. Key words and phrases. Complex dynamics; laminations; Mandelbrot set; Julia set. The first and the third named authors were partially supported by NSF grant DMS The fourth named author was partially supported by RFBR grant , RScF grant , and AG Laboratory NRU-HSE, MESRF grant ag. 11.G

2 2 A. BLOKH, L. OVERSTEEGEN, R. PTACEK, AND V. TIMORIN d with connected Julia set J(P ). A crucial role in studying its dynamics is played by the conformal isomorphism Φ P between C \ D and the complement U of the filled Julia set K(P ) asymptotic to the identity at infinity. The images of radial rays under Φ P are called external rays of P and serve as one of the main tools. By a theorem of Carathéodory, if J(P ) is locally connected (and hence connected), then Φ P can be extended to a continuous map Φ P : C \ D U, under which S maps onto J(P ). By Thurston [Thu85], the map P J(P ) is then conjugate to a self-mapping f P, induced by z d S, of the quotient space J P of S = Bd(D) under the equivalence P which identifies points of S if and only if Φ P sends them to the same point of J(P ). Denote the quotient map from S to S/ P by π P. The equivalence relation P is called the lamination generated by P while J P is called the topological Julia set of P and the map f P is called the topological polynomial (of P ). Thus, in this case P on J(P ) is conjugate to its topological polynomial. If for each P -class one takes its convex hull in D and then considers the union of all edges of all such convex hulls, then the collection L P of these chords is called the geolamination (generated by P ). Geolaminations give a geometric interpretation for laminations and topologize laminations, reflecting limit transitions among them. Both laminations and their geolaminations can be defined intrinsically (without using polynomials). In that case some geolaminations will not directly correspond to an equivalence relation on S but the family of all geolaminations will become closed. This allows one to work with limits of geolaminations and therefore study limits of polynomials (which do not necessarily have locally connected Julia sets). Thurston [Thu85] models polynomials by laminations and their geolaminations. By extension, he models families of polynomials by families of geolaminations. In the quadratic case this yields a certain tagging of laminations and geolaminations so that the corresponding family of tags (called minors) is itself a geolamination generating a special lamination called the quadratic minor lamination and denoted QML. This gives rise to the quotient space S/QML of S modeling the boundary of the Mandelbrot set M 2, i.e. of the family of all polynomials z 2 + c with connected Julia set (also called the quadratic connected locus). The induced quotient space of D serves as a combinatorial model for M 2. In short, one can say that parameterization of families of (geo)laminations yields combinatorial models of the corresponding families of polynomials. In this paper we follow similar logic and suggest an approach to describing models of certain families of cubic polynomials contained in the cubic connectedness locus. Call polynomials with only repelling cycles dendritic. Dendritic polynomial have a rich combinatorial and topological structure of external rays

3 COMBINATORIAL MODELS 3 with rational arguments landing at the same point and forming planar cuts. The family of collections of arguments of all such finite collections of rays is considered by Jan Kiwi in [Kiw04]. Its existence is largely responsible for the fact that the notion of the lamination P of P introduced above for polynomials with locally connected Julia sets can be extended to the dendritic case. This is due to Kiwi [Kiw04] who proved that in that case P J(P ) is monotonically semiconjugate by a map Ψ P to its topological polynomial f P on its topological Julia set J P which in this case is a dendrite (explaining our terminology). Moreover, by Kozlovskii-van Strien [KvS06] in non-renormalizable cases J(P ) is locally connected. We aim at showing that these individual nice properties result in nice properties of families of cubic dendritic polynomials so that the family of all dendritic polynomials with connected Julia sets can be modeled by a suitable combinatorial space. However first consider the classic quadratic case. Let P c (z) = z 2 + c be a dendritic polynomial with connected Julia set (the parameter values c, for which all these assumptions are fulfilled, are also called dendritic). Then c J(P c ), and by [Kiw04] the set of angles given by π 1 Pc (Ψ Pc (c)) is finite. Denote the convex hull of this set by G c. An important result, going back to Thurston, implies that G c and G c are either equal or disjoint for any two dendritic parameters c and c. Moreover, the mapping c G c is upper semicontinuous in a natural sense (if a sequence of dendritic parameters c n converges to a dendritic parameter c, then the limit set of the corresponding convex sets G cn is a subset of G c ). We call G c the tag associated to c. It follows that the set of dendritic parameters projects continuously to the quotient space of the union of all tags defined by the partition of this union into individual tags. We generalize these results to the cubic case as an application of new combinatorial machinery introduced here (in fact, many of our tools are developed for polynomials of any degree). Also, we adopt a new approach to the choice of tags. To explain our approach, we need the following notion. Let P be a dendritic polynomial of any degree with connected Julia set. Then, given a point z J(P ), we associate to it the convex hull G P,z of the set π 1 P (Ψ P (z)) (if P is fixed and the situation is not ambiguous we will write G z instead of G P,z ). In other words, we project z to the point Ψ P (z) of the model topological Julia set J P and then consider all angles from S associated to Ψ P (z) in the sense of the lamination P. The set G z is a finite polygon, a chord, or a point; it will be called the set associated to z and should be viewed as a combinatorial object corresponding to z. For any points z w J(P ), the sets G z and G w either coincide or are disjoint. We need the concept of a (critically) marked cubic polynomial, i.e. a triple (P, c, w) where P is a cubic polynomial with critical points c and w

4 4 A. BLOKH, L. OVERSTEEGEN, R. PTACEK, AND V. TIMORIN (perhaps, c = w). If c w, then the triple (P, c, w) and the triple (P, w, c) are viewed as two distinct marked cubic polynomials. Let MD 3 be the space of all marked cubic dendritic polynomials. A co-critical point associated to a critical point τ of a cubic polynomial P is the only point τ τ such that P (τ ) = P (τ) (if the critical points of P are distinct) or τ itself (if P has a unique critical point). Then to every marked dendritic polynomial (P, c, w) we associate the corresponding mixed tag Tag(P, c, w) = G c G P (w) D D. This defines the mixed tag Tag(P, c, w) for all marked cubic dendritic polynomials. Let Tag(MD 3 ) + = MD 3 Tag(P, c, w) D D. The sets Tag(P, c, w) form a partition of Tag(MD 3 ) + and generate the corresponding quotient space of Tag(MD 3 ) + denoted by CML (for cubic mixed lamination). We prove that Tag : MD 3 CML is continuous and thus CML can serve as a combinatorial model for MD 3. Theorem A. Mixed tags of marked polynomials from MD 3 are disjoint or coincide. The map Tag : MD 3 CML is continuous. Thus, there is a continuous function from the space of marked cubic dendritic polynomials to the model space of their tags defined through (geo)- laminations associated with marked polynomials from MD 3. This can be viewed as a partial generalization of [Thu85] to cubic polynomials. 2. GEOLAMINATIONS AND THEIR PROPERTIES In this section we give basic definitions, list some known results on geolaminations, and establish some new facts about them Basic definitions. For a collection R of chords of D set R = R +. A geolamination is a collection L of (perhaps degenerate) chords of D called leaves which are pairwise disjoint in D such that L + = l L l is closed, and all points of S are elements of L. We linearly extend σ d over leaves of L; clearly, this extension is continuous and well-defined Sibling invariant geolaminations. Let us introduce the notion of a (sibling) σ d -invariant geolamination which is a slight modification of an invariant geolamination introduced by Thurston. Definition 2.1 (Invariant geolaminations [BMOV13]). A geolamination L is (sibling) σ d -invariant provided: (1) for each l L, we have σ d (l) L, (2) for each l L there exists l L so that σ d (l ) = l. (3) for each l L such that σ d (l) is a leaf, there exist d pairwise disjoint leaves l 1,..., l d in L so that l = l 1 and σ d (l i ) = σ d (l) for all i = 1,..., d.

5 COMBINATORIAL MODELS 5 We call the leaf l in (2) a pullback of l and the leaves l 2,..., l d in (3) siblings of l. In a broad sense a sibling of l is a leaf with the same image but distinct from l. Definition 2.1 is slightly more restrictive than Thurston s definition of an invariant geolamination. By [BMOV13], a σ d - invariant geolamination L is invariant in the sense of Thurston [Thu85] and, in particular, gap invariant: if G is a gap of L and H is the convex hull of σ d (G S), then H is a point, a leaf of L, or a gap of L, and in the latter case, the map σ d Bd(G) : Bd(G) Bd(H) of the boundary of G onto the boundary of H is the positively oriented composition of a monotone map and a covering map. From now on by (σ d -)invariant geolaminations we mean sibling σ d -invariant geolaminations and consider only such geolaminations. Theorem 2.2 (Theorem 3.21 [BMOV13]). The family of sets L + of all invariant geolaminations L is closed in the Hausdorff metric. Clearly, L + i L + (understood as convergence of compact subsets of D) implies that the collections of chords L i converge to the collection of chords L (i.e., each leaf of L is the limit of a sequence of leaves from L i, and each converging sequence of leaves of L i converges to a leaf of L). Thus, from now on we will write L i L if L + i L + in the Hausdorff metric. Two distinct chords of D are called linked if they intersect in D (we will also say that these chords cross each other). A gap G is called infinite (finite, uncountable) if G S is infinite (finite, uncountable). Uncountable gaps are also called Fatou gaps. By the degree of a gap G we mean the number m of components in the full preimage in G of a point of σ d (G) S except when a gap (leaf) G collapses to a point in which case the degree of G is the number of points in G S. Often we then call G an m-to-1 gap. Definition 2.3. We say that a is a chord of a geolamination L when a is unlinked with all leaves of L. A critical chord (leaf) ab of L is a chord (leaf) of L such that σ d (a) = σ d (b). A gap is all-critical if all its edges are critical. A gap G is said to be critical if the degree of G is greater than one. A critical set is either a critical leaf or a critical gap. By Thurston [Thu85], there is a canonical barycentric extension of the map σ d to the entire closed disk D. First σ d is extended linearly over all leaves of an invariant geolamination L. Then one defines a map on barycenters of gaps which map onto one another if the boundaries of gaps map onto one another, and then one uses a standard construction using cones in order to extend the map over all interiors of gaps of L. When talking about σ d on D, we always have some invariant geolamination in mind and mean Thurston s barycentric extension described above.

6 6 A. BLOKH, L. OVERSTEEGEN, R. PTACEK, AND V. TIMORIN Laminations as equivalence relations. A lot of geolaminations naturally appear in the context of invariant equivalence relations on S (laminations) satisfying special conditions. Definition 2.4 (Laminations). An equivalence relation on the unit circle S is called a lamination if either S is one -class (such laminations are called degenerate), or the following holds: (E1) the graph of is a closed subset of S S; (E2) the convex hulls of distinct equivalence classes are disjoint; (E3) each equivalence class of is finite. Definition 2.5 (Laminations and dynamics). An equivalence relation is called (σ d -)invariant if: (D1) is forward invariant: for a -class g, the set σ d (g) is a -class; (D2) for any -class g, the map σ d : g σ d (g) extends to S as an orientation preserving covering map such that g is the full preimage of σ d (g) under this covering map. For an invariant lamination consider the topological Julia set S/ = J and the topological polynomial f : J J induced by σ d. The quotient map π : S S/ = J semi-conjugates σ d with f J. A lamination admits a canonical extension over C: classes of this extension are either convex hulls of classes of, or points which do not belong to such convex hulls. By Moore s Theorem the quotient space C/ is homeomorphic to C. The quotient map π : S S/ extends to the plane with the only non-trivial point-preimages (fibers) being the convex hulls of non-degenerate -classes. One can extend f to a branched-covering map f : C C of degree d called a topological polynomial too. The complement K of the unique unbounded component U (J ) of C \ J is called the filled topological Julia set. For a closed convex set H, straight segments from Bd(H) are called edges of H. Define the canonical geolamination L generated by as the collection of edges of convex hulls of all -classes and all points of S. By [BMOV13], the geolamination L is σ d -invariant Other useful notions. Considering objects that may appear in laminations, we do not have to fix the laminations containing these objects. Definition 2.6. By a periodic gap or leaf we mean a gap or leaf G for which there exists the least number n (called the period of G) such that σd n(g) = G. Then we call the map σn d : G G the remap. Given two points a, b S we denote by (a, b) the positively oriented arc from a to b (i.e., moving from a to be b within (a, b) takes place in the counterclockwise direction). For a closed set G S we call components of S \ G holes. If l = ab is an edge of G = CH(G ), then we denote by

7 COMBINATORIAL MODELS 7 H G (l) the component of S \ {a, b} disjoint from G and call it the hole of G behind l. The relative interior of a gap is its interior in the plane; the relative interior of a segment is the segment minus its endpoints. Definition 2.7. If A S is a closed set such that all the sets CH(σd i (A)) are pairwise disjoint, then A is called wandering. If there exists n 1 such that all the sets CH(σd i (A)), i = 0,..., n 1 have pairwise disjoint relative interiors while σd n (A) = A, then A is called periodic of period n. If there exists m > 0 such that all CH(σd i (A)), 0 i m + n 1 have pairwise disjoint relative interiors and σd m (A) is periodic of period n, then we call A preperiodic of period n and preperiod m. If A is wandering, periodic or preperiodic, and for every i 0 and every hole (a, b) of σd i (A) either σ d (a) = σ d (b), or the positively oriented arc (σ d (a), σ d (b)) is a hole of σ i+1 d (A), then we call A (and CH(A)) a (σ d )-laminational set; we call CH(A) finite if A is finite. A (σ d -)stand alone gap is defined as a laminational set with non-empty interior. Denote by < the positive (counterclockwise) circular order on S = R/Z induced by the usual order of R. Note that this order is only meaningful for sets of cardinality at least three. For example, we say that x < y < z provided that moving from x in the positive direction along S we meet y before meeting z. Definition 2.8 (Order preserving). Let X S be a set with at least three points. We call σ d order preserving on X if σ d X is one-to-one and, for every triple x, y, z X with x < y < z, we have σ d (x) < σ d (y) < σ d (z) General properties of geolaminations. Lemma 2.9 (Lemma 3.7 [BMOV13]). If ab and ac are two leaves of a geolamination L such that σ d (a), σ d (b) and σ d (c) are all distinct points then the order among points a, b, c is preserved under σ d. We prove a few corollaries of Lemma 2.9 Lemma If L is a geolamination, l = ab is a leaf of L and the point a is periodic, then b is (pre)periodic of the same period. Proof. We may assume that a is fixed but b is not fixed. Then, by Lemma 2.9, the circular order among the points b i = σd i (b) is the same as the order of subscripts unless b i = b i+1 for some i. It follows that b i converge to some limit point, a contradiction with the expansion property of σ d. We will need the following elementary lemma. Lemma Suppose that x S is such that the chords σ i d (x)σi+1 d (x), i = 0, 1,... are pairwise unlinked. Then x (and hence the leaf xσ d (x) = l) is (pre)periodic.

8 8 A. BLOKH, L. OVERSTEEGEN, R. PTACEK, AND V. TIMORIN Proof. The sequence of leaves from the lemma is the σ d -orbit of l in which consecutive images are concatenated and no two leaves are linked. If for some i the leaf σd i (x)σi+1 d (x) = σd i (l) is critical then σi+1 d (l) = {σ i+1 d (x)} is a σ d -fixed point, which proves the claim in this case. Assume now that l is not (pre)critical. If x is not (pre)periodic, then, by topological considerations, leaves σd n (l) must converge to a limit leaf (or point) n. Clearly, n is σ d -invariant. However σ d is expanding, a contradiction. Lemma 2.11 easily implies Lemma Lemma Let L be a geolamination. Then the following holds. (1) If l is a leaf of L and for some n > 0 the leaf σd n (l) is concatenated to l then l is (pre)periodic. (2) If l has a (pre)periodic endpoint then l is (pre)periodic. (3) If two leaves l 1, l 2 from geolaminations L 1, L 2 share the same (pre)periodic endpoint then they are (pre)periodic with the same eventual period of their endpoints. Proof. Let l = uv. First, assume that σd n (u) = u. Then (1) follows from Lemma Second, assume that σd n (u) = v. Then (1) follows from Lemma Statements (2) and (3) follow from (1) and Lemma A similar conclusion can be made for edges of periodic gaps. Lemma Any edge of a periodic gap is (pre)periodic or (pre)critical. Proof. Let G be a fixed gap and l be its edge which is not (pre)critical. The length s n of the hole H G (σd n(l)) of G behind the leaf σn d (l) grows with n as long as s n stays sufficiently small (it is easy to see that the correct bound on s n is that s n < 1 ). This implies that the sequence {s d+1 i} will contain infinitely many numbers greater than or equal to 1. Since there are only d+1 1 finitely many distinct holes of G of length greater or bigger, this implies d+1 that l is (pre)periodic. Given v S, let E(v) be the set of all endpoints u of leaves uv of L (if E(v) accumulates on v, then we include v into E(v)). Lemma If v is not (pre)periodic then E(v) is at most finite. If v is (pre)periodic then E(v) is at most countable. Proof. The first claim is proven in [BMOV13, Lemma 4.7]. The second claim follows from Lemma Properties of individual wandering polygons were studied in [Kiw02]; properties of collections of wandering polygons were studied in [BL02]; their existence was established in [BO08]. The most detailed results on wandering polygons and their collections are due to Childers [Chi07].

9 COMBINATORIAL MODELS 9 Let us describe the entire σ d -orbit of a finite periodic laminational set. Proposition Let P be a σ d -periodic finite laminational set and X be the union of the forward images of P. Then, for every connected component R of X, there is an m-tuple of points a 0 < a 1 < < a m 1 < a m = a 0 in S such that R consists of eventual images of P containing a i a i+1 for i = 0,..., m 1. If m > 1, then the remap of R is a combinatorial rotation sending a i to a i+1. Note that the case m = 1 is possible. In this case, R consists of several images of P sharing a common vertex a 0. Observe that there is a natural cyclic order among the images of P. The remap of R is some cyclic permutation of these images, not necessarily a combinatorial rotation. Proof. Set P k = σd k(p ). Let k be the smallest positive integer such that P k intersects P 0 ; we may suppose that P k P 0. There is a vertex a 0 of P 0 such that a 1 = σd k(a 0) is also a vertex of P 0. Clearly, both a 1 and a 2 = σd k(a 1) are vertices of P k. Set a i = σd ki(a 0). Then we have a m = a 0 for some m > 0. Let Q be the convex hull of the points a 0,..., a m 1. This is a convex polygon, or a chord, or a point. If m > 1, then a i and a i+1 are the endpoints of the same edge of Q (otherwise the polygons P ki would cross in D). Set R = m 1 i=0 P ki. If m = 1, then the sets P ki share the vertex a 0. If m > 1, then it follows from the fact that the boundary of Q is a simple closed curve that every chord a i a i+1 is an edge of P ki shared with Q, sets P ki are disjoint from the interior of Q, and the remap σd k of R is a combinatorial rotation acting transitively on the vertices of Q. To prove that R is disjoint from R j = σ j d (R) for j < k suppose that R j intersects some P ki. Note that the map σd k fixes both R and R j. It follows that R j intersects all P ki, hence contains Q, a contradiction. It is well-known [Kiw02] that any infinite gap G of a geolamination L is (pre)periodic. By a vertex of a gap or leaf G we mean any point of G S. Lemma Let G be a periodic gap of period n and set K = Bd(G). Then σd n K is either a monotone map of the Jordan curve K onto itself, or the composition of a covering map and a monotone map of K. If σd n K is of degree one then either (1) or (2) holds. (1) The gap G has finitely many σd n -periodic edges or vertices. Between any pair of adjacent σd n -periodic points a and b of K, there are one or more concatenated σd n -critical edges at a or b (say, at a) which collapse to a. The gap G has countably many vertices. (2) The map σd n K is monotonically semiconjugate to an irrational circle rotation so that each fiber of this semiconjugacy is a finite concatenation of (pre)critical edges of G.

10 10 A. BLOKH, L. OVERSTEEGEN, R. PTACEK, AND V. TIMORIN Proof. We will prove only the very last claim. Denote by φ the semiconjugacy from (2). Let T K be a fiber of φ. By Lemma 2.13 all edges of G are (pre)critical. Hence if T contains infinitely many edges, then the forward images of T will hit critical leaves of σd n infinitely many times as T cannot collapse under a finite power of σd n. This would imply that an irrational circle rotation has periodic points, a contradiction. Lemma 2.16 implies Corollary Corollary Suppose that G is a periodic gap of a geolamination L, whose remap has degree one. Then at most countably many pairwise unlinked leaves of other geolaminations can be located inside G. Proof. Any chord located inside G has its endpoints at vertices of G. Since in case (1) of Lemma 2.16 there are countably many vertices of G, we may assume that case (2) of Lemma 2.16 holds. Applying the semiconjugacy φ from this lemma we see that if a leaf l is located in G and its endpoints do not map to the same point by φ, then l will eventually cross itself. If there are uncountably many leaves of geolaminations inside G then among them there must exist a leaf l with endpoints in distinct fibers of φ. By the above some forward images of l cross each other, a contradiction Geolaminations with qc-portraits. Here we define geolaminations with quadratically critical (qc-)portraits and discuss linked or essentially coinciding geolaminations with qc-portraits. First we motivate our approach. Thurston defines the minor m of a σ 2 -invariant lamination as the image of a longest leaf M of L. The leaf M is called a major of L. If m is nondegenerate, L has two majors which both map to m. In the quadratic case the majors are uniquely determined by the minor. Even though in the cubic case one could define majors and minors similarly, unlike in the quadratic case these minors do not uniquely determine the corresponding majors. The simplest way to see that is to consider distinct pairs of critical leaves with the same images. One can choose two all-critical triangles with socalled aperiodic kneadings as defined by Kiwi in [Kiw04]. By [Kiw04], this would imply that any choice of two disjoint critical leaves, one from either triangle, will give rise to the corresponding geolamination; clearly, these two geolaminations are very different even though they have the same images of their critical leaves, i.e., the same minors. Thus, we should be concerned with critical sets, not only their images. We study how ordered collections of critical sets of geolaminations are located with respect to each other. The fact that critical sets may have different degrees complicates such study. So, it is natural to adjust our geolaminations to make sure that the critical sets of two geolaminations which

11 COMBINATORIAL MODELS 11 are associated to one another are of the same type. As such we choose (generalized) critical quadrilaterals. Definition A (generalized) critical quadrilateral Q is the circularly ordered 4-tuple [a 0, a 1, a 2, a 3 ] of marked points a 0 a 1 a 2 a 3 a 0 in S so that a 0 a 2 and a 1 a 3 are critical chords (called spikes). In other words, critical quadrilaterals [a 0, a 1, a 2, a 3 ], [a 1, a 2, a 3, a 0 ], [a 2, a 3, a 0, a 1 ] and [a 3, a 0, a 1, a 2 ] are viewed as equal. We want to comment upon our notation. By (X 1,..., X k ) we always mean a k-tuple, i.e. an ordered collection of elements X 1,..., X k. On the other hand by {X 1,..., X k } we mean a collection of elements X 1,..., X k with no fixed order. Since, for critical quadrilaterals, we need to emphasize the circular order among its vertices, we choose the notation [a 0, a 1, a 2, a 3 ] distinct from either of the two just described notations. For brevity, we will often use the expression critical quadrilateral when talking about the convex hull of a critical quadrilateral. In fact, it is easy to see that if all vertices of a critical quadrilateral are distinct, then the quadrilateral is uniquely defined by its convex hull. However, if the convex hull of a critical quadrilateral is a triangle, this is no longer true. Indeed, let T = CH(a, b, c) be an all-critical triangle. Then [a, a, b, c] is a critical quadrilateral, but so are [a, b, b, c] and [a, b, c, c]. A collapsing quadrilateral is a critical quadrilateral, whose σ d -image is a leaf. A critical quadrilateral Q has two intersecting spikes and is either a collapsing quadrilateral, a critical leaf, an all-critical triangle, or an allcritical quadrilateral. If all its vertices are pairwise distinct, we call Q nondegenerate, otherwise Q is called degenerate. Vertices a 0 and a 2 (a 1 and a 3 ) are called opposite. Considering geolaminations, all of whose critical sets are critical quadrilaterals, is not very restrictive: we can tune a given geolamination by inserting new leaves into its critical sets in order to construct a new geolamination with all critical sets being critical quadrilaterals. Lemma The family of all critical quadrilaterals is closed. The family of all critical quadrilaterals that are critical sets of geolaminations is closed too. Proof. The first claim is trivial. The second one follows from Theorem 2.2 and the fact that if L i L, then the critical quadrilaterals of geolaminations L i converge to critical quadrilaterals that are critical sets of L. In the quadratic case, Thurston s parameterization [Thu85] associates to a geolamination the image of its critical set. If the images of the critical sets of two geolaminations intersect, then some edges of these images intersect. Consider these intersecting edges (minors); their full pullbacks are collapsing quadrilaterals (these quadrilaterals are convex hulls of pairs of majors

12 12 A. BLOKH, L. OVERSTEEGEN, R. PTACEK, AND V. TIMORIN and are contained in the original critical sets). In that case, the mutual location of the two collapsing quadrilaterals is rather specific: their vertices alternate on the circle. This motivates Definition Definition Let A and B be two quadrilaterals. Say that A and B are strongly linked if the vertices of A and B can be numbered so that a 0 b 0 a 1 b 1 a 2 b 2 a 3 b 3 a 0 where a i, 0 i 3, are vertices of A and b i, 0 i 3 are vertices of B. Equivalently, A and B are strongly linked if no hole of either quadrilateral contains more than one vertex of the other one. Strong linkage is a closed condition: if two variable critical quadrilaterals are strongly linked and converge, then they must converge to two strongly linked critical quadrilaterals. An obvious case of strong linkage is between two non-degenerate critical quadrilaterals whose vertices alternate on the circle so that all the inequalities in Definition 2.20 are strict. Yet even if both critical quadrilaterals are non-degenerate, some inequalities may be nonstrict which means that some vertices of both quadrilaterals may coincide. For example, two coinciding critical leaves can be viewed as strongly linked critical quadrilaterals, or an all-critical triangle A with vertices x, y, z and its edge B = yz can be viewed as strongly linked quadrilaterals if the vertices are chosen as follows: a 0 = x, a 1 = a 2 = y, a 3 = z and b 0 = b 1 = y, b 2 = b 3 = z. If a critical quadrilateral Q is a critical leaf or has all vertices distinct, then Q as a critical quadrilateral has a well-defined set of vertices; the only ambiguous case is when Q is an all-critical triangle. To study collections of critical quadrilaterals we need a few notions and a lemma. If a few chords can be concatenated to form a Jordan curve, or if there are two identical chords, then we say that they form a loop. In particular, one chord does not form a loop while two equal chords do. If an ordered collection of chords (l 1,..., l k ) contains no chords forming a loop we call it a no loop collection. Lemma A family of no loop collections of critical chords is closed. Proof. Suppose that there is a sequence of no loop collections of critical chords N i = (l i 1,..., l i s) such that N i N = (l 1,..., l s ) where all chords l i are critical. We need to show that N is a no loop collection. By way of contradiction assume that, say, chords l 1 = a 1 a 2,..., l k = a k a 1 form a loop in which the order of points a 1,..., a k is positive. We claim that this loop cannot be the limit of no loop collections of critical chords; clearly this contradicts the convergence assumption that N i N. Loosely, we base the argument upon the fact that if G S is a union of finitely many sufficiently small circle arcs such that all edges of the convex hull

13 COMBINATORIAL MODELS 13 G = CH(G ) are critical, then in fact all circle arcs in G are degenerate, so that G is a finite polygon. A more formal proof follows. Consider chords l i 1 = b i 1d i 1,..., l i k = bi k di k such that points bi j converge to a j and points d i j converge to a j+1 (j + 1 is understood here and in the rest of the argument modulo k) as i. Then for a well-defined collection of integers m 1,..., m k we have that d i j = b i j + m j 1 d. Since N i is a no loop collection and all leaves in N i are unlinked, for each 1 j k we have d i j b i j+1 and there exists at least one 1 j k such that d i j < b i j+1. Adding up these inequalities we see that b i 1 < d i k, a contradiction. Given a collection of d 1 critical quadrilaterals, we can choose one spike in each of them and call this collection of d 1 critical chords a complete sample of spikes. Call a no loop collection of d 1 critical chords a full collection. If L is a geolamination which corresponds to a lamination and all its critical sets are critical quadrilaterals, then any complete sample of spikes is a full collection. This inspires yet another definition. Definition 2.22 (Quadratic criticality). Let (L, QCP) be a geolamination with a (d 1)-tuple QCP of critical quadrilaterals that are gaps or leaves of L such that any complete sample of spikes is a full collection. Then QCP is called a quadratically critical portrait (qc-portrait) for L while the pair (L, QCP) is called a geolamination with qc-portrait (if the appropriate geolamination L for QCP exists but is not emphasized we simply call QCP a qc-portrait). The space of all qc-portraits is denoted by QCP d. The family of all geolaminations with qc-portraits is denoted by LQCP d. If c 1,..., c d 1 is a complete sample of spikes and C is a component of D \ c i then σ d restricted to the boundary of C is one-to-one except for critical chords contained in the boundary of C. Corollary The spaces QCP d and LQCP d are compact. Proof. Let (L i, QCP i ) (L, C); by Theorem 2.2 here in the limit we have an invariant geolamination L and an ordered collection C of d 1 critical quadrilaterals (the latter follows from Lemma 2.19). Let C = (C j ) d 1 j=1 be the limit critical quadrilaterals. Choose a collection of spikes l j of quadrilaterals of C. Suppose that there is a loop formed by some of these spikes. Then by construction there exist collections of spikes from qc-portraits QCP i converging to (l 1,..., l d 1 ). Since by definition these are full collections of critical chords, this contradicts Lemma Hence (l 1,..., l d 1 ) is a full collection of critical chords too which implies that C is a qc-portrait for L and proves that QCP d and LLP d are closed spaces. The following lemma describes geolaminations admitting a qc-portrait.

14 14 A. BLOKH, L. OVERSTEEGEN, R. PTACEK, AND V. TIMORIN Lemma A geolamination L has a qc-portrait if and only if all its critical sets are collapsing quadrilaterals, critical leaves or all-critical gaps. Proof. If L has a qc-portrait, then the claim of the lemma follows by definition. Assume now that all critical sets of L are collapsing quadrilaterals and all-critical sets. Then L may have several critical leaves (some of them are edges of all-critical gaps, some are edges of other gaps, some are not edges of any gaps at all). We can always choose a maximal by cardinality family of critical leaves of L without loops. Also, select each critical quadrilateral which is not all-critical. Include all selected sets in the family of pairwise distinct sets C = (C 1,..., C m ) consisting of critical leaves and critical quadrilaterals which are not all-critical. All selected sets are critical quadrilaterals with well-defined vertices and hence well-defined spikes. We claim that C is a qc-portrait. To this end we need to show that m = d 1 and that any collection N of spikes of sets from C is a no loop collection. First let us show that any such collection N contains no loops. Indeed, suppose that N contains a loop l 1 C 1,..., l r C r. By construction we may assume that, say, C 1 = [a, x, b, y] is a critical quadrilateral which is not all-critical and l 1 = ab is contained in the interior of C 1 except for points a and b. The spikes l 2,..., l k 1 form a chain of concatenated critical chords which has, say, b as its initial point and a as its terminal point. Since these spikes come from sets C 2,..., C r distinct from C 1, they have to pass through either x or y as a vertex, a contradiction. Thus, N contains no loops which implies that the number m of chords in N is at most d 1. Assume now that m < d 1 and bring it to a contradiction. Indeed, if m < d 1 then we can find a component U of D \ N + on which σ d is not monotone. We claim that there exists a critical chord l of L inside U which does not connect points in Bd(U) otherwise connected by a chain of critical edges in Bd(U). Consider all arcs A Bd(U) such that σ d is not monotone on A, and the endpoints of A are connected by a leaf of L. Call such arcs non-monotone. Non-monotone arcs exist: by the assumptions there exist leaves l of L inside U, and at least one of the two arcs in the boundary of U which connects the endpoints of l must be non-monotone. The intersection of a decreasing sequence of non-monotone arcs is a closed arc with endpoints connected with a leaf of L such that either this leaf is the desired critical leaf of L, or the intersection arc is still non-monotone. Thus, it is enough to show that if there exists the minimal by inclusion nonmonotone arc A 0 then there exists the desired critical chord of L. Let l 0 be the leaf connecting the endpoints of A 0. Then A 0 l 0 is a Jordan curve enclosing a Jordan disk T, and A 0 cannot be a union of spikes. If l 0 is critical we are done. Otherwise by the assumption l 0 cannot be approached by leaves of L from within T, thus l 0 is an edge of a gap G T such that

15 COMBINATORIAL MODELS 15 on all holes of G contained in A 0 the map σ d is monotone. Let us show that G must be critical. Indeed, suppose otherwise. Then σ d maps G onto σ d (G) in a one-to-one fashion, and σ d (G) is a gap such that each component of U \ G which shares an edge, say, m with G, has the boundary which maps onto the hole of σ d (G) located behind σ d (m) united with σ d (m). This shows that σ d A0 is monotone, a contradiction. Hence G is critical and we can choose a critical chord l G. Clearly, l does not connect points in Bd(U) otherwise connected by a chain of critical edges in Bd(U). If l is a leaf of L then l can be added to C without creating a loop. This contradicts the choice of the collection C which contains maximal by cardinality collection of critical leaves of L without loops. Hence l is a chord inside a gap G of L. By the previous paragraph, neither component of D \ l can contain a chain of critical leaves from C. Consider two cases. Suppose first that G is all-critical. Then by the above either component of Bd(G) \ l contains at least one edge of Bd(G) not included in N (and hence not included in C either). Consider all such edges of G and show that at least one of them can be added to C without creating loops. Indeed, suppose that adding to C any edge l of G which is not yet a set from C creates a loop in C. This would mean that there already exists a chain of leaves which are sets from C connecting the endpoints of l. Since by the assumption this holds for all edges of G, then in fact there is already a loop of critical leaves from C, a contradiction. On the other hand, if G is not all-critical then it is a collapsing quadrilateral which cannot be contained in U, a contradiction. Thus, m < d 1 leads to a contradiction in all cases and hence m = d 1 as desired. Observe that there might exist several qc-portraits for L from Lemma For example, consider a σ 4 -invariant geolamination L with two all-critical triangles T 1 = CH(a, b, c), T 2 = CH(a, c, d) sharing an edge l = ac. The proof of Lemma 2.24 leads to a qc-portrait consisting of any three edges of T 1, T 2 not equal to l in some order (recall that for each critical leaf its structure as a quadrilateral is unique). However it is easy to check that the collection ([a, b, b, c], [a, a, c, c], [a, c, d, d]) is a qc-portrait too. Notice, that in the definition of a complete sample of spikes we do not allow to use more than one spike from each critical set, hence the fact that the same spike appears twice in [a, a, c, c] does not result into a loop. Given a qc-portrait QCP, any complete sample of spikes is a full collection of critical chords. If QCP includes sets which are not leaves, there are several complete samples of spikes as the choice of spikes is ambiguous. This is important for Subsection 4.1 where we introduce and study the socalled smart criticality and its applications to linked geolaminations with qc-portraits introduced below. First we need a technical definition.

16 16 A. BLOKH, L. OVERSTEEGEN, R. PTACEK, AND V. TIMORIN Definition A critical cluster of L is by definition a convex subset of D, whose boundary is a union of critical leaves. A critical leaf is itself a critical cluster. For example, consider the case discussed after Lemma There, a σ 4 -invariant geolamination L has two all-critical triangles sharing a critical edge; the union of these triangles is a critical cluster of L. Definition 2.26 (Linked geolaminations). Let L 1 and L 2 be geolaminations with qc-portraits QCP 1 = (C1) i d 1 i=1 and QCP 2 = (C2) i d 1 i=1 and a number 0 k d 1 such that: (1) for every i with 1 i k, the sets C1 i and C2 i are either strongly linked critical quadrilaterals or share a spike; (2) for each j > k the sets C j 1 and C j 2 are contained in a common critical cluster of L 1 and L 2 (in what follows these clusters will be called special critical clusters and leaves contained in them will be called special critical leaves). Then we use the following terminology: (a) if in (1) for every i with 1 i k, the quadrilaterals C1 i and C2 i share a spike we say that QCP 1 and QCP 2, (as well as (L 1, QCP 1 ) and (L 2, QCP 2 )) coincide in essence (or essentially coincide, or are essentially equal), (b) if in (1) there exists i with 1 i k such that the quadrilaterals C1 i and C2 i are strongly linked and do not share a spike, we say that QCP 1 and QCP 2 (as well as (L 1, QCP 1 ) and (L 2, QCP 2 )) are linked. The critical sets C1 i and C2, i 1 i d 1 are called associated (critical sets of geolaminations with qc-portraits (L 1, QCP 1 ) and (L 2, QCP 2 )) Some special types of geolaminations. Below, we discuss perfect geolaminations and dendritic geolaminations Perfect geolaminations. A geolamination L is perfect if no leaf of L is isolated. Every geolamination contains a maximal perfect sublamination (clearly, this sublamination contains all degenerate leaves). Indeed, consider L as a metric space of leaves with the Hausdorff metric and denote it by L. Then L is a compact metric space which has a maximal perfect subset L c called the perfect sublamination of L. The process of finding L c was described in detail in [BOPT14]. Lemma 2.27 is left to the reader. Lemma The collection L c is an invariant perfect geolamination. For every l L c and every neighborhood U of l there exist uncountably many leaves of L c in U.

17 COMBINATORIAL MODELS 17 Observe that if there are three leaves of L c which share a vertex then, since by Lemma 2.14 there are at most countably many leaves of L c emanating from one point, we see that L c has isolated leaves, a contradiction. Thus there are at most two leaves of L c coming out of one point. Therefore, any leaf of L c is a limit of an uncountably many leaves of L c disjoint from l. If l is critical, this implies that σ d (l) is a point separated from the rest of the circle by images of those leaves. Thus, a critical leaf l is either disjoint from all other leaves or gaps of L c or is an edge of an all-critical gap of L c disjoint from all other leaves or gaps of L c. Together with the fact, that for any point at most two leaves come out of it, this implies Lemma Lemma Let L be a perfect geolamination. Then the critical sets of L are pairwise disjoint and are either critical leaves, or all-critical gaps, or critical sets which map exactly k-to-1, k > 1, onto their images Dendritic geolaminations with critical patterns. The main applications of our results will concern dendritic laminations defined below. Definition A lamination and its geolamination L are called dendritic if the topological Julia set J is a dendrite. The family of all dendritic geolaminations is denoted by LD d. Lemma 2.30 is well-known. Lemma Dendritic geolaminations L are perfect. Dendritic geolaminations are closely related to polynomials. Let D be the space of all polynomials with only repelling periodic points and D d be the space of all such polynomials of degree d. By Jan Kiwi s results [Kiw04] if a polynomial P with connected Julia set J(P ) has no Siegel or Cremer periodic points (i.e., irrationally indifferent periodic points whose multiplier is of the form e 2πiθ for some irrational θ) then there exists a special lamination P, determined by P, with the following property: P J(P ) is monotonically semiconjugate to f P J P. Moreover, all P -classes are finite and the semiconjugacy is one-to-one on all (pre)periodic points of P. Strong conclusions about the topology of the Julia sets of non-renormalizable polynomials P D follow from [KvS06]. Building upon earlier results by Kahn and Lyubich [KL09a, KL09b] and by Kozlovskii, Shen and van Strien [KSvS07a, KSvS07b], Kozlovskii and van Strien generalized results of Avila, Kahn, Lyubich and Shen [AKLS09] and proved in [KvS06] that if all periodic points of P are repelling and P is non-renormalizable then J(P ) is locally connected; moreover, by [KvS06] two such polynomials that are topologically conjugate are in fact quasi-conformally conjugate. Thus, in this case f P J P is a precise model of P J(P ). Finally, for a given dendritic lamination it follows from another result of Jan Kiwi [Kiw05]

18 18 A. BLOKH, L. OVERSTEEGEN, R. PTACEK, AND V. TIMORIN that there exists a polynomial P with = P. Thus, by [Kiw05] associating polynomials from D with their laminations P and geolaminations L P = L P, one maps polynomials from D d onto LD d. To study the association of polynomials with their geolaminations we need Lemma 2.31 (it is stated as a lemma in [GM93] but goes back to Douady and Hubbard [DH8485]). Lemma 2.31 ([GM93, DH8485]). Let P be a polynomial, I be the set of all (pre)periodic external rays landing at the P n -th preimage x n of a repelling periodic point x, and let x n be not (pre)critical. Then the set I is finite, and for any polynomial P sufficiently close to P, there are a corresponding repelling periodic point x close to x and a (P ) n -th preimage x n of x close to x n such that the family I of all (pre)periodic rays, landing at x n, consists of rays uniformly (with respect to the spherical metric) close to the corresponding rays of I with the same external arguments. We also need the following simple lemma. Lemma Suppose that is a dendritic lamination. Then each leaf of L can be approximated by (pre)periodic leaves. Proof. Consider the topological polynomial f. Choose an arc I J. By [BL02] we can find k > 0 such that I and f (I) k are non-disjoint. Consider the union T of all f -images k of I (which is connected) and take its closure K. Then K J is an f k -invariant dendrite. Any periodic point x K corresponds to a finite gap of with periodic edges which are fixed by σd m for some m > 0. Hence there are short open pairwise disjoint arcs (x, s ) (x, s) K such that all points y (x, s ) are repelled away from x but have images which belong to (x, s). By Theorem of [BFMOT10] there are infinitely many periodic cutpoints in K. Since T is connected and dense in K, it follows that T contains periodic points. Hence I contains (pre)periodic points. Clearly, this implies the lemma. We will use qc-portraits to parameterize (tag) dendritic geolaminations. An obstacle to this is the fact that a geolamination L with a k-to-1 critical set such that k > 2 does not admit a qc-portrait for L. However using Lemma 2.24 it is easy to see that in this case one can insert critical quadrilaterals in critical sets of higher degree in order to tune L into a geolamination with a qc-portrait. This motivates the following. Definition Let L have pairwise disjoint critical sets D 1,..., D k. Let L L 1 and QCP = (E 1,..., E d 1 ) be a qc-portrait for L 1. Then there is a unique (d 1)-tuple Z = (C 1,..., C d 1 ) such that for every 1 i d 1 we have E i C i and there is 1 j k with C i = D j. Then Z is called

19 COMBINATORIAL MODELS 19 the critical pattern of QCP in L. Observe that each D j is repeated in Z exactly m j 1 times, where m j is the degree of D j. In general, given a geolamination L with pairwise disjoint critical sets D 1,..., D k, by a geolamination with a critical pattern we mean a pair (L, Z) where Z = (C 1,..., C d 1 ) is a (d 1)-tuple of sets provided for every 1 i d 1 there is a 1 j k with C i = D j and, for every j = 1,..., k, each D j is repeated in Z exactly m j 1 times, where m j is the degree of D j. Then Z is called a critical pattern for L. The space of all dendritic geolaminations with critical patterns is denoted by LCPD d. By changing the order of the critical sets, various critical patterns for the same geolamination can be obtained. In the dendritic case the connection between critical patterns and geolaminations can be studied using results of Jan Kiwi [Kiw04]. One of the results of [Kiw04] can be stated as follows: if L is a dendritic geolamination and L is a different geolamination such that L and L share a collection of d 1 critical chords with no loops among them, then L L. Since all gaps of L are finite this means that L \ L consists of countably many leaves inserted in certain gaps of L. Observe that if a sequence of geolaminations with critical patterns (L i, Z i ) converges, then, by Theorem 2.2, the limit L of geolaminations L i is itself a σ d -invariant geolamination. Moreover, it is easy to see that then critical patterns Z i converge to the limit collection of d 1 critical sets of L. Together with results from [Kiw04] this implies the following lemma given without proof. Lemma Suppose that a sequence of geolaminations with critical patterns (L i, Z i ) converges in the sense of the Hausdorff metric to a geolamination L with a collection of limit critical sets C 1,..., C d 1. Suppose that there exists a dendritic geolamination L with a critical pattern Z = (Z 1,..., Z d 1 ) such that C i Z i, 1 i d 1. Then L L. For an integer m > 0, we use a partial order by inclusion among m- tuples: we write (A 1,..., A m ) (B 1,..., B m ) if and only if A i B i for all i = 1,..., m. Thus m-tuples and k-tuples with m k are always incomparable. Lemma 2.34 says that if critical patterns converge into a critical pattern of a dendritic geolamination L, then the corresponding geolaminations themselves converge over L. The notion of a geolamination with critical pattern is related to the notion of a (critically) marked polynomial [Mil12], i.e., a polynomial P with an ordered collection CM of its critical points each of which is listed according to its multiplicity (so that there are d 1 points in CM). Critically marked polynomials do not have to be dendritic (in fact, the notion is used by Milnor and Poirier for hyperbolic polynomials, i.e., in the situation diametrically opposed to that of dendritic polynomials). Evidently, the space of critically