Density of orbits in laminations and the space of critical portraits

Thurston introduced $\si_d$-invariant laminations (where $\si_d(z)$ coincides with $z^d:\ucirc\to \ucirc$, $d\ge 2$). He defined \emph{wandering $k$-gons} as sets $\T\subset \ucirc$ such that $\si_d^n(\T)$ consists of $k\ge 3$ distinct points for all $n\ge 0$ and the convex hulls of all the sets $\si_d^n(\T)$ in the plane are pairwise disjoint. Thurston proved that $\si_2$ has no wandering $k$-gons and posed the problem of their existence for $\si_d$,\, $d\ge 3$. Call a lamination with wandering $k$-gons a \emph{WT-lamination}. Denote the set of cubic critical portraits by $\A_3$. A critical portrait, compatible with a WT-lamination, is called a \emph{WT-critical portrait}; let $\WT_3$ be the set of all of them. It was recently shown by the authors that cubic WT-laminations exist and cubic WT-critical portraits, defining polynomials with \emph{condense} orbits of vertices of order three in their dendritic Julia sets, are dense and locally uncountable in $\A_3$ ($D\subset X$ is \emph{condense in $X$} if $D$ intersects every subcontinuum of $X$). Here we show that $\WT_3$ is a dense first category subset of $\A_3$. We also show that (a) critical portraits, whose laminations have a condense orbit in the topological Julia set, form a residual subset of $\A_3$, (b) the existence of a condense orbit in the Julia set $J$ implies that $J$ is locally connected.

Definition 1.2. For a topological space X a set A ⊂ X is continuumwise dense (abbreviated condense) in X if A ∩ Z = ∅ for each non-degenerate continuum Z ⊂ X. A map f : X → X is also called condense if there exists x 0 ∈ X such that {f n (x 0 ) | n ≥ 0} is condense in X.
It is not hard to see that condensity is much stronger than density. For example, if J is a Julia set from the real quadratic family which is not homeomorphic to an interval, the set of endpoints is dense in J, but not condense. Moreover, in this case the set of transitive points (i.e., points with dense orbit in J) is a subset of the endpoints of J, so such maps are not condense.
To state the results of [3] precisely, we must indicate in which parameter space we are working. Polynomials are naturally associated to critical portraits, introduced by Yuval Fisher in his Ph.D. thesis [11]. Let σ d : S → S be the angle d-tupling map σ d (z) = z d . A degree d critical portrait, loosely speaking, is a maximal collection Θ = {Θ 1 , . . . , Θ n } of sets of angles in S which are pairwise disjoint, pairwise unlinked (i.e., have disjoint convex hulls in D when angles are interpreted as points in S), and such that σ d (Θ i ) is a point for each Θ i ∈ Θ (it is easy to see that (|Θ i | − 1) = d − 1).
This notion is used to capture the location of critical points. The set of all critical portraits of degree d is denoted A d , and is naturally endowed with a topology (see Definition 2.4 for details). We say that a critical portrait Θ corresponds to a polynomial P with dendritic Julia set if for each Θ i ∈ Θ there is a distinct critical point c i ∈ J P such that the external rays whose angles are in Θ i land at c i (see Section 2.3 for more information). Now we state the main theorem of [2].

Theorem 1.3 ([2])
. A 3 contains a dense locally uncountable set {Θ α | α ∈ A} of critical portraits such that for each α ∈ A the following holds: • Θ α corresponds to a polynomial P α with dendritic Julia set J Pα , • {P α | JP α } are pairwise non-conjugate, and • J Pα contains a wandering vertex with condense orbit.
The aim of this paper is to further investigate the notions and objects studied in Theorem 1.3, such as condensity and the set of critical portraits which correspond to polynomials with wandering vertices. To explain our results, we recall constructions from [14,3]: given a polynomial P with connected Julia set J P , one can construct a corresponding locally connected continuum J ⊂ C (called a topological Julia set) and branched covering map f : C → C (called a topological polynomial) so that P is monotonically semiconjugate to f (i.e., there exists a monotone map m : C → C such that m•P = f •m) and J = m(J P ).
We refer to f | J as the locally connected model of P . It is known [6] that in some cases J is a single point.
Let us describe the organization of the paper and the main results. After discussing preliminary notions and history in Section 2, we study properties of condense maps in Section 3. In particular we show in Theorem 3.6 that polynomials which admit condense orbits either in their Julia sets (or in some circumstances their locally connected models) have locally connected Julia sets. In Section 4 we prove that the set of cubic critical portraits corresponding to polynomials with condense orbits in their Julia sets is residual in A 3 (Theorem 4.1), while the set of critical portraits which correspond to polynomials with wandering vertices is meager (Theorem 4.3).

Laminations.
In what follows, we parameterize the circle as S = R/Z, so the total arclength of S is 1. The positive direction on S is the counterclockwise direction, and by the arc (p, q) in the circle we mean the positively oriented arc from p to q. A (strictly) monotone map g : (p, q) → S is a map (strictly) monotone at each point of (p, q) in the sense of positive direction on S. By Ch(A) we denote the convex hull of a set A ⊂ C and by |B| we denote the cardinality of the set B.
Laminations are combinatorial structures on the unit circle, introduced by Thurston [24] as a tool for studying individual complex polynomials P : C ∞ → C ∞ and the space of all of them. In what follows, we use a restricted formulation of this concept. Specifically, [24] defines a lamination as a collection of chords in the unit disk, satisfying certain dynamical properties. These collections are intended to reflect the pattern of external rays landing in the Julia set. We instead use the formulation contained in [14], explained in detail below, based on pinched disk models (see [8] for more information in this vein, and also [5]).
Let P be a degree d polynomial with a locally connected (and hence connected) Julia set J P ; we will recall how to associate an equivalence relation ∼ P on S to P , called the d-invariant lamination generated by P . The filled-in Julia set K P is compact, connected, and full, so its complement C ∞ \ K P is conformally isomorphic to the open unit disk D. By [18,Theorem 9.5], there is a particular conformal isomorphism Ψ : D → C ∞ \ K P so that Ψ conjugates σ d (z) = z d on D to P | C∞\KP (i.e., Ψ(z d ) = (P | C∞\KP • Ψ)(z) for z ∈ D). When J P is locally connected, Ψ extends to a continuous map Ψ : D → C ∞ \ K P which semiconjugates z → z d on D to P | C∞\KP . Let ψ : S → J P denote the restriction Ψ| S . Define the equivalence ∼ P on S so that x ∼ P y if and only if ψ(x) = ψ(y); this equivalence relation is the aforementioned d-invariant lamination generated by P . The quotient space S/ ∼ P = J ∼P is homeomorphic to J P and the induced map f ∼P : Kiwi [14] extended this construction to polynomials P with no irrationally neutral cycles and introduced a similar d-invariant lamination ∼ P . Then J ∼P = S/ ∼ P is locally connected and P | JP is semi-conjugate to f ∼P by a monotone map m : J p → J ∼P , i.e., a map m whose point preimages are connected. This was extended in [3] to all polynomials P with connected J P . The lamination ∼ P combinatorially describes the dynamics of P | JP .
One can introduce abstract laminations (frequently denoted by ∼) as equivalence relations on S having properties in common with laminations generated by polynomials as above. Consider an equivalence relation ∼ on the unit circle S. Equivalence classes of ∼ will be called (∼-)classes and will be denoted by boldface letters. A ∼-class consisting of two points is called a leaf ; a class consisting of at least three points is called a gap (this is more restrictive than Thurston's definition in [24]). Fix an integer d > 1. Then ∼ is said to be a d-invariant lamination if: (E1) ∼ is closed: the graph of ∼ is a closed set in S × S; (E2) ∼-classes are pairwise unlinked: if g 1 and g 2 are distinct ∼-classes, then their convex hulls Ch(g 1 ), Ch(g 2 ) in the unit disk D are disjoint; (E3) ∼-classes are either totally disconnected (and hence ∼ has uncountably many classes) or the entire circle S is one class; (D1) ∼ is forward invariant: for a class g, the set σ d (g) is also a class; (D2) ∼ is backward invariant: for a class g, its preimage σ −1 d (g) = {x ∈ S : σ d (x) ∈ g} is a union of classes; and (D3) for any gap g, the map σ d | g : g → σ d (g) is a covering map with positive orientation, i.e., for every connected component (s, Then we call f ∼ a topological polynomial, and J ∼ a topological Julia set.

Bounds for wandering classes. J. Kiwi [13] extended the No Wandering Triangles Theorem by showing that a wandering gap in a d-invariant lamination is at most a d-gon.
Thus all infinite ∼-classes (and Jordan curves in J ∼ ) are preperiodic. In [17] G. Levin showed that laminations with one critical class have no wandering gaps. For a lamination ∼, let k ∼ be the size of a maximal collection of non-degenerate ∼-classes whose σ d -images are points and whose orbits are infinite and pairwise disjoint. Also, let N ∼ be the number of cycles of infinite ∼-classes plus the number of cycles of Jordan curves in J ∼ .

Theorem 2.1 ([5]). Let ∼ be a d-invariant lamination and let Γ be a non-empty collection of wandering
In particular, in the cubic case if Γ is non-empty, then it must consist of one non-precritical ∼-class with three elements, all ∼-classes are finite, J ∼ is a dendrite, and both critical classes are leaves with disjoint forward orbits.

Critical portraits.
Following [24] and [9,10] we look at the set C d from infinity and consider the shift locus, which is the set S d of polynomials whose critical points escape to infinity. The set S d is the unique hyperbolic component of P d consisting of polynomials with all cycles repelling. It is not known if all polynomials with all cycles repelling belong to the set S d . Looking at C d from infinity means studying locations of polynomials in S d depending on their dynamics and using this to describe the polynomials belonging to S d ∩ C d . A key tool in studying C d is critical portraits, introduced in [11], and widely used afterward (see, e.g., [1,21,12] and [15]). We now recall some standard material; here we closely follow [15,Section 3]. Call a chord with endpoints

Definition 2.2.
A critical portrait is a collection Θ = {Θ 1 , . . . , Θ n } of finite subsets of S such that the following hold: 1. the boundary of the convex hull Ch(Θ i ) of every set Θ i consists of critical chords (under σ d ); 2. the sets Θ 1 , . . . , Θ n are pairwise unlinked (that is, convex hulls of the sets Θ i are pairwise disjoint); and 3. ( The sets Θ 1 , . . . , Θ n are called the initial sets of Θ (or Θ-initial sets). Denote by A(Θ) the union of all angles from the initial sets of Θ. The convex hulls of the Θ-initial sets divide the rest of the open unit disk into components. In Definition 2.3, points of S \ A(Θ) are declared equivalent if they belong to the boundary of one such component; we do not assume that Θ is a critical portrait because we need this equivalence later in a more general situation. Definition 2.4 (compact-unlinked topology [15]). Define the space A d as the set of all critical portraits endowed with the compact-unlinked topology generated by the subbasis Note for example that A 2 is the quotient of S with antipodal points identified, so it is homeomorphic to the unit circle. For a critical portrait Θ, a lamination ∼ is called Θ-compatible if all Θ-initial sets are contained in ∼-classes; if there is a Θ-compatible WT-lamination, Θ is said to be a WT-critical portrait. The trivial lamination, identifying all points of S, is compatible with any critical portrait.
To define critical portraits with aperiodic kneading, let us introduce the notion of a one-sided itinerary for t ∈ S (see [15]). Given a critical portrait Θ = {Θ 1 , . . . , Θ d } with Θ-unlinked classes L 1 (Θ), . . . , L d (Θ) and θ ∈ S, define i + (θ) (respectively, i − (θ)) as the sequence (i 0 , i 1 , . . . ), with i j ∈ {1, . . . , d} such that there are y n ց θ (respectively, y n ր θ) with σ j d (y n ) ∈ L ij (Θ) for n sufficiently large. Also, define the itinerary i(θ) as a sequence I 0 I 1 . . . such that each I j is the set from Θ ∪ {L 1 (Θ), . . . , L d (Θ)} to which σ j (θ) belongs. An angle θ ∈ S is said to have a periodic kneading if i + (θ) or i − (θ) is periodic. A critical portrait Θ is said to have aperiodic kneading if no angle from A(Θ) has periodic kneading. The family of all degree d critical portraits with aperiodic kneading is denoted by AP d . ([14, 15]). The lamination ∼ Θ is the smallest closed equivalence relation identifying any pair of points x, y ∈ S where i + (x) = i − (y). By Kiwi [14,15], for any critical portrait Θ the relation ∼ Θ is a Θ-compatible lamination; it is said to be generated by Θ.

Definition 2.5
Critical portraits reflect the landing patterns of the external rays at the critical points. By Kiwi [15], a nice correspondence between critical portraits of degree d and the set S d ∩ C d associates to each critical portrait Θ ∈ A d a connected set I(Θ) ⊂ S d ∩ C d , called the impression of Θ, such that the dynamics of a polynomial in I(Θ) is closely related to the properties of Θ. The relation is especially nice when Θ has aperiodic kneading. The following fundamental result of Kiwi [14,15] explicitly lists properties of critical portraits with aperiodic kneading and their connections to polynomials. Theorem 2.6. Let Θ ∈ AP d . Then ∼ Θ is the unique Θ-compatible invariant lamination. The quotient J ∼Θ is a non-degenerate dendrite, and all ∼-classes are finite. Furthermore, there exists a polynomial P whose Julia set J P is a non-separating continuum in the plane and P | JP is monotonically semiconjugate to f ∼Θ | J∼ Θ . The semiconjugating map m Θ,P = m : J P → J ∼Θ maps impressions of external angles to points and maps the set of Ppreperiodic points in J P bijectively to the set of f ∼Θ -preperiodic points. Moreover, J P is locally connected at all P -preperiodic points.
In the situation of Theorem 2.6 polynomials P such that P | JP is monotonically semiconjugate to f ∼Θ | J∼ Θ are said to be associated to the critical portrait Θ.

2.4.
Monotone models for connected Julia sets. As was explained in Section 1, the main results of [14,3] yield a locally connected model for the restriction of a polynomial to its connected Julia set. We will need a detailed version of these results stated below in Theorem 2.7. 1. J ∼ = M P (J P ) and J P ⊂ M −1 P (J ∼ ) ⊂ K P . 2. M P sends impressions of J P to points. 3. m P = M P | JP is the finest monotone map of J P onto a locally connected continuum (i.e., if ψ : J P → T is a monotone map onto a locally connected continuum T , then there is a monotone map ψ ′ : J ∼ → T such that ψ = ψ ′ • m P ). 4. M P semiconjugates P to a branched covering map g P : C → C under which J ∼ is fully invariant so that g P | J∼ is conjugate to the topological polynomial f ∼ .

Remark 1.
Suppose that Θ ∈ AP d is associated to the polynomial P ; let us show that the lamination ∼ Θ defined in Theorem 2.6 and the lamination ∼ P defined in Theorem 2.7 coincide. Indeed, by Theorem 2.7 there exists a monotone map ψ ′ : J ∼P → J ∼Θ . If this map is not a homeomorphism, it will collapse a non-degenerate subcontinuum Q ⊂ J ∼P to a point x ∈ J ∼Θ . Since impressions map to points of J ∼P , infinitely many distinct impressions of external rays are contained in the fiber m −1 Θ,P (x) which by Theorem 2.6 implies that the ∼ Θ -class corresponding to x is infinite. This contradicts Theorem 2.6, which states that ∼ Θ -classes are finite. Theorem 2.7 establishes the semiconjugacy m P on the entire complex plane, so that m P -images of external rays to J P are curves in C accumulating on points of J ∼P . For x ∈ J ∼P , the set m −1 P (x) ∩ J P is the union of impressions of angles α such that m P (R α ) lands on x . The order of x in J ∼P is the number of components of J ∼P \ {x} and can be either a finite number or infinity. By Theorem 2.7 if the order of x in J ∼P is finite then it equals the number of angles with impressions in m −1 P (x) (or equivalently the number of angles whose impressions intersect m −1 P (x)). If the order of x in J ∼P is infinite, then there are infinitely many angles with impressions in m −1 P (x).

Condensity.
We begin with a few lemmas concerning the dynamics of a condense topological polynomial. If J is a dendrite, by [a, b] J we mean the unique arc in J connecting the points a, b ∈ J. A continuum X ⊂ C is called unshielded if it coincides with the boundary of the unique unbounded component of C \ X. Note that all connected Julia sets of polynomials and all topological Julia sets are unshielded continua. A point x ∈ X is called a cutpoint of X if X \ {x} is not connected. In what follows a lamination ∼ such that f ∼ is condense is called condense; also, a critical portrait compatible with a condense lamination is said to be condense.
Lemma 3.1. If X ⊂ C is an unshielded locally connected continuum and A ⊂ X is connected and dense in X, then A is condense in X and contains all cutpoints of X.
Proof. If Z ⊂ X is a closed set with X \ Z disconnected, then all components of X \ Z are open. Hence all such components intersect A. Since A is connected, this implies that A ∩ Z = ∅. Suppose that A is not condense in X. Then there exists an arc I ⊂ X disjoint from A. Note that X \ I is open and connected (by virtue of containing A). Therefore X \ I is path connected and locally path connected. It follows that there exists a simple closed curve S ⊂ X which contains a non-degenerate subsegment I ′ with endpoint a ′ , b ′ of I. The curve S encloses a topological disk U . Clearly, any two-point set {a, b} ⊂ S separates X (two external rays landing at a and b and an arc inside U from a to b disconnect C).
Let us now study condensity in the context of laminations. We call a lamination ∼ degenerate if the whole S forms a ∼-class (and so J ∼ is a point); we call ∼ trivial if all ∼-classes are singletons (and J ∼ = S).

Lemma 3.2.
Let ∼ be a condense lamination. Then either ∼ is degenerate, or ∼ is trivial, or J ∼ is a dendrite.
Proof. Suppose J ∼ is non-degenerate and let x ∈ J ∼ be a point with condense orbit. If J ∼ is not a dendrite, then it contains a Jordan curve. By [5] it follows that J ∼ contains a periodic Jordan curve B of period, say, k. Since x must enter B, it follows that the union of Since J ∼ is a topological Julia set, it is easy to see that then J ∼ is the unit circle and the lamination ∼ is trivial.
Observe that in this lemma we do not assume that f is condense.
Proof. Under the hypotheses, A 0 = f nk ∼ (K) is a connected subset of J ∼ , and so are the In the case that f n ∼ (K) ⊂ K, it follows that A 0 ⊂ K; that K is closed and A 0 is dense implies that K = J ∼ .
The next lemma shows that condense maps resemble transitive maps. Recall that any topological polynomial on a dendrite must have fixed cutpoints (see, e.g., [24,4]). Proof. Since every subcontinuum of J ∼ contains an interval, it is clear that (3) and (2) are equivalent. If a point x ∈ J ∼ has condense orbit and K ⊂ J ∼ is a continuum, then x must enter K, and the orbit of K is dense. This shows that (1) implies (2). Moreover, by Lemma 3.3, (1) implies (4).
Let us show that (2) and (4) are equivalent. Suppose that (2) holds and let K be a periodic continuum K. Then K has to have a dense orbit which by Lemma 3.3 implies that K = J ∼ . Suppose that (4) holds and let L ⊂ J ∼ be a continuum. By [5]  Let us show that (2) implies (1). If J ∼ has a bounded complementary domain U , then we may assume that Bd(U ) is periodic. By Lemma 3.3 we conclude that Bd(U ) = J ∼ , so f ∼ is conjugate to z → z d and condense. Therefore we may assume that J ∼ is a dendrite. Let {A i | i ≥ 0} be a countable collection of closed arcs such that any continuum K ⊂ J ∼ contains some A s . For convenience, we choose the sequence {A i } so that no element of the sequence contains an endpoint of J ∼ .
Let I ⊂ J ∼ be an arc; we will show for each s ≥ 0 that B s = {x ∈ I | f k ∼ (x) ∈ A s for some k} is an open and dense subset of I. Let α denote a fixed cutpoint of J ∼ . It follows that, for i sufficiently large, α ∈ f i (I). This is because no subcontinuum of J ∼ is wandering, i.e., there exists s, n such that f s ∼ (I) ∩ f s+n ∼ (I) = ∅ [5]. By Lemma 3.3, for some M ≥ 0 we have α ∈ f s+Mn (I); since α is fixed, α ∈ f i (I) for all i ≥ s + M n.
There exist components K of J ∼ \ A s such that every arc intersecting K and containing α also contains a subinterval of A s . Since every continuum in J has a dense orbit, there exists k ≥ 0 such that α ∈ f k ∼ (I) and f k ∼ (I) ∩ K = ∅. Hence, f k ∼ (I) intersects A s in an open subset. Since f k ∼ is finite-to-one, this implies that an open subset of I maps into A s . Since we can repeat this argument on any subinterval of I, B s is a dense open subset of I.
By the Baire Category Theorem, s≥0 B s is then a residual (and hence non-empty) subset of I; this is the set of points in I which eventually map into each A s , and therefore into every subcontinuum of J ∼ as desired.
Powers of condense maps are condense, too. Proof. By Lemma 3.4 we need to show that any continuum K ⊂ J ∼ has dense f s ∼ -orbit in J ∼ . By Lemma 3.2 we only need to consider the case that J ∼ is a dendrite. Let α ∈ J ∼ be a fixed cutpoint. By Lemma 3.3 there exists i ≥ 0 such that α ∈ f i ∼ (K); since α is fixed we may assume that i = ks for some integer k. Clearly, (f s ∼ ) k+1 (K) ∩ (f s ∼ ) k (K) = ∅, since it contains α. By Lemma 3.3, ∞ j=0 f js ∼ (f ks ∼ (K)) is a connected condense subset of J, so the f ks ∼ -orbit of K is condense. Since K was an arbitrary continuum in J ∼ , f s ∼ is condense by Lemma 3.4. Theorem 3.6. Let P be a polynomial with connected Julia set. Then the following claims hold.
1. Suppose that the finest model J ∼ of J P , given by a lamination ∼, is non-degenerate, all points of J ∼ are of finite order, and f ∼ is condense. Then J P is locally connected and P | JP is conjugate to f ∼ . 2. Suppose that P | JP is condense. Then P has no proper periodic subcontinua (in particular, P is non-renormalizable), J P is locally connected and P is conjugate to g P from Theorem 2.7.
Observe, that by this theorem P | JP satisfies Lemmas 3.2 -3.5. Observe also, that by Theorem 2.6 (1) holds for polynomials associated with condense critical portraits having aperiodic kneading.
Proof. (1) Let m : J P → J ∼ be the finest monotone map to a locally connected continuum defined in Theorem 2.7. Since the order of any periodic point p ∈ J ∼ is finite, by [3, Lemma 37] the set m −1 (p) is a repelling or parabolic periodic point. Hence, P has no Cremer points: if U were a periodic Siegel domain of P , then m(Bd(U )) would be a classes, a contradiction with Claim A. This again implies that we can chose two distinct sides (positive and negative) such that the corresponding one-sided itineraries of a and b coincide. Thus, in any case a ∼ Θ b and so T is contained in a ∼ Θ -class T ′ .
By Theorem 2.6, T ′ is finite. Since Θ ∈ K, T ′ is not wandering by Theorem 2.1, and T ′ is not precritical by Lemma 4.2. Hence, T ′ is preperiodic, and either T itself is preperiodic or its future images cross each other inside D. As the latter is impossible by continuity, we may assume that there exist powers s and t > 0 such that σ s 3 (T) = σ s+t 3 (T). Again by continuity σ s 3 (T i ) and σ s+t 3 (T i ) approach σ s 3 (T) in the Hausdorff metric while the area of Ch(T) is at least 1/n. For geometric reasons this contradicts that σ t 3 (T i ) and σ s+t 3 (T i ) are disjoint for all i. Therefore, Θ / ∈ K. We have established that W n is nowhere dense in A 3 , so ∞ n=1 W n = WT 3 is a first category subset of A 3 .