Laminations from the Main Cubioid

According to a recent paper \cite{bopt13}, polynomials from the closure $\bar{\rm PHD}_3$ of the {\em Principal Hyperbolic Domain} ${\rm PHD}_3$ of the cubic connectedness locus have a few specific properties. The family $\mathrm{CU}$ of all polynomials with these properties is called the \emph{Main Cubioid}. In this paper we describe the set $\mathrm{CU}^c$ of laminations which can be associated to polynomials from $\mathrm{CU}$.

(Communicated by the associate editor name) Abstract. Polynomials from the closure of the principal hyperbolic domain of the cubic connectedness locus have some specific properties, which were studied in a recent paper by the authors. The family of (affine conjugacy classes of) all polynomials with these properties is called the Main Cubioid. In this paper, we describe a combinatorial counterpart of the Main Cubioid -the set of invariant laminations that can be associated to polynomials from the Main Cubioid.
conjugacy class. Given a polynomial P , we denote its class by [P ]. The Principal Hyperbolic Domain PHD d in M d consists of all classes of hyperbolic polynomials with Julia set homeomorphic to a circle. Equivalently, [P ] is in PHD d if all critical points of P are in the immediate basin of attraction of some attracting fixed point. An important question then is to describe the set of all classes of polynomials that belong to the closure PHD d of PHD d . Here we address this question for d = 3 (i.e., in the cubic case).
The key object in Thurston's combinatorial model is the notion of a lamination (full definitions are provided in Section 2, see Definitions 2.1, 2.2, 2.4, and 2.5, while in Subsection 1.2 we will only give loose descriptions). Laminations provide combinatorial models for connected polynomial Julia sets. Thurston's work [31] can be seen as consisting of two parts: he defined a set of laminations that are good candidates to be models of quadratic Julia sets, and then he described the set of such laminations, which led to a combinatorial model for the Mandelbrot set.
In this paper, we will make a similar first step for PHD 3 with the hope that our classification eventually yields a suitable combinatorial model. Polynomials from PHD 3 satisfy certain dynamical properties. These dynamical properties in turn force the corresponding laminational models to have certain properties. We will consider all laminations that satisfy these properties and provide a simple classification of these laminations which revolves around a reduction to laminations with model polynomials from PHD 2 .
The cubic Mandelbrot set M 3 or its parts have been studied before. P. Lavaurs in his thesis titled Systèmes dynamiques holomorphes, Explosion de points périodiques paraboliques (Université Paris-Sud, Orsay, 1989) proved that M 3 is not locally connected. Epstein and Yampolsky [15] proved that the bifurcation locus in the space of real cubic polynomials is not locally connected either. This makes defining a combinatorial model of M 3 very delicate. Buff and Henriksen [10] presented copies of quadratic Julia sets, including Julia sets which are not locally connected, in slices of M 3 ; however, these copies are disjoint from the closure of PHD 3 . Moreover, Mc-Mullen [20] has shown that slices of M 3 contain lots of copies of M 2 . In addition, Gauthier [16] has shown that M 3 contains copies of M 2 × M 2 . Observe also that in his thesis titled Local connectivity in a family of cubic polynomials (Cornell University, 1992) D. Faught considered the slice A of M 3 consisting of polynomials with a fixed critical point and showed that A contains countably many homeomorphic copies of M 2 and is locally connected everywhere else. In particular, A intersects the boundary of PHD 3 along a Jordan curve.
Roesch [30] generalized Faught's results to higher degrees. Zakeri [33] described some important Jordan curves in the boundary of PHD 3 , whose points are represented by polynomials with both critical points on the boundary of the same Siegel disk. Milnor and Poirier [24] gave a classification of hyperbolic components in M d ; in particular, he proved that the topology of many hyperbolic components can be reduced to that of PHD 3 . In a recent paper [28], Petersen and Tan Lei introduced an analytic coordinate system on PHD 3 that reflects dynamical properties of the corresponding polynomials. The authors planned a sequel [29] to this paper, in which the boundary of PHD 3 would be discussed. After the main results of our paper had been obtained, we discovered that our work may have some overlap with [29].
1.2. Introduction to Laminations. Thurston [31] gave a combinatorial model for the entire Mandelbrot set. It has been conjectured that the Mandelbrot set is homeomorphic to Thurston's model; in fact, this conjecture is equivalent to local connectivity of M 2 . Although a global homeomorphism is not known, some parts of the Mandelbrot set can be shown to be homeomorphic to the corresponding parts of the model. For example, CA is homeomorphic to CA c , and both sets are homeomorphic to the closed disk (see, e.g., [12]).
For higher degree Mandelbrot sets M d even conjectural models are missing. To begin with, it is natural to model PHD d . As a first step to solving this problem in the cubic case, we study individual polynomials from PHD 3 . Similar to Thurston [31], we use laminations.
We will write C for the plane of complex numbers, C for the Riemann sphere, and D = {z ∈ C | |z| < 1} for the open unit disk. A lamination is a closed equivalence relation ∼ on S 1 = {z ∈ C | |z| = 1}, whose classes are finite sets, such that the convex hulls of different classes have disjoint relative interiors, see Definition 2.1. A lamination is (σ d -)invariant if classes map to classes under σ d : S 1 → S 1 , z → z d in a covering fashion. See Definition 2.2, which makes this more precise. If a polynomial P has a locally connected Julia set J, then there is a lamination ∼ P identifying pairs of angles if the corresponding external rays land at the same point. The quotient J ∼P = S 1 /∼ P is homeomorphic to J, and the self-mapping f ∼P of J ∼P induced by σ d is conjugate to P | JP ; the map f ∼P and the set J ∼P are called a topological polynomial and a topological Julia set, respectively. Laminations can play a role for some polynomials whose Julia sets are not locally connected. Then P | JP and f ∼P | J∼ P are not conjugate, however, they are semiconjugate by a monotone map (a continuous map, whose fibers are continua). Topological Julia sets and polynomials make sense for any σ d -invariant lamination, not just those described above.
It is very useful to associate to each lamination ∼ important geometric objects defined below. We will identify S 1 with R/Z. For a pair of angles a, b ∈ R/Z, we will write ab for the chord (a straight line segment in C) connecting the points of S 1 corresponding to angles a and b. If G is the convex hull Ch(G ′ ) of some closed set G ′ ⊂ S 1 , then we write σ d (G) for the set Ch(σ d (G ′ )). The set G ′ = G ∩ S 1 is called the basis of G. The boundary of G will be denoted by Bd(G).

Definition 1.1 (Leaves). If
A is a ∼-class, call a chord ab in Bd(Ch(A)) a leaf of ∼. All points of S 1 are also called (degenerate) leaves. The family L ∼ of all leaves of ∼ is called the geo-lamination (geometric lamination, or geodesic lamination) generated by ∼. The union of all leaves of L ∼ is denoted by L + ∼ . In general, collections of leaves with properties similar to those of collections L ∼ are also called invariant geo-laminations, see Definition 2.5 for a more precise formulation. In fact, it is these collections that Thurston introduced and studied in [31]. Thus, laminations (and their geo-laminations) on the one hand, and geolaminations in general, on the other hand, are studied in complex dynamics. The motivation of this can be as follows. The direct association between polynomials and laminations was described above. However, not every polynomial P can be directly associated to a suitable lamination. Hence it is natural to consider polynomials close to P for which such association is possible and then associate to P the appropriately defined limit of their laminations. To define such a limit one has to consider geo-laminations. However, limits of geo-laminations, which were associated to laminations, are not easily associated to laminations. This motivates the usage of "abstract" geo-laminations, i.e., geo-laminations not associated with any lamination. Definition 1.2 (Gaps). Let L be an invariant geo-lamination, e.g., we may have L = L ∼ for some invariant lamination ∼. The closure in C of a non-empty component of D \ L + is called a gap of L. Edges of a gap G are defined as leaves of L on the boundary of G. A gap is said to be finite (infinite) if its basis is finite (infinite). Infinite gaps of L are also called Fatou gaps.
The map σ d : G ′ → σ d (G ′ ) extends to Bd(G) as a composition of a monotone map and a covering map of some degree m. Then m is called the degree of G.
Such a gap G is said to be a periodic Siegel gap if the degree of σ n d | G is 1. If the degree of G is 2, then the gap G is said to be quadratic. If the period of G is 1, then G is said to be invariant. Theorem A ( [9]). If [P ] ∈ PHD d , then P has a fixed non-repelling point, no repelling periodic cutpoints, and at most one non-repelling periodic point with multiplier different from 1.
A polynomial P with [P ] ∈ PHD 3 has at most two non-repelling cycles. One of them must be a fixed point (as we approximate P with polynomials g, whose classes belong to PHD 3 , the attracting fixed points of g converge to a non-repelling fixed point of P ). If there is a non-repelling cycle of period greater than 1, then by Theorem A this cycle must have multiplier 1. Moreover, all fixed neutral points of f but one must have multiplier 1. In this paper, we will consider all polynomials which satisfy the conclusions of Theorem A. Definition 1.3. The main cubioid CU is defined as the set of all classes of cubic polynomials that have a fixed non-repelling point, no repelling periodic cutpoints and at most one non-repelling periodic point with multiplier different from 1.

If [P ]
∈ CU and J P is locally connected, then Definition 1.3 forces the corresponding lamination ∼ P to have certain properties (see Lemma 1.4). By an edge of a ∼ P -class, we mean an edge of the convex hull of that class. If an edge is periodic, its vertices may have a larger period than the period of the edge (for instance a class with vertices 1 4 , 3 4 is fixed under σ 3 but its vertices are of period two). Periodic gaps/leaves of a geo-lamination L are naturally associated with a rotation number (unless these are periodic Fatou gaps of degree greater than 1). They are said to be rotational if the rotation number is not zero. A thorough treatment is given in Section 2. Lemma 1.4. If a polynomial P with [P ] ∈ CU has locally connected Julia set, then the lamination ∼ P has at most one rotational periodic set (hence this set must be fixed ). Moreover, each periodic non-degenerate ∼ P -class G has a cycle of edges of vertex period n, at which a Fatou gap of period n is attached to G.
Proof. Let g be a rotational periodic set of ∼ P , and let ψ P denote the semiconjugacy between σ 3 : S 1 → S 1 and P : J P → J P obtained as the composition of the quotient map from S 1 to S 1 / ∼ P and the natural conjugacy between f ∼P and P | JP . If g is finite, it maps under the semiconjugacy ψ P to a periodic point x ∈ J P , which must be a cutpoint of J P . By Theorem A, the point x cannot be repelling. Hence, and since g is rotational, the point x is parabolic with multiplier different from 1. If g is infinite, then by definition g is a periodic Siegel gap which, since J P is locally connected, corresponds to a periodic Siegel domain of P which contains a periodic Siegel point. Now from Theorem A it follows that g is unique. Indeed, by the above each rotational periodic set g of ∼ P generates a periodic non-repelling point of multiplier distinct from 1. Yet by Theorem A the polynomial P has at most one periodic non-repelling point of multiplier distinct from 1. Hence there is at most one rotational set g. This implies the first claim of the lemma. Since each periodic non-degenerate ∼ P -class G corresponds to a parabolic periodic point, the second claim follows.

Definition 1.5 (Combinatorial Main Cubioid). Define the Combinatorial Main
Cubioid CU c as the family of all cubic laminations ∼ with at most one rotational periodic set (the set must be fixed because if its period were greater than 1, then there would be at least two such sets) and such that each periodic non-degenerate ∼-class G has a cycle of edges of vertex period n at which Fatou gaps of period n are attached to G.

1.4.
Main results. In this paper we classify all laminations in CU c . In doing so it is important to keep in mind that any cubic lamination either has two critical sets of degree 2 (so that either set maps two-to-one on its image) or one critical set of degree 3 (which maps onto its image three-to-one).
To state our results, we need to discuss invariant quadratic gaps of cubic laminations. Let U be such a gap. We show that the gap U has a unique major edge (or simply major ) M = ab such that the arc (a, b) is of length at least 1 3 and contains no points of U . It turns out that M can be either periodic (then U is said to be of periodic type) or critical (then U is said to be of regular critical type).
In either case there is a specific canonical lamination ∼ U associated to U . In the regular critical case ∼ U is defined as follows: a ∼ U b if there is N 0 such that σ N 3 (a) and σ N 3 (b) are endpoints of the same edge of U , and the set {σ i 3 (a), σ i 3 (b)} is not separated by U for i = 0, . . . , N − 1. Basically, this means that there is a gap V attached to U along its major and folding on top of U under σ 3 ; the rest of ∼ U consists of well-defined pullbacks of V , which accumulate to points of S 1 . Now, let U be of periodic type. Then one can define a periodic quadratic gap V attached to U along its major M U = M . The gap V has the same period n as M and can be defined as follows. Take all points a ∈ S 1 such that, for any i 0, the point σ i 3 (a) is separated from U by σ i 3 (M ) or is an endpoint of σ i 3 (M ). Then V is the convex hull of the set of all such points a. We call V = V (U ) the vassal gap of U . Now, define the lamination ∼ U as follows: a ∼ U b if there exists N 0 such that σ N 3 (a) and σ N 3 (b) are endpoints of the same edge of U or the same edge of V , and the chord σ i Basically, ∼ U includes U , the orbit of V , and the rest of ∼ U consists of well-defined pullbacks of V which accumulate to points of S 1 .
Let us now give a heuristic description of laminations from CU c . Namely, a lamination ∼ from CU c can be thought of as a result of an at most two-step process. First, an invariant quadratic Fatou gap U is created, together with its canonical lamination ∼ U . This could end the whole process. However it could also happen that afterwards the gap U is (weakly) tuned (see Definition 1.9 and Definition 1.10) by a quadratic lamination from the Combinatorial Main Cardioid (since σ 3 | U ′ is semiconjugate to σ 2 by a map collapsing all edges of U , it is easy to define tuning in this setting). We show that basically this mechanism describes all laminations from CU c . Although the classification given in the Main Theorem is the main result, we do obtain as a corollary that Definition 1.5 implies an even stronger condition on attached Fatou gaps. Corollary 1.6. A lamination ∼ belongs to CU c if and only if it has at most one rotational periodic (hence fixed) set and, for each leaf ℓ of ∼ of vertex period n, there is a Fatou gap of period n attached to ℓ.
To give the precise statement of our main result, we have to introduce a few definitions. The first one relates different laminations. Definition 1.7. Let G be a gap of some lamination. A lamination ∼ coexists with G if every leaf of ∼ intersecting an edge ℓ of G in D coincides with ℓ. A lamination ∼ coexists with a lamination ≃ if no leaf of ∼ intersects a leaf of ≃ in D unless the two leaves coincide. Now we can formalize some situations in which one lamination modifies another one.
Let G be a Fatou gap of some lamination. If all edges of G are leaves of ∼ then we say that ∼ tunes the gap G.
The terminology can be better understood if we think of L ∼ as being obtained by adding new leaves to L ≃ ; since these leaves can only be added inside gaps of L ∼ , we can think of this as tuning of gaps of ∼ which explains the terminology.
Given a periodic Fatou gap G of some unspecified lamination, consider a map ψ G : Bd(G) → S 1 that collapses all edges of G to points; ψ G maps Bd(G) onto the unit circle S 1 . If G is periodic of period n, then ψ G : Bd(G) → S 1 semiconjugates σ n d | Bd(G) to either irrational rotation (if G is a Siegel gap) or to σ k (if G is of degree k > 1). Now, suppose that ∼ tunes a periodic quadratic gap G. Then leaves of L ∼ contained in G map under ψ G to chords of D. In Section 3, we show that these chords can be viewed as leaves of a geo-lamination generated by an invariant lamination; this lamination is denoted by ψ G (∼). Definition 1.9. Suppose that ∼ tunes a periodic quadratic gap G. Then we say that ∼ tunes G according to the lamination ψ G (∼). If G is a gap of a lamination ≃, and ∼ tunes ≃, then for brevity we also say that ∼ tunes ≃ according to the lamination ψ G (∼) (in general, in this last case the behavior of ∼ outside G, even though compatible with ≃, is not completely defined by the way ∼ tunes G). Clearly, distinct laminations can tune the same quadratic invariant gap.
A weaker (and finer) case of tuning is described in the next definition. Let us emphasize that in it, ∼ only coexists with a periodic quadratic gap U (i.e., U is not necessarily a gap of ∼). Definition 1.10. If ∼ coexists with a periodic quadratic gap U of some unspecified lamination, and ψ U (∼) coincides with a quadratic lamination ≍, then we say that ∼ weakly tunes U according to the lamination ≍. If U is a periodic gap of a lamination ≃ and ∼ coexists with ≃, then for brevity we also say that ∼ weakly tunes ≃ (on U ) according to the lamination ≍.
We are ready to state our main theorem. Recall the vassal gap V (U ) was defined in the second paragraph of Subsection 1.4.
Main Theorem. Let ∼ be a non-empty lamination from CU c . Then there exists an invariant quadratic gap U for which (1) or (2) takes place.
1. The lamination ∼ coexists with the canonical lamination ∼ U and weakly tunes ∼ U on U according to a quadratic lamination ≍ from CA c so that edges of U are not leaves of ∼. Moreover, U can be chosen to be of regular critical type (and, if ∼ is not canonical, then U must be of regular critical type). 2. The lamination ∼ tunes the canonical lamination ∼ U according to a quadratic lamination ≍ from CA c (possibly empty), and if U is of periodic type, then the vassal gap V (U ) is a gap of ∼.
The rest of the paper is structured as follows. In Section 2, we give precise definitions of laminations and related terminology and state technical results, including Theorem 2.13 from [3], which provides a method for finding fixed objects in laminations. Section 3 investigates the structure of quadratic invariant gaps and the related notion of canonical laminations of quadratic invariant gaps, upon which the classification of cubioidal laminations rests. Tuning and weak tuning, processes which insert laminations into infinite gaps of other laminations, are also developed there. Invariant rotational sets play a prominent role in the definition of the Combinatorial Main Cubioid. In Section 4, we define and study various canonical laminations. All of these notions are combined in Section 5, where the full classification of CU c in terms of (weak) tuning quadratic invariant gaps is given.
Acknowledgements. During the work on this project, the fourth named author (V.T.) has visited Max Planck Institute for Mathematics (MPIM), Bonn. During the first revision of the paper the first named author (A.B.) was also visiting Max Planck Institute for Mathematics (MPIM), Bonn. A.B. and V.T. are very grateful to MPIM for inspiring working conditions. We all would also like to thank the referee for useful remarks.

2.
Preliminaries. Let a, b ∈ S 1 . By [a, b], (a, b), etc, we mean the closed, open, etc positively oriented circle arcs from a to b, and by |I| the length of an arc I in S 1 normalized so that the length of S 1 is 1. In this section, we will introduce classic results due to Douady and Hubbard [13,14] and Thurston [31]. These results allow one to study connected filled-in Julia sets by means of studying their complements in the complex plane.

2.1.
Laminations. For a compactum X ⊂ C, let U ∞ (X) be the unbounded component of C \ X containing infinity. If X is connected, there exists a Riemann mapping Ψ X : C \ D → U ∞ (X); we always normalize it so that Ψ X (∞) = ∞ and Ψ ′ X (z) tends to a positive real limit as z → ∞. Consider a polynomial P of degree d 2 with Julia set J P and filled-in Julia set K P . Extend z d : C → C to a map θ d on C. If J P is connected, then Ψ JP = Ψ : C \ D → U ∞ (K P ) is such that Ψ • θ d = P • Ψ on the complement of the closed unit disk [13,14,22]. If J P is locally connected, then Ψ extends to a continuous function Ψ : C \ D → C \ K P , and Ψ • θ d = P • Ψ on the complement of the open unit disk; thus, we obtain a continuous surjection Ψ : Bd(D) → J P (the Carathéodory loop, see [11]).
Let J P be locally connected, and set ψ = Ψ| S 1 . Following Thurston [31] (see also [13,14]), define an equivalence relation ∼ P on S 1 by x ∼ P y if and only if ψ(x) = ψ(y), and call it the (σ d -invariant) lamination of P (since Ψ defined above semiconjugates θ d and P , the map ψ semiconjugates σ d and P | J(P ) which implies that ∼ P is invariant). Equivalence classes of ∼ P are pairwise unlinked : their Euclidian convex hulls are disjoint. The topological Julia set S 1 / ∼ P = J ∼P is homeomorphic to J P , and the topological polynomial f ∼P : J ∼P → J ∼P is topologically conjugate to P | JP . One can extend the conjugacy between P | JP and f ∼P : J ∼P → J ∼P to a conjugacy on the entire plane.
An equivalence relation ∼ on the unit circle, with similar properties to those of ∼ P above, can be introduced abstractly without any reference to the Julia set of a complex polynomial (see [4]).
Definition 2.1 (Laminations). An equivalence relation ∼ on the unit circle S 1 is called a lamination if it has the following properties: (E1) the graph of ∼ is a closed subset in S 1 × S 1 ; (E2) if t 1 ∼ t 2 ∈ S 1 and t 3 ∼ t 4 ∈ S 1 , but t 2 ∼ t 3 , then the open straight line segments in C with endpoints t 1 , t 2 and t 3 , t 4 are disjoint; (E3) each equivalence class of ∼ is finite.
Recall that, for a closed set A ⊂ S 1 , we denote its convex hull by Ch(A). Then by an edge of Ch(A) we mean a closed subsegment of the straight line connecting two points of the unit circle which is contained in the boundary of Ch(A). By an edge of a ∼-class we mean an edge of the convex hull of that class.

Definition 2.2 (Laminations and dynamics). A lamination
(D1) ∼ is forward invariant: for a class g, the set σ d (g) is a class too; (D2) for any ∼-class g, the map σ d : g → σ d (g) extends to S 1 as an orientation preserving covering map such that g is the full preimage of σ d (g) under this covering map. Definition 2.2 (D2) has an equivalent version. Given a closed set Q ⊂ S 1 , a (positively oriented) hole (a, b) of Q (or of Ch(Q)) is a component of S 1 \ Q. Then (D2) is equivalent to the fact that for a ∼-class g either σ d (g) is a point or, for each positively oriented hole (a, b) of g, the positively oriented arc (σ d (a), σ d (b)) is a hole of σ d (g). From now on, we assume in this paper that, unless stated otherwise, ∼ is a σ d -invariant lamination.
Given ∼, consider the topological Julia set S 1 / ∼= J ∼ and the topological polynomial f ∼ : J ∼ → J ∼ induced by σ d . Using Moore's Theorem, embed J ∼ into C and extend the quotient map pr ∼ : S 1 → J ∼ to C with the only non-trivial fibers being the convex hulls of non-degenerate ∼-classes.
is conjugate either to an irrational rotation of S 1 or to σ k with some k > 1 [4]. In the case of irrational rotation, U is called a Siegel domain. The complement of the unbounded component of C \ J ∼ is called the filled-in topological Julia set and is denoted by K ∼ . Equivalently, K ∼ is the union of J ∼ and its bounded Fatou domains. If the lamination ∼ is fixed, we may omit ∼ from the notation. By default, we consider f ∼ as a self-mapping of J ∼ . In what follows, for a collection R of sets, denote the union of all sets from R by R + .
In the Introduction (Definition 1.1), we defined leaves of a lamination ∼ and the geo-lamination L ∼ associated with ∼. Extend σ d (keeping the notation) linearly over all individual chords in D, in particular, over leaves of L ∼ . Note that even though the extended σ d is not well defined on the entire disk, it is well defined on L + ∼ .
Recall that for a gap/leaf U we denote U ∩ S 1 by If m can be chosen to be 0, then U is called periodic, otherwise U is called preperiodic (hence, preperiodic implies non-periodic). Also, by a (pre)periodic gap/leaf we mean gap/leaf which is either periodic or preperiodic. A Fatou gap is the pr ∼ -preimage of the closure of a Fatou domain. Similarly, a Siegel gap is the pr ∼ -preimage of a Siegel domain. Equivalently, these are gaps with infinite bases. By [18], a Fatou gap G is (pre)periodic under σ d .

Definition 2.3 (Critical leaves and gaps). A leaf of a lamination
We define precritical and (pre)critical objects similarly to how (pre)periodic and preperiodic objects are defined above.
For example, a periodic Siegel gap is non-critical even though the first return map is not one-to-one on its basis (because there must be critical leaves in the boundaries of gaps from its orbit).

Geometric laminations.
Laminations, understood as equivalence relations, can be described in a geometric fashion, as was done in the original approach by Thurston [31]. Thurston studied collections of chords in D similar to L ∼ , for a given σ d -invariant lamination ∼, with no lamination given. Definition 2.4 (Geometric laminations, cf. [31]). A geometric pre-lamination L is a set of (possibly degenerate) chords in D such that any two distinct chords from L meet at most in a common endpoint; L is called a geometric lamination (geolamination) if all points of S 1 are elements of L, and L + is closed. Elements of L are called leaves of L. By a degenerate leaf (chord) we mean a singleton in S 1 .
In the Introduction (Definition 1.2), we defined gaps of geo-laminations. Now let us discuss geo-laminations in the dynamical context. Recall that given a chord ℓ = ab of the unit disk we define σ d (ℓ) as the chord σ d (a)σ d (b) and extend σ d linearly over ab.
Definition 2.5 (Invariant geo-laminations, cf. [31]). A geometric lamination L is said to be an σ d -invariant geo-lamination if the following conditions are satisfied: is a positively oriented composition of a monotone map and a covering map (thus, if G is a gap with finitely many edges, all of which are critical, then its image is a singleton).
Some invariant geo-laminations are not generated by laminations (see, e.g., [31], where Thurston considers geo-laminations with countable concatenations of leaves forming the boundary of a gap). We will use a special extension σ * d,L = σ * d of σ d to the closed unit disk associated with L. On S 1 and all leaves of L, we set σ * d = σ d . Define σ * d on the interiors of gaps using a standard barycentric construction [31]. For brevity, we sometimes use σ d instead of σ * d . We will mostly use the map σ * d if L = L ∼ for some invariant lamination ∼.

2.3.
Laminational sets and their basic properties. So far we have dealt with (geo-)laminations. However, we also consider subsets of D that have the properties of leaves and gaps of geo-laminations while no actual geo-lamination is specified. A number of facts can be proven for such sets, and we establish some of them in this subsection.
Definition 2.6. Let f : X → X be a self-mapping of a set X. For a set G ⊂ X, let the return time (to G) of x ∈ G be the least positive integer n x with f nx (x) ∈ G, or infinity if there is no such integer. Set n = min x∈G n x , define the set D G = {x ∈ G : n x = n}, and call the map f n : D G → G the remap (first return map of G). Also, we define refixed points in G as points x ∈ G such that f n (x) = x. Similarly, we talk about reorbits of points in G.
For example, if G is the boundary of a periodic Fatou domain of period n of a topological polynomial f ∼ , and the images f j By the relative interior of a set in the plane, we mean the interior of this set in its affine hull. Thus, the relative interior of a gap of some lamination is its interior, while the relative interior of a chord is the chord minus the endpoints.

Definition 2.7. Throughout this definition we assume that
If there exists m > 0 such that all Ch(σ i d (A)), 0 i m + n − 1 have pairwise disjoint relative interiors and σ m d (A) is periodic of period n, then we call A preperiodic. Observe that the above applies to sets A regardless of whether they are a part of a (geo-)lamination or not.
If A is wandering, periodic or preperiodic, and for every i 0 and every hole , then we call A (and Ch(A)) a (σ d -)laminational set ; we call both A and Ch(A) finite if A is finite. A (σ d -)stand alone gap G is defined as a laminational set with non-empty interior (note that a gap of a geo-lamination always has nonempty interior). In other words, a stand alone G is of the form Ch(A) for some closed set A ⊂ S 1 of more than two points such that the above listed properties hold for A.
In what follows, whenever we say that G is a "gap" we mean that at least G is a stand alone gap (it will be clear from the context if there is a lamination or a geo-lamination such that G is a part of it). Also, abusing the language, we will sometimes identify closed sets A ⊂ S 1 and their convex hulls (again, it will be clear from the context what kind of set we consider).
The basis G ′ = G ∩ S 1 of a gap G coincides with the union A ∪ B of two welldefined sets, where A is a maximal Cantor subset of G ′ or an empty set and B is countable. Assume that A = ∅, and define a map ψ G : S 1 → S 1 that collapses all holes of A to points. Suppose that G is m-periodic. It is well-known that ψ G can be chosen so that it semiconjugates σ m d | G ′ to an irrational rotation of the circle or to the map σ k , where k 2. Indeed, by Definition 2.7, the map of the circle, to which σ d is semi-conjugate under ψ G , is locally 1-to-1 and orientation preserving. On the other hand, the fact that σ d is locally expanding implies that the induced map of the circle does not have wandering arcs (see, e.g., [4] where similar arguments were used in classifying different types of gaps of laminations). This implies the above claim.
Accordingly, if A = ∅, we call a stand alone periodic gap G a stand alone periodic Fatou gap of degree k if in the above construction the map ψ G semiconjugates σ m d | G ′ to σ k , k 2. Also, we call a stand alone periodic gap G a stand alone periodic Siegel gap if, in the above construction, the map ψ G semiconjugates σ m d | G ′ to an irrational rotation. Moreover, for periodic laminational sets G with finite basis G ′ and for periodic stand alone Siegel gaps G we can define the rotation number τ G . If the rotation number is not equal to zero, the set G is said to be rotational. If such G is invariant, we call it an invariant rotational set.
Lemma 2.8 (Lemma 2.16 [8]). Suppose that ℓ = xy is a laminational chord such that there exists a component Q of the complement of its orbit in the disk D whose closure contains σ n d (ℓ) for all n 0. Then the chord ℓ is either (pre)critical or (pre)periodic.
Let U be the convex hull of a closed subset A of S 1 . For every edge ℓ of U , let H U (ℓ) denote the hole of U whose endpoints coincide with the endpoints of ℓ. In this situation, we define |ℓ| U as |H U (ℓ)|. Notice that if U is a chord, then it has two holes (on opposite sides of U ).
Suppose in addition that A ⊂ S 1 is a laminational set. Mostly, the holes of A map increasingly onto the holes of σ d (A). However, if the length of a hole is at least 1 d , then the map σ d wraps the hole around the circle one or more times. Thus, holes H of U such that |H| 1 d are the only holes which the map σ d does not take one-to-one onto their images. Definition 2.9. Let G be the convex hull of a closed subset of S 1 . If ℓ is an edge of G such that its hole H G (ℓ) is not shorter than 1 d , then ℓ is called a (σ d -)major edge of G (or simply a (σ d -)major of G), and H G (ℓ) is called a (σ d -)major hole of G.
It is useful to work with a wider class of sets, which we now introduce.
For a chord ℓ, let orb σ d (ℓ) denote the union of all chords in the forward orbit of ℓ under σ d .
follows that I ⊂ σ d (I) and that the endpoints of σ d (I) do not belong to I. Hence I contains a σ d -fixed point.
Let us now prove the lemma. By the above, σ 2 changes orientation on at most one hole. Since ℓ is periodic and its endpoints are refixed, σ 2 changes orientation an even number of times. Hence σ 2 never changes orientation on holes of Z. This implies that Z is semi-laminational. Now, if a hole of Z is shorter than 1 2 , then it doubles in length under σ 2 while still being mapped onto its image one-to-one. Hence there exists a hole H of Z whose length is at least 1 2 . If we draw a critical leaf c with endpoints in H, we see that the entire orbit of ℓ consists of leaves with endpoints in the complement of H. Now, it is well known (see, e.g., [31]) that a periodic orbit of σ 2 contained in a given half-circle (in our case, this is any half-circle containing S 1 \ H) is the point 0, or the pair {1/3, 2/3}, or the set of vertices of a finite invariant stand alone σ 2 -gap. Moreover, a given half-circle contains exactly one periodic σ 2 -orbit. It follows that the endpoints of ℓ are vertices of an invariant σ 2 -gap Z. Clearly, ℓ then has to be an edge of Z (if ℓ is a diagonal of Z, then distinct images of ℓ intersect inside D).
The class of semi-laminational sets is wider than the class of invariant laminational sets because Definition 2.10 allows for circle arcs to be parts of semilaminational sets. For example, take a σ d -invariant stand alone Fatou gap G of degree k > 1 such that there is a periodic orbit Q of edges of G. Let H 1 , . . . , H n be the holes of G behind edges from Q.
The assumption that a Fatou gap like G exists means that d must be greater than 2. In Lemma 3.9 we study semi-laminational sets for cubic laminations.
Majors of semi-laminational sets play an important role because, as we see below in Lemma 2.12, all edges of semi-laminational sets have majors of these semilaminational sets in their forward orbits. Proof. Let ℓ be an edge of G. The case when ℓ is invariant (i.e. such that σ d (ℓ) = ℓ) is left to the reader. Otherwise if |H G (ℓ)| < 1/d, then H G (ℓ) maps onto the hole σ d (H G (ℓ)) one-to-one. The fact that ℓ is not invariant implies that σ d (H G (ℓ)) is disjoint from H G (ℓ). Hence H G (ℓ) contains no fixed points. On the other hand, suppose that |H G (ℓ)| 1/d. Then σ d (H G (ℓ)) covers the entire S 1 while the images of the endpoints of H G (ℓ) are outside H G (ℓ). This implies that there exists a fixed point a ∈ H G (ℓ). To prove the second claim, choose an edge ℓ of G. For any i set Hence there exists the least n such that |T n | 1 d . Then the leaf σ n d (ℓ) is a major of G, as desired.
2.4. Fixed points and invariant sets. Theorem 2.13 allows one to find specific invariant sets in some parts of the disk. We state it in the language of laminations. Then at least one of the following claims holds: 1. X contains an invariant gap of ∼ of degree k > 1; 2. X contains an invariant rotational set.
3. Invariant quadratic gaps and their canonical laminations. By cubic laminations we mean σ 3 -invariant laminations. By a quadratic gap we mean a stand alone periodic Fatou gap U of degree 2. In this section, we assume that U is a σ 3 -invariant quadratic gap and study its properties. We then define canonical laminations, which correspond to these gaps and describe other laminations that refine the canonical ones. Throughout the rest of the paper we will often write σ instead of σ 3 .

Invariant quadratic gaps.
Recall that, given a gap U with an edge ℓ, we write |ℓ| U for |H U (ℓ)|. If U is given, we may drop the subscript U from the notation. only if ℓ = 0 1 2 . Proof. The existence of a major ℓ follows from Lemma 2.12. Observe that, if a set A ⊂ S 1 lies in the complement of two disjoint closed arcs in S 1 of length 1/3 each, then the restriction of σ to A is injective. This implies that all holes H = H(ℓ) of U are shorter than 1 3 and that |ℓ| U < 2 3 (here we use the fact that σ| U ′ is two-to-one). Clearly, |ℓ| U can be equal to 1 3 (just take ℓ = 1 3 2 3 and assume that U is the convex hull of the set of all points x ∈ S 1 with orbits outside the arc ( 1 3 , 2 3 )). This situation corresponds to case (1) of the lemma. Since σ expands the length by the factor of 3, it follows that all holes H = H(ℓ) of U are shorter than 1 9 . Suppose that |ℓ| U > 1 3 . Then σ(ℓ) is eventually mapped to ℓ by Lemma 2.12, hence ℓ is periodic.
with equality possible only if k = 1 in which case clearly ℓ = 0 1 2 (observe that under the assumptions of the lemma the leaf ℓ maps onto itself by σ k 3 so that each endpoint of ℓ maps to itself). This corresponds to case (2) of the lemma. Lemma 3.2. In case (2) of Lemma 3.1, there exists a unique leaf ℓ * disjoint from ℓ with endpoints in H(ℓ) such that σ(ℓ * ) = σ(ℓ) and such that one of the following holds: 1. we have ℓ = 0 1 2 , the only possible holes of U ∪ ℓ * of length 1 6 are 0, and all other holes of U ∪ ℓ * are of length at most 1 18 ; 2. we have ℓ = 0 1 2 , every hole of U ∪ ℓ * is shorter than 1 6 except for the hole of ℓ * that is disjoint from U and has length greater than 5 18 .
Proof. Let ℓ = ab and observe first that . Consider now two cases. First, assume that ℓ = 0 1 2 . Then it is easy to see that the claims of the lemma hold. Since 0 1 2 is a unique major edge ℓ of an invariant quadratic gap with |ℓ| = 1 2 , from now we may assume that ℓ < 1 2 . Consider a hole H of U ∪ ℓ * . If H is a hole of U , then it maps forward monotonically (and hence expanding by the factor of 3) a few times before it maps onto H(ℓ). Since, by the above, |H(ℓ)| , and, similarly, |(a * , b)| < 1 6 . Finally, consider the arc (b * , a * ). Its image is the arc complementary to the arc (σ(a), σ(b)). Since we assume that ℓ = 0 1 2 , it follows that (σ(a), σ(b)) is a hole of U distinct from H(ℓ). Hence the length of (σ(a), σ(b)) is less than 1 6 , which implies that the length of (σ(b), σ(a)) is more than 5 6 , and hence the length of (b * , a * ) is more than 5 18 . From now on, the major ℓ of U is denoted by M (U ). Lemma 3.1 implies a simple description of the basis U ′ of U if M (U ) is given. and its convex hull V (U ) = Ch(N (U )) (this notation is used in several lemmas below). We call V (U ) the vassal (gap) of U (see Figure 1). . This generates the quadratic gap V (U ) contained in the strip between ab and a * b * . In the language of one-dimensional dynamics (see [26,27,25,32] and the book [1]) one can say that closed intervals [a, b * ] and [a * , b] form a 2-horseshoe of period k. A standard argument shows that N (U ) is a Cantor set. The remaining claim easily follows.
We also define another type of gap called a caterpillar (gap). This is a periodic gap Q with the following properties: • The boundary of Q consists of a periodic (possibly degenerate) leaf ℓ 0 = xy of period k called a head of the caterpillar gap Q , a σ k -critical leaf ℓ −1 = yz concatenated to it, and a countable concatenation of leaves ℓ −n accumulating at x (the leaf ℓ −r−1 is concatenated to the leaf ℓ −r , for every r = 1, 2, . . . ). • We have σ k (x) = x, σ k ({y, z}) = {y}, and σ k maps each ℓ −r−1 to ℓ −r (all leaves are shifted by one towards ℓ 0 except for ℓ 0 , which maps to itself, and ℓ −1 , which collapses to the point y).
Similar gaps are already useful for quadratic laminations (see [31] where the invariant gap with edges 0 1 2 , 1  Proof. Let us describe one of the caterpillar gaps in question. Since the the hole (a, b) of U behind M (U ) has length between 1 3 and 2 3 , there exists a unique point a * ∈ (a, b) such that σ 3 (a * ) = σ 3 (a) and there exists a unique point b * ∈ (a, b) such that σ 3 (b * ) = σ 3 (b). Let ℓ 0 = M (U ) and ℓ −1 = aa * (clearly, aa * is a σ 3 -critical leaf). Choose the unique σ k 3 -preimage a −2 of a −1 = a * such that a < a −1 < a −2 < b. Put ℓ −2 = a −1 a −2 . Consider further pullbacks ℓ −n of ℓ 0 so that they form a concatenation of leaves converging to the point b. Then the required σ k 3 -fixed caterpillar gap is the convex hull of the union of leaves ∞ i=0 ℓ −i . The other desired caterpillar gap with critical leaf bb * can be constructed similarly.
We call the caterpillar gaps from Lemma 3.5 canonical caterpillar gaps of U . A critical edge c of a canonical caterpillar gap defines it, so this caterpillar gap is denoted by C(c). We denote its basis by C ′ (c). To study related invariant quadratic gaps we first prove the following general lemma, in which we adopt a different point of view. Namely, any leaf ℓ that is not a  Observe that, since c is a critical leaf, its σ-image is a point.
Proof. It is easy to see that the set Π(c) is closed and forward invariant; it is nonempty because it contains at least one fixed point (indeed, as was noticed before the lemma, the circle arc L(c) in the case at hand is of length 2 3 , and each circle arc of length 2 3 contains at least one σ 3 -fixed point). Let x ∈ Π(c). If x = σ(c), then, of its three preimages, one belongs to S 1 \ L(c) while two others are in L(c), and hence, by definition, also in Π(c). Suppose that x = σ(c) (and so, since by the assumption x ∈ Π(c), the orbit of c is contained in L(c)). Then the entire triple σ −1 (σ(c)) is contained in L(c) and, again by definition, in Π(c).
To prove the next claim of the lemma, we prove that any hole I of Π(c) except for the hole T , whose closure contains the endpoints of c, maps to a hole of Π(c). Indeed, otherwise there is a point y ∈ I such that σ(y) is a point of Π(c). Since I ⊂ L(c), we have y ∈ Π(c), a contradiction. Consider the hole T = (a, b) defined above. If T = S 1 \ L(c), there is nothing to prove as in this case the leaf c is a critical edge of G(c) that maps to a point of Π(c). Suppose now that T = S 1 \ L(c) and that there is a point z ∈ (σ(a), σ(b)) ∩ Π(c). Then there is a point t ∈ T ∩ L(c) with σ(t) = z and hence t ∈ Π(c), a contradiction. Thus, (σ(a), σ(b)) is a hole of Π(c), and σ(a)σ(b) is an edge of G(c). All this implies that G(c) is an invariant gap, and it follows from the definition that it is quadratic. The last claim easily follows.
Below we will use the notation G(c) = Ch(Π(c)). Let us relate invariant quadratic gaps defined in terms of periodic majors (see Lemma 3.4) and caterpillar gaps C(c) (see Lemma 3.5) to the gaps G(c). To state Lemma 3.7, we need the following concept. By Lemmas 3.1 and 3.3, all holes of U map onto H(M (U )) after finitely many steps in a monotone fashion. Suppose that I is a hole of U and n 0 is a positive integer such that σ n (I) = H(M (U )) and σ n | I is monotone. Then for any point x ∈ H(M (U )) we call a unique point x ∈ I with σ n ( x) = x the first pullback of x to I. Proof. Clearly U ′ ⊂ Π(c). We may assume that the circle arc [a, b] is ordered in the positive direction from a to b. Let ℓ = b * a * be the non-periodic edge of V (U ) such that σ(a * ) = σ(a), σ(b * ) = σ(b); suppose that ℓ ∩ c = ∅. The order of points on the circle arc [a, b] is as follows: a < b * < a * < b. By construction of V (U ), we have that I = H V (U) (ℓ) coincides with (b * , a * ), which in particular implies that the length of I is less than 1 3 (because the length of (a, a * ) is 1 3 ). Moreover, I contains no points of Π(c). Since c is disjoint from ℓ and has endpoints in H(M (U )), we may assume that x ∈ (a, b * ) and y ∈ (a * , b). Thus, we have J = (x, y) ⊃ I. Note that the restriction of σ k to H(M (U )) \ J is a one-to-one expanding map to H(M (U )). It follows that all points of H(M (U )) eventually map to J , therefore, they do not belong to Π(c) either. By Lemma 3.3, any point x ∈ U ′ eventually maps to H(M (U )). Thus, Π(c) = U ′ . The case, where c = aa * or c = b * b, is left to the reader. Lemma 3.8 complements Lemma 3.7.
Lemma 3.8. If c = xy is a critical leaf with a periodic endpoint, say, y such that x, y ∈ Π(c) then there exists a quadratic invariant gap W with a periodic major M (W ) = zy of period k, the point z is the closest to x in L(c) = (y, x) point that is σ k -fixed, and z is of period k.
Proof. Assume that the period of y is k. Consider the gap G(c). By the invariance of G(c), there is an edge c −1 of G(c) attached to c at x which maps to c under σ k . Moreover, |c −1 | G(c) = 3 −k−1 . This can be continued infinitely many times so that the m-th edge of G(c), which maps to c under σ mk , is a leaf c −m such that its hole is of length 3 −mk−1 . Clearly, the concatenation A of leaves c, c −1 , . . . converges to a point z ∈ S 1 which is σ k -fixed. Set zy = M .
Since, by Lemma 3.6, the gap G(c) is quadratic, there are many preimages of A on the boundary of G(c). Replace them all by the corresponding preimages of M (e.g., replace A by M ). It follows that the newly constructed gap W is a quadratic invariant stand alone Fatou gap with major M = M (W ) as desired. The last claim of the lemma easily follows.
Let us summarize the above results. Let c be a critical leaf. We will now define an invariant stand alone quadratic Fatou gap U (c) determined by c. Even though in the beginning of this section we announced that we will consider a given invariant stand alone quadratic Fatou gap U , we choose the notation U (c) for the gap determined by c to reflect the fact that U (c) has the same properties (i.e., that it is an invariant stand alone quadratic Fatou gap). In what follows we will use the notation L(c), Π(c) and G(c) introduced above.
The orbit of a critical leaf c can be of three types. First, the orbit of c can be contained in L(c) so that no endpoint of c is periodic. Then we set U (c) = G(c) and call U (c) and the leaf c regular critical. Second, an endpoint of c can be periodic with the orbit of c contained in L(c). By Lemmas 3.7 and 3.8, then Π(c) consists of U ′ , C ′ (c), and first pullbacks of C ′ (c) to holes of U for some invariant quadratic gap U = U (c) with a periodic major M (U ) (the gap U can be defined as the convex hull of all non-isolated points in Π(c)). This defines the gap U (c) in that case. Then we call the gap G(c) an extended caterpillar gap, and the critical leaf c a caterpillar critical leaf. Third, there can be n > 0 with σ n (c) ∈ L(c). Denote by n c the smallest such n. Then G(c) has a periodic major of period n c and we set U (c) = G(c) and call this gap a gap of periodic type. Only regular critical gaps or gaps of periodic type can be invariant quadratic gaps of laminations (a critical leaf with a periodic endpoint would imply that the corresponding infinite canonical caterpillar is contained in one class, a contradiction with condition (E3) of Definition 2.1).
We show below in Lemma 3.9 that properties listed in item (2) of Lemma 3.1 are basically sufficient for a periodic leaf ℓ to be the major of a quadratic invariant gap of periodic type. The arguments in Lemma 3.9 are related to those in Lemma 2.11. Lemma 3.9. Let ℓ = xy be a σ-periodic leaf of period r with σ r -fixed endpoints for which there is a unique component Z of D\orb σ (ℓ) such that any two iterated images of ℓ are disjoint or coincident edges of Z. Then a hole of Z with length greater than 1 3 exists if and only if Z is a semi-laminational set. In that case an eventual σ-image of ℓ that corresponds to the major hole of Z is a major of a quadratic invariant gap of periodic type.
Observe that Z has r holes each of which is located behind an image of ℓ and has length equal to neither 1 3 nor 2 3 . Recall that if a Fatou gap G is invariant, then the quotient map ψ G : Bd(G) → S 1 is defined as a map collapsing all edges of G to points and mapping G to the unit circle.
Proof. First assume that Z is semi-laminational. Then, by the above remark and Lemma 2.12, at least one hole H of Z must be longer than 1 3 . We may assume that H = H Z (ℓ). Choose a critical leaf c whose non-periodic endpoints are in (x, y). Let us show that G(c) is of periodic type and ℓ coincides with the major M of the gap G(c). Indeed, suppose otherwise. By definition the endpoints of ℓ belong to G ′ (c) and ψ G(c) maps ℓ to a leaf ψ G(c) (ℓ) such that the leaf ψ G(c) (ℓ) and its σ 2 -images satisfy conditions of Lemma 2.11. Hence ψ G(c) (ℓ) and its σ 2 -images are the edges of a finite σ 2 -invariant gap. In particular, they are not pairwise disjoint. Now, ψ G(c) collapses only preimages of M . If G(c) is of regular critical type (i.e., M = c has no periodic endpoints), it will follow that ℓ and its σ-images are not pairwise disjoint, a contradiction. If G(c) is of periodic type (i.e., M is a periodic leaf) then, if ℓ and its σ-images miss endpoints of M , we have that ψ G(c) is oneto-one on the endpoints of ℓ and its σ-images. Hence ℓ and its σ-images are not pairwise disjoint, a contradiction.
Suppose finally that, say, ℓ = M shares an endpoint x with M = xz, z = y. Since ℓ and its σ-images are pairwise disjoint, y does not belong to the same periodic orbit as x. On the other hand, ψ G(c) -images of leaves from the σ-orbit of ℓ are the edges of a finite σ 2 -invariant gap. Thus, y belongs to some σ-image of M and so the orbit of y coincides with the orbit of z. However this is impossible as by the construction z ∈ H while H cannot contain points of σ-images of ℓ. The rest of the lemma follows from the above and the already obtained description of quadratic invariant gaps of σ.
By the above proven lemmas, each gap W = G(c) of periodic type has a periodic major M (W ) = xy of period n c with endpoints in L(c); moreover, x and y are the closest in L(c) points to the endpoints of c that are σ nc -fixed (in fact, they are periodic of period n c ). Proof. In the regular critical and periodic cases, it suffices to prove that the set W ′ has no isolated points. Indeed, by Lemma 2.12, an isolated point in W ′ must eventually map to an endpoint of M (W ). Thus it remains to show that the endpoints of M (W ) are not isolated. This follows because the endpoints of M (W ) are periodic, and suitably chosen pullbacks of points in W ′ to W ′ under the iterates of the remap of W ′ will converge to the endpoints of M (W ). The case of an extended caterpillar gap follows from Lemma 3.7.

3.2.
Canonical laminations of invariant quadratic gaps. Let us associate a specific lamination with each invariant quadratic gap. We do this in the spirit of [31], where pullback laminations are defined for maximal collections of critical leaves. Since ∼-classes are finite, an invariant lamination cannot contain any caterpillar gaps. Hence we consider only regular critical gaps and gaps of periodic type.
Let U be a stand alone quadratic invariant gap of regular critical type with critical major M (U ). Edges of U have uniquely defined iterated pullbacks disjoint from U which define an invariant lamination. More precisely, we define a lamination ∼ U as follows: a ∼ U b if there is N 0 such that σ N (a) and σ N (b) are endpoints of the same edge of U , and the set {σ i (a), σ i (b)} is not separated by U for i = 0, . . . , N − 1. Loosely, one can say that points a, b "travel" together visiting the same holes of U until at some moment they simultaneously map to the endpoints of an edge of U . Clearly, all ∼ U -classes are either points or leaves and ∼ U is an invariant lamination. Now, let U be of periodic type and V be its vassal. Define a lamination ∼ U as follows: a ∼ U b if there exists N 0 such that σ N (a) and σ N (b) are endpoints of the same edge of U or the same edge of V , and the chord σ i (a)σ i (b) is disjoint from U ∪ V for i = 0, . . . , N − 1. Note that V is a gap of ∼ U . It is easy to check that the canonical lamination of a quadratic periodic gap U does not have periodic non-degenerate classes that are not edges of U . Lemma 3.11. If U is a stand alone invariant quadratic gap of regular critical type, then ∼ U is the unique invariant lamination such that U is one of its gaps.
In Definition 1.7, we defined coexistence of a gap (of some unspecified lamination) and a lamination. The definition holds verbatim if a gap of some lamination is replaced with a stand alone gap. We also defined coexistence of two laminations. Lemma 3.12. Suppose that a cubic invariant lamination ∼ coexists with a stand alone invariant quadratic gap U of regular critical type. Then ∼ also coexists with the canonical lamination ∼ U of U .
Proof. Suppose that a leaf ℓ of ∼ crosses a leaf ℓ U of ∼ U in D. By the assumption of the lemma, both ℓ and ℓ U must have their endpoints in the closure of some hole of U . Every hole of U maps one-to-one onto its image. It follows that σ(ℓ) and σ(ℓ U ) also intersect in D. However, any leaf of ∼ U eventually maps to an edge of U , a contradiction.
Proof of Lemma 3.11. Suppose that ∼ is a cubic invariant lamination and U is a gap of ∼; then ∼ coexists with U , hence, by Lemma 3.12, the lamination ∼ coexists with the canonical lamination ∼ U . If a leaf ℓ of ∼ is not a leaf of ∼ U , then ℓ is in some pullback of U with respect to ∼ U . Hence the leaf ℓ eventually maps to U . Since U is a gap of ∼, the leaf ℓ eventually maps to an edge of U . By definition it follows that ℓ is a leaf of ∼ U . The definition of a lamination now implies that all leaves of ∼ U are leaves of ∼.
The proof of Lemma 3.13 is similar to that of Lemmas 3.11 and 3.12.

ALEXANDER BLOKH, LEX OVERSTEEGEN, ROSS PTACEK AND VLADLEN TIMORIN
Lemma 3.13. Let U be an invariant quadratic gap of periodic type. Then ∼ U is the unique invariant lamination such that U and the vassal V (U ) are its gaps. If a cubic invariant lamination ∼ coexists with U and V (U ), then ∼ coexists with the canonical lamination ∼ U of U .

3.3.
Tuning. In this subsection we discuss the notion of coexistence of two laminations and make it more explicit. We will define the notion of tuning which is stronger than coexistence of laminations. Right after Definition 2.7 we introduced a monotone map ψ G : Bd(G) → S 1 ; for an invariant stand alone gap G of degree k > 1, we have that σ d | Bd(G) is semiconjugate to σ k by means of ψ G . If a lamination ∼ coexists with G, then we want to obtain an induced lamination ψ G (∼) which is invariant under σ k . We need the following notions.
Definition 3.14 ( [6]). Let L be a geo-lamination such that the σ d -images of its leaves are again leaves of the same geo-lamination. Any two disjoint leaves of L with the same σ d -image are called sibling leaves, or siblings. Suppose that any leaf of L has at least one σ d -preimage, and, for any leaf ℓ 1 of L with non-degenerate σ d -image, there are d − 1 leaves ℓ 2 , . . . , ℓ d of L such that the leaves ℓ i , i = 1, . . . , d are pairwise disjoint and map to σ d (ℓ 1 ). This property is called the Sibling Property and L is then called a sibling invariant lamination.
Let us explain the difference between Definition 3.14 and the so-called leaf invariance (see Definition 2.5(1)). In Definition 2.5(1), one begins with a non-degenerate leaf ℓ and postulates the existence of d pairwise disjoint preimage-leaves of ℓ. In Definition 3.14 we begin with any leaf ℓ 1 whose image is a non-degenerate leaf and postulate the existence of d − 1 pairwise disjoint (and disjoint from ℓ 1 ) leaves with the same image σ d (ℓ 1 ). This is a little stronger than Definition 2.5(1) as it does not follow from Definition 2.5(1) that we will be able to find siblings of any leaf mapped to ℓ. A surprising fact is that this subtle difference proves to be sufficient to imply all other properties of invariant geo-laminations.
More precisely, the main result of [6] is that sibling invariant laminations are invariant geo-laminations. An advantage of using sibling invariant laminations is that checking if a geo-lamination is sibling invariant requires considering only leaves of the lamination. Also, geo-laminations generated by laminations are sibling invariant [6], and it is proved in [6] that if sibling invariant laminations L i are such that continua L + i converge (in the Hausdorff metric) to a continuum K, then in fact there exists a sibling invariant geo-lamination L such that L + = K. Now we consider the union of all leaves of two coexisting laminations ∼ and ≃. By the above, this gives rise to a sibling invariant lamination L ∼ ∪ L ≃ . To show that this generates a lamination, we need more tools.
Definition 3.15 (Proper lamination [6]). Two leaves with a common endpoint v and the same non-degenerate image are said to form a critical wedge (with vertex v). A lamination L is proper if it contains no critical leaf with a periodic endpoint and no critical wedge with a periodic vertex.
A geo-lamination L has period matching property if any leaf of L with a periodic endpoint of period n is such that its other endpoint is also of period n. Lemma 3.16 follows from the definitions. Lemma 3.16. Suppose that ∼ is a lamination and L ∼ is the geo-lamination generated by ∼. Then L ∼ has period matching property. Also, any geo-lamination with period matching property is proper.
By Lemma 3.16, given two coexisting laminations ∼ and ≃, their geo-laminations L ∼ and L ≃ have period matching property. Then L ∼ ∪L ≃ also has period matching property and, hence, is a proper geo-lamination. Conversely, suppose that L is an invariant geo-lamination. Define an equivalence relation ≈ L by declaring that x≈ L y if and only if there exists a finite concatenation of leaves of L connecting x and y.  We defined tuning in Definition 1.8. Take an invariant lamination ≃ that coexists with an invariant stand alone quadratic gap U ; e.g., it may be that ≃ tunes U . We want to show that then ψ U transports ≃ to a quadratic invariant lamination, which we will denote ψ U (≃). Take a non-critical leaf ℓ of ≃ inside U . It has two sibling leaves which are disjoint. Clearly, one of them, say, a, is contained in U . If ψ U (a) and ψ U (ℓ) are non-disjoint, then either ψ U (ℓ) = ψ U (a) is a critical leaf or the two chords ψ U (ℓ), ψ U (a) form a critical wedge, a wedge subtended by a diameter. We need to show that the latter case is impossible. In this case, ℓ and a share endpoints with an edge e of U . There are no leaves of ≃ separating ℓ \ S 1 from a \ S 1 in D, since any such leaf would have to cross e. Therefore, ℓ and a are edges of the same gap G of ≃. Since some edges of G are sibling leaves, the gap G must be quadratic.
The sibling e * of e in G is clearly an edge of U connecting the endpoints of ℓ and a different from the endpoints of e.
As L ≃ is proper, it follows that ψ U (L ≃ ) is proper. Indeed, by Lemma 3.16, if an endpoint of a leaf ℓ of L ≃ is periodic, then ℓ is periodic. Consider a leaf ℓ of L ≃ and the leaf ψ U ( ℓ). If an endpoint of ψ U ( ℓ) is periodic, then ℓ has a σ 3 -periodic endpoint, hence ℓ is periodic, hence ψ U ( ℓ) is periodic. Thus the lamination ψ U (L ≃ ) = L is proper and, by Theorem 3.17, one can construct the lamination ≈ L which generates a geo-lamination L ≈L .
Let us compare L with L ≈L and show that they almost coincide.
Lemma 3.18. Let L be a proper quadratic geo-lamination. Then L ⊃ L ≈L . Moreover, L \ L ≈L consists of the grand orbit of a critical leaf and/or the grand orbit of a critical quadrilateral of L that is strictly inside a finite critical gap of L ≈L .
Proof. To prove that L ⊃ L ≈L , it suffices to show that the equivalence relation ≈ L produces leaves which the geo-lamination L already contains. By Thurston's No Wandering Triangle theorem [31], any finite non-precritical gap of a quadratic lamination is pre-periodic or periodic. Moreover, by [31], the vertices of a periodic gap must form one cycle. Hence any chord inside such a gap will cross itself and cannot be a leaf of any lamination. Therefore, if a gap G of ≈ L is not a gap of L, then it must be critical or pre-critical. Suppose that G is critical. If L has more than a critical leaf or/and a critical quadrilateral inside G, then the image of G is still a gap and has at least one chord inside. However, σ 2 (G) cannot be pre-critical, and, by the previous case, this is impossible.
Hence, any quadratic proper sibling lamination L can be cleaned (if necessary) by means of removing critical leaves/quadrilaterals of it contained inside appropriate finite gaps of L ≈L (as described above) as well as removing their pullbacks. This results into the geo-lamination generated by ≈ L . In particular, we can clean the geo-lamination ψ U (L ≃ ) constructed above and in this way relate it to a certain quadratic lamination ≍. Strictly speaking, the lamination ≍ coincides with ≈ ψU (L≃) (first the lamination ≃ generates the geo-lamination L ≃ , then the geo-lamination L ≃ is transported to the circle by the map ψ U , and then the geo-lamination ψ U (L ≃ ) generates the lamination ≈ ψU (L≃) ), however for brevity in what follows we will simply denote ≍ by ψ U (≃).
The above arguments allowed us to define the lamination ψ U (≃). They were based on the fact that U is an invariant quadratic stand alone gap. Literally the same arguments apply if U is a stand alone periodic quadratic gap (i.e., a periodic Fatou gap of degree two). Hence, in the periodic case, given a lamination ≃ that coexists with U , we can also define the lamination ψ U (≃). Lemma 3.19 proves in the case of tuning the claims established in Lemmas 3.11, 3.12 and 3.13. The proof is analogous to the proofs of Lemma 3.11 and 3.12 and is left to the reader. Lemma 3.19. Suppose that ≈ is a lamination which tunes an invariant quadratic gap U . Then the following holds.
1. If U is of regular critical type, then ≈ tunes the canonical lamination ∼ U . 2. If U is of periodic type and ≈ tunes V (U ) as well, then in fact ≈ tunes the canonical lamination ∼ U .
This provides a more explicit description of how a lamination can tune an invariant quadratic gap.

3.4.
Cubic laminations with no rotational gaps or leaves. We now describe all cubic laminations with no periodic rotational gaps or leaves (a rotational leaf is a periodic leaf with non-refixed endpoints). Recall that for an invariant quadratic gap of periodic type U we let M * (U ) denote the sibling leaf of M (U ) in V (U ). Recall from [6] that, for any invariant lamination ∼, the corresponding geo-lamination L ∼ is sibling invariant. Observe that we do not assume U to be a gap of ∼.
Proof. Since M is a leaf of ∼, then by the Sibling Property so is M * . Now, if there is a gap G of ∼ such that M and M * are edges of G, then, by definition of V (U ), we see that G ⊂ V (U ) is a σ k 3 -invariant gap. Applying the map ψ V (U) , we get a σ 2invariant gap ψ V (U) (G) that contains the angle 0. Clearly, then ψ V (U) (Bd(G)) = S 1 and hence G = V (U ).
Suppose that M and M * are not on the boundary of a single gap of ∼. Then there must be a leaf ℓ of L ∼ that separates M \ S 1 from M * \ S 1 in D. Note that one of its siblings ℓ * is also contained in the strip between M and M * . We claim that (ℓ ∪ ℓ * ) ∩ (M ∪ M * ) = ∅. To see this, note that if ℓ and M share an endpoint, then ℓ is not critical because L ∼ is proper. Hence σ k (ℓ) ∪ ℓ ∪ M is a tripod, which is impossible. Indeed, σ k (ℓ) cannot coincide with ℓ, since there are no k-periodic points between M and M * . Hence M is approached by leaves of ∼ separating M from M * (as such we can choose pullbacks of ℓ or ℓ * ). Choose one such leafq of ∼ close enough to M and then choose the leafq * with the same σ-image asq, separatingq from M * (q * is a leaf of ∼ by the Sibling Property). By Theorem 2.13, there exists a σ k -fixed gap or leaf Q in the closed strip S where S = S(q,q * ) is the open strip betweenq andq * . It follows that the entire orbit of Q is located in the same parts of the circle, where the orbit of V (U ) is located, which implies that Q ⊂ V (U ). Applying ψ V (U) to Q and using well-known facts about quadratic laminations and their invariant sets, we see that Q is rotational, as desired.
Let us characterize laminations with no rotational sets. Lemma 3.21. Suppose that a cubic invariant lamination ∼ has no periodic rotational gaps or leaves. Then either ∼ is empty, or it coincides with the canonical lamination of an invariant quadratic gap.
Proof. Suppose that a non-empty lamination ∼ has no rotational sets. Then by Theorem 2.13 there is an invariant gap U of degree 1 < k 3. If k = 3, then ∼ is empty, hence k = 2. If U is of regular critical type, then, by Lemma 3.11, the lamination ∼ is the canonical lamination ∼ U . Let U be of periodic type. Since M (U ) is a leaf of L ∼ , then by Lemma 3.20, so is M * (U ). Now, if there is a gap G of ∼ such that M (U ) and M * (U ) are edges of G, then, by Lemma 3.20, we have G = V (U ), and, by Lemma 3.13, the lamination ∼ coincides with the canonical lamination ∼ U . Suppose that M (U ) and M * (U ) are not contained in the same gap of ∼. Then, by Lemma 3.20, there exists a rotational gap or leaf of ∼, a contradiction.

Coexistence of quadratic invariant gaps and other laminational sets.
Here we show how invariant quadratic gaps can coexist with each other as well as how they can coexist with other laminational sets (we consider gaps of laminations, i.e., invariant quadratic gaps which are regular critical or of periodic type). We are motivated here by the desire to provide a model for specific families of laminations and laminational sets (such as the family of all quadratic invariant gaps) which should be helpful in the description of the entire cubic Mandelbrot set M 3 . Some of these lemmas are used in [9].
Let us discuss two special quadratic invariant gaps which often play the role of exceptions to the claims proven below. Let 0 1 2 = Di be the unique chord in D with σ-invariant endpoints. Let FG a be the convex hull of all points with orbits above Di and FG b be the convex hull of all points with orbits below Di. Then Di is the major of both gaps. However as a major of FG b it should be viewed so that the positive direction on Di is from 0 to 1 2 , and if Di is considered as the major of FG a , then the positive direction on Di is from 1 2 to 0. Recall that when talking about a Jordan curve K which encloses a simply connected domain W on the plane, by the positive direction on K one means the counterclockwise direction with respect to W , i.e., the direction of a particle moving along K so that W remains on its left.
Let U be an invariant quadratic gap. If U is of regular critical type, then we set M * (U ) = M (U ); if U is of periodic type, then we set M * (U ) to be the leaf that is not an edge of U and that has the property σ(M * (U )) = σ(M (U )). We summarize a few simple facts If a lamination has the gap U , it must have the leaf M * (U ). Let S U be the closed strip in D between M (U ) and M * (U ), and set H(M (U )) = H(U ). In the regular critical case, we have 1 3 = |H(U )|; in the periodic case, by Lemma 3.1, we have Then an edge ℓ of A connects two points t 1 , t 2 of the boundary of a gap T of the canonical lamination ∼ U of U and passes through the interior of T . This includes the possibility that ℓ crosses two edges of T in D. We show that σ q (A) separates M (U ) from M * (U ) for some q. Indeed, as long as T maps onto its image one-to-one, the images of ℓ connect the corresponding images of t 1 , t 2 . By the properties of ∼ U there is the least n with σ n (T ) = U or σ n (T ) = V . The leaf σ n (ℓ) connects two points of Bd(σ n (T ))) and passes through the interior of σ n (T ). Hence σ n (T ) = U , the leaf σ n (ℓ) connects two points of Bd(V ) and passes through the interior of V . Let ψ V collapse all edges of V ; then ψ V semiconjugates σ k | Bd(V ) to σ 2 and maps σ n (ℓ) to a chord inside D. By properties of σ 2 , the chord σ q (ℓ) = e separates M (U ) and M * (U ) for some q n. The claim aboutê immediately follows (the critical set D = U of ∼ is located between e andê).
Let Lemma 3.25. Let ∼ i (i = 1, 2) be two laminations. Suppose that A is a nondegenerate class or an infinite gap of both laminations, and that each ∼ i has a quadratic invariant gap U i , i = 1, 2. Then U 1 = U 2 except for the case when A is a gap or a class of the lamination that has both FG a and FG b as gaps; in the latter case, U 1 and U 2 may coincide with FG a and FG b (in any order).
Proof. Suppose that A is a non-degenerate class or an infinite gap of a lamination ∼ that has an invariant quadratic gap U . We show how to recover U knowing A, except in the case, where both FG a and FG b are gaps of ∼. This will imply the statement of the lemma.
For every leaf or gap T of ∼, let ||T || denote the length of the largest hole of T . Set µ = inf n 0 ||σ n (A)||. There may or may not be a leaf or gap B in the forward orbit of A with the property ||B|| = µ. If there is no such B, then there exists a leaf ℓ of ∼ such that ||ℓ|| = µ, and there is a sequence of leaves or gaps in the forward orbit of A accumulating on ℓ. In this latter case, we set B = ℓ. Suppose that no image of A has 0 1 2 as its edge, and let H be the largest hole of B. We claim that, in this case, U ′ can be recovered as the set of all points in H, whose forward orbits stay in H.
To prove the claim, let M denote the major of U and, as before, let M * denote its sibling not in U . Observe that, by our assumptions and by Lemma 3. There are one or two majors of G. We classify invariant rotational gaps by types. This classification mimics Milnor's classification of hyperbolic components in slices of cubic polynomials and quadratic rational functions [23,21]. Namely, we say that • the gap G is of type A (from "Adjacent") if G has only one major (whose length is at least 2 3 ); • the gap G is of type B (from "Bi-transitive") if G has two majors that belong to the same periodic cycle; • the gap G is of type C (from "Capture") if it is not type B, and one major of G eventually maps to the other major of G; • the gap G is of type D (from "Disjoint") if there are two majors of G, whose orbits are disjoint. Clearly, it follows from the definitions that finite rotational gaps cannot be of type C (since then σ| Bd(G) is not one-to-one). Also, if G is of type B, then σ| Bd(G) is one-to-one, and hence G must be finite.

Finite rotational sets.
A classification of finite rotational sets (under the name of fixed point portraits) can be found in [17]. We now give some examples illustrating a part of this classification concerning the degree 3 case.
Let G be a finite invariant rotational set (as we fix G in this section, we may omit using G in the notation). By [18], there are at most two periodic orbits (of the same period denoted in this section by k) forming the set of vertices of G. If vertices of G form two periodic orbits, points of these orbits alternate on S 1 . 2 and is longer than 2 3 . Let G = ℓ = ab be an invariant leaf. We can think of G as a gap with empty interior and two edges ab and ba, and deal with all finite invariant sets in a unified way. Let us list all non-degenerate invariant leaves ab. Either points a, b are fixed, or they form a two-periodic orbit. In the first case, we have the leaf 0 1 2 = Di, in the second case, we have one of the leaves 1 . Informally, we regard Di as an invariant rotational set of type D (even though its rotation number is 0).
Let G be a finite invariant laminational set with m edges ℓ 0 , . . . , ℓ m−1 . For each i, let FG i be the convex hull of all points x ∈ H G (ℓ i ) with σ j (x) ∈ H G (σ j (ℓ i )) for every j 0 (compare this to the definition of a vassal in Section 3). It is straightforward to see that FG i are infinite stand alone gaps such that FG i maps to FG j if ℓ j = σ(ℓ i ). These gaps are called the canonical Fatou gaps attached to G. The gap FG i is critical if and only if the corresponding edge ℓ i is a major.  Suppose first that G is of type B or D. Then by definition there are two major edges of G, which are denoted by M 1 and M 2 . Let H 1 and H 2 be the corresponding holes. By Lemma 2.12, we may assume that 0 ∈ H 1 and 1 2 ∈ H 2 . Since M 1 and M 2 are periodic, their lengths are strictly greater than 1 3 . Let U 1 and U 2 be the canonical Fatou gaps attached to M 1 and M 2 , respectively. Thurston's pullback construction [31] yields an invariant lamination formed by the pullbacks of G disjoint from the interiors of U 1 and U 2 . More precisely, we can define a lamination ∼ G as follows: two points a and b on the unit circle are equivalent if there exists N 0 such that σ N (a) and σ N (b) are vertices of G, and the chords σ i (a)σ i (b) are disjoint from G and from the interior of U 1 ∪ U 2 for i = 0, . . . , N − 1. It is straightforward to check that ∼ G is indeed an invariant lamination. This lamination is called the canonical lamination associated with G.
Assume now that G is of type A. Let M be the major edge of G, and U the corresponding canonical Fatou gap attached to G at M . The canonical lamination ∼ G of G is defined similarly to those for types B and D. Namely, two points a and b on the unit circle are equivalent with respect to ∼ G if there exists N 0 such that σ N (a) and σ N (b) are vertices of G, and the chord σ i (a)σ i (b) is disjoint from G and from the interior of U for i = 0, . . . , N − 1.
Lemma 4.4 is similar to Lemma 3.11. It is based on Thurston's pullback construction of laminations.  Let G be an invariant Siegel gap of rotation number τ . Then G may have one or two critical majors of length 1 3 , or one critical major of length 2 3 . It is also possible that G has a non-critical major. However, a non-critical major eventually maps to the critical major by Lemma 2.8 (in this case, G is of type C). Thus an infinite rotational gap G can have type A, C or D.
We now construct the canonical lamination for a Siegel gap G of type D with critical edges L and M . Consider well-defined pullbacks of G attached to G at L and M . Then apply Thurston's pullback procedure to these gaps. As holes in the union of bases of these gaps are shorter than 1 3 , the pullbacks of the gaps converge in diameter to 0. Alternatively, we can define ∼ G as follows: two points a and b in the unit circle are equivalent if there exists N > 0 such that σ N (a) and σ N (b) lie on the same edge of G, and the chords σ i (a)σ i (b) are disjoint from G for i = 0, . . . , N − 1.
We will not define canonical laminations of type A or C Siegel gaps.
Lemma 4.6. Suppose that G is a type D stand alone invariant gap of Siegel type, and ∼ is an invariant cubic lamination with gap G. Then ∼ coincides with ∼ G .
The proof goes almost verbatim as in Lemma 3.11. It is based on Thurston's pullback construction of laminations. 5. The description of the Combinatorial Main Cubioid. If G is a finite σ 2 -invariant rotational set, we define the canonical lamination of G as the only quadratic invariant lamination with a cycle of Fatou gaps attached to edges of G (it represents a parabolic quadratic polynomial z 2 + c, whose parameter c belongs to the Main Cardioid). Similarly, if G is a stand alone invariant Siegel gap with respect to σ 2 , we define the canonical lamination of G as the unique quadratic invariant lamination that has G as its gap.
Proposition 5.1. A non-empty quadratic lamination ∼ with at most one periodic (hence fixed) rotational set G coincides with the canonical lamination of G. The Combinatorial Main Cardioid CA c consists of quadratic laminations with at most one periodic rotational set.
Proof. By Theorem 2.13, the lamination ∼ has an invariant rotational set G 2 . If G 2 is a Siegel gap, then ∼ is the canonical lamination of G 2 because all pullbacks of G 2 are uniquely defined (cf. Lemma 3.11). Let G 2 be a finite gap or leaf. By [18], the laminational set G 2 has a unique major M 2 separating G 2 from 0, and the edges of G 2 form one cycle (of period r). Let V 2 be the Fatou gap of the canonical lamination of G 2 which has M 2 as one of its edges. Let M * 2 be the sibling of M 2 ; then M * 2 is an edge of V 2 . By Theorem 2.13 and because ∼ has at most one periodic rotational gap or leaf, the strip between M 2 and M * 2 contains a σ r 2 -invariant Fatou gap U of ∼ of degree greater than 1. In fact, r is the period of U as U passes through every hole of G 2 before returning. Hence U ⊂ V 2 , which immediately implies that U = V 2 , and ∼ is the canonical lamination of G 2 .
Let us go back to cubic laminations ∼ (recall that σ 3 is denoted by σ). Gaps U (c) for critical chords c are defined right after Lemma 3.8. Proof. We need to show that a leaf of ∼ does not intersect an edge of U (c) = U in D unless it coincides with it. By way of contradiction, suppose that a leaf ℓ of ∼ intersects an edgej = ℓ of U in D. Let us show that then σ(ℓ) intersects σ(j) in D. Indeed, there exists a component of D \ (c ∪ d) whose closure contains both ℓ andj. Hence, σ(j) = σ(ℓ). The only case when σ(ℓ) and σ(j) do not intersect in D under the circumstances is as follows: ℓ andj contain distinct endpoints of one of the critical chords, c or d. Let us show that this is impossible.
By the assumptions on d the leaf ℓ does not contain an endpoint of d. Suppose that each of the chords ℓ andj contains an endpoint of c. Then an endpoint of c belongs to U (c), and since c has no periodic endpoints, c is regular critical. Hence the only edge of U (c) containing an endpoint of c is c, andj = c. But then c and ℓ do not intersect in D because c and leaves of ∼ distinct from c are disjoint inside D by the assumption. Thus σ(ℓ) and σ(j) = σ(ℓ) intersect in D. By induction this implies that for any n 0, σ n (j) = σ n (ℓ) intersect in D.
Since no image of ℓ can intersect c while not coinciding with c, this implies that c cannot be regular critical. Then the whole orbit ofj stays strictly on one side of c which implies that so does the whole orbit of ℓ. Hence (see Subsection 3.1) the endpoints of ℓ belong to U ′ (c), and ℓ =j cannot intersectj in D, a contradiction. Denote the major of U (c) by M . The last claim of the lemma follows from the fact that if M is a leaf of ∼ then ∼ tunes U (c) (because ∼ is backward invariant and coexists with d) while if M is not a leaf of ∼ then no edge of U (c) can be a leaf of ∼ (because ∼ is forward invariant and by Lemma 3.3 which states that all edges of U (c) are preimages of M ). this yields that M must be an edge of G, a contradiction with M not being a leaf of ∼.
A similar argument holds in the case, where G is a Siegel gap. Let d be a critical edge of G. There is some other critical leaf or gap C. Let c be a critical chord in C. As before, c may be chosen to have non-periodic endpoints and G ⊂ U (c). Since no leaves of ∼ other than d intersect d, then c and d satisfy the conditions of Lemma 5.2.
We are ready to prove the Main Theorem stated in the Introduction. The following statement makes it more precise (we refer to the notation introduced in the Main Theorem). Let ∼ be a lamination from CU c . Then by Lemma 5.4 ∼ coexists with a quadratic invariant gap U . The gap U in the Main Theorem can be chosen as in Lemma 5.4.
Theorem 5.5. Assume the conditions of Lemma 5.4 and adopt the notation from its conclusion. If the ψ U -image of the major M of U does not eventually map (by σ 2 ) to a periodic Fatou gap of ≍= ψ U (∼), then case (2) of the Main Theorem holds. This is also the case, when U is of periodic type and the lamination ∼ is not the canonical lamination of a finite invariant rotational gap or leaf of type A or B.
Proof of the Main Theorem and of Theorem 5.5. Suppose that ∼ is not a canonical lamination of an invariant quadratic gap. Then, by Theorem 2.13, there exists an invariant rotational gap or leaf G. Suppose that G is of type D (finite or Siegel). Then, by Lemma 4.5 and Lemma 4.6, the lamination ∼ must be the canonical lamination of G. Choose a major M of G. It follows that the same major M defines also an invariant quadratic gap U . It is easy to see that then ∼ tunes the canonical lamination ∼ U according to an appropriate quadratic invariant lamination from CA c , which corresponds to case (2). From now on, we may assume that G is not of type D.
By Lemma 5.4, the lamination ∼ coexists with a quadratic invariant stand-alone gap U , either ∼ tunes U or no edge of U belongs to ∼ and U can be chosen to be of regular critical type (moreover, if ∼ is not canonical, then U must be of regular critical type). Furthermore, G ⊂ U . Suppose that the map ψ U projects the restriction of ∼ onto U to a quadratic invariant lamination ≍= ψ U (∼). By Definition 1.10, the lamination ∼ weakly tunes U according to the lamination ≍. Since ∼ has a unique rotational set, so does ≍. By Proposition 5.1, the lamination ≍ comes from the Combinatorial Main Cardioid CA c .
If U is of periodic type and its canonical lamination is tuned by ∼ then, if the gap V (U ) is of period greater than 1, it cannot be tuned by ∼ as this would create a rotational set of period greater than 1. Now, the only cases when V (U ) is of period 1 are when U = FG a , V (U ) = FG b , or U = FG b , V (U ) = FG a . In the former case, if ∼ is not empty inside U , then again V (U ) cannot be non-trivially tuned by ∼ as this would create two rotational sets of ∼. Suppose now that ∼ is empty inside U . Then it may happen that V (U ) is non-trivially tuned by ∼. Similarly to the above, this tuning must be according to some quadratic lamination ≍ from CA c . In that case we simply declare that U = V (U ) = FG b and V ( U ) = FG a . Clearly, this is possible, and with this choice of the gap tuned by ∼, the lemma holds. The case U = FG b , V (U ) = FG a is similar. Now we need to prove the remaining claims of the theorem. By Lemma 5.4, to see whether ∼ tunes U , we need to see whether the major M of U belongs to ∼. Suppose that the point ψ U (M ) does not eventually map (by σ 2 ) to a periodic infinite gap of ≍. Then well-known properties of quadratic laminations from the Combinatorial Main Cardioid CA c imply that ψ U (M ) is separated from the rest of the circle by a sequence of leaves of ≍. Hence M is the limit of appropriately chosen leaves of ∼, and so M itself is a leaf of ∼. Thus, if ψ U (M ) does not eventually map (by σ 2 ) to a periodic infinite gap of ≍, then ∼ tunes U .
Finally, let us prove the last claim of the theorem. We need to prove that if U is of periodic type and ∼ is not canonical, then case (2) must hold. However, this immediately follows from Lemma 5.4 as in the case of weak tuning and noncanonical lamination this lemma states that U must be of regular critical type. This completes the proof.
The statement of the Main Theorem is somewhat involved. However it leads to a more explicit description if one thinks of constructing a non-empty lamination ∼ from CU c . Indeed, for definiteness assume that ∼ has a finite rotational gap G. Observe that for canonical laminations of type D the explanation as how ∼ fits into the description from the Main Theorem is given in the proof. Otherwise, just like in the arguments of some of our theorems, consider both critical sets of ∼. One of them is attached to G. The other one can be either (a) a vassal gap of some invariant quadratic gap U of periodic type, or (b) a critical leaf which is a major of regular critical type of some quadratic invariant gap U , or (c) the same as the first one (canonical lamination of type A), or (d) an infinite gap-preimage of the first one (canonical laminations of type B or C), or (e) a finite gap-preimage of G.
In cases (a) or (b) the lamination ∼ tunes the canonical lamination of U according to a lamination from the Main Cardioid. In the other cases the basis of the second critical set contains endpoints of a critical leaf which is itself a major of regular critical type of some quadratic invariant gap U ; moreover, all other edges of U are also present as diagonals (but not as edges) of other gaps of ∼. The construction of such ∼ can be viewed as a three step process: first, we take the canonical lamination of an invariant quadratic gap U , then U is tuned according to a quadratic lamination from the Main Cardioid, and then finally edges of U and their preimages are erased giving rise to ∼ (whether we get a lamination described in (c), (d) or (e) above depends on the relation between the major of U , the gap G and gaps of the canonical lamination of G attached to G).
In conclusion, we prove Corollary 1.6 from the Introduction, which allows for a shorter definition of laminations from the Combinatorial Main Cubioid CU c .
Proof of Corollary 1.6. The "if" part of the claim follows immediately from definitions. To prove the "only if" part of this corollary one simply has to go over different types of laminations listed in Theorem 5.5 (or in our explanation right after this theorem). Indeed, first we observe that by definition if ∼ belongs to CU c then it has at most one rotational periodic (hence fixed) set. Now, consider the second part of the claim. It is obvious for canonical laminations of quadratic invariant gaps or for canonical laminations of finite gaps of type D. In the case when the lamination ∼ is obtained as described in Case (1) of Theorem 5.5 -or, equivalently, in cases (c), (d) or (e) above -the only periodic leaves of ∼ are edges of the periodic rotational gap G (and only in the case when G is finite), so the claim follows. Finally, if Case (2) of Theorem 5.5 applies, then, in addition to the edges of G, the lamination ∼ may also have periodic leaves ℓ that form the orbit of a major M of periodic type generating an invariant quadratic gap U from Theorem 5.5. However, in that case, there must exist an infinite gap attached to each such leaf ℓ that itself belongs to the orbit of the vassal gap attached to M . Thus the claim holds in this case too.