Non-degenerate locally connected models for plane continua and Julia sets

Suppose that a $X$ is an \emph{unshielded} plane continuum (i.e., $X$ coincides with the boundary of the unbounded complementary component of $X$). Then there exists a \emph{finest monotone} map $m:X\to L$, where $L$ is a locally connected continuum (i.e., $m^{-1}(y)$ is connected for each $y\in L$, and any monotone map $\varphi:X\to L'$ onto a locally connected continuum is a composition $\varphi=\varphi'\circ m$ where $\varphi':L\to L'$ is monotone). Such finest locally connected model $L$ of $X$ is easier to understand because $L$ is locally connected (in particular it can be described by a picture) and represents the finest but still understandable decomposition of $X$ into possibly complicated but pairwise disjoint \emph{fibers} (point-preimages) of $m$. However, in some cases (i.e., in case $X$ is indecomposable) $L$ is a singleton. In this paper we provide sufficient conditions for the existence of a non-degenerate model depending on the existence of certain subcontinua of $X$ and apply these results to the connected Julia sets of polynomials.


Introduction
A natural approach to studying a topological space X is to model X using simpler and easier to deal with spaces. By this we mean finding a factor space of X such that both the quotient map m : X → L and the model space L are manageable. In this paper we consider only plane continua; in that setting we view monotone maps and locally connected continua as manageable. This leads to the concept of the finest locally connected model under a monotone map of a plane continuum X.
The concept was inspired by Jan Kiwi who approached the problem of modeling from the point of view of (complex) dynamical systems. To state Kiwi's results we need a few definitions.
Definition 1.1 (Semiconjugacy of maps). Two maps f : X → X and g : Y → Y are said to be semiconjugate if there exists a map ψ : X → Y such that ψ • f = g • ψ. In other words, the following diagram is commutative: We also need to define a concept of a monotone map.
Definition 1.2 (Monotone map). A map f : X → Y is monotone provided for each y ∈ Y , f −1 (y) is connected.
In what follows let C be the complex plane and let C be the complex sphere. In his paper [Kiw04] Kiwi proves that if a polynomial P with connected Julia set J(P ) has no periodic points with multipliers which are complex numbers of modulus 1 and irrational argument then P can be semiconjugate to a so-called topological polynomial f P : C → C. The semiconjugacy ϕ : C → C is a monotone map which is oneto-one outside the Julia set J(P ); thus, basically ϕ collapses some subcontinua of J(P ) (fibers of ϕ) to points. The topological polynomial f P is a branched covering map such that ϕ(J(P )) is a locally connected continuum with well-understood structure and dynamics described by so-called laminations.
As mentioned above, Kiwi's approach to the problem was based upon dynamical systems' considerations. Later on in [BCO11] it was discovered that an approach based upon continuum theory yields results that extend those of [Kiw04] while also being applicable in a purely topological setting. We need a few definitions. Definition 1.3. Let X be a continuum. A continuum Y is a finest locally connected model for X if there exists a monotone map m : X → Y so that for any monotone map f : X → Z, where Z is a locally connected continuum, there exists a monotone map g : Y → Z so that g • m = f ; then we will call the map m a finest monotone map.
We consider this notion on the plane in the context of so-called unshielded continua.
Definition 1.4. Given a compact set X in the plane, let U ∞ X denote the unbounded complementary domain of X. The set TH(X) = C \ U ∞ X is called the topological hull of X. A compact set X in the plane is unshielded provided X coincides with the boundary ∂(U ∞ X ) of the unbounded complementary domain U ∞ X of X. Observe that any subcontinuum of an unshielded continuum is unshielded.
The following theorem shows that a finest locally connected model and a finest monotone map are well-defined for unshielded plane continua (in [BCO13] the result was extended to plane compacta).
Theorem 1.5 ( [BCO11]). Every unshielded plane continuum X has a finest locally connected model Y and a finest monotone map m. Moreover, any two finest locally connected models of an unshielded continuum X are homeomorphic. Furthermore, m can be extended to a monotone map C → C which maps ∞ to ∞, in C \ X collapses only those complementary domains to X whose boundaries are collapsed by m, and is a homeomorphism elsewhere in C \ X.
By Theorem 1.5 we can talk about the finest locally connected model of an unshielded continuum and the finest monotone map. It follows that if an unshielded plane continuum X has the finest locally connected model which is non-degenerate then its topological hull TH(X) also has a non-degenerate model.
In particular, the connected Julia set of a polynomial admits a finest locally connected model. However, in some cases the finest locally connected model is a single point; in this case we say that the finest locally connected model is degenerate while otherwise we call such model nondegenerate. Obviously, if the finest model is degenerate, all information regarding the continuum X is lost while otherwise some of the structure of X is preserved in its model. This shows the importance of the fact that the finest locally connected model of an unshielded continuum is non-degenerate. In the present paper we will study conditions under which the finest locally connected model is non-degenerate. Moreover, in the final section we apply this result to polynomial dynamics.

Statement of main results and applications
In this section we assume knowledge of basic concepts of continuum theory and complex dynamics (all necessary definitions are given in detail later in the sections of the paper containing the proofs of our main results). Denote the open unit disk by D and the disk at infinity (i.e., C \ D) by D ∞ . We will identify the unit circle S = ∂D = ∂D ∞ with R/Z and call the induced order on S the circular order. Note that the circular order is not defined for a pair of points in S, but if x, y, z ∈ S are three distinct points, then x < y < z in the circular order if, when traveling from x in the positive direction along S, we encounter y before z. Thus, from now on a single point x ∈ S will be denoted by the corresponding angle, i.e. by a number α ∈ [0, 1) with If X is a plane continuum, then by the Riemann mapping theorem there exists a conformal map ψ X : D ∞ → U ∞ X with derivative converging to a real number as |z| → ∞. External rays of X foliate U ∞ X and serve as a major tool in studying the topology of ∂(U ∞ X ). Definition 2.1 (External rays). Let X be a plane continuum. By an external ray of X we mean the image of the radial line segment with argument 2πα under the Riemann map ψ X ; in what follows, this image will be denoted by R X (α). In other words, If we do not want to emphasize the argument we denote an external ray of X by R X . We say that the external ray R X (α) lands on We will mostly consider external rays for unshielded plane continua X (in that case X = ∂(U ∞ X )) such as connected Julia sets of complex polynomials, however sometimes we work with external rays of other plane continua (e.g., we consider external rays of connected filled Julia sets). Observe that the unbounded complementary domain U ∞ X of a continuum X coincides with the unbounded complementary domain U ∞ TH(X) of its topological hull. Therefore we can (and will) interchangeably talk about external rays of X and/or external rays of TH(X).
Definition 2.2 (Strategically placed subcontinua). Suppose that Y is a subcontinuum of an unshielded continuum X in the complex plane. Then we say that Y is strategically placed in X provided that there exists a dense set A(Y, X) = A ⊂ S so that: (1) for each α ∈ A, R Y (α) lands on a point y α ∈ Y , (2) the set of points there exists a circle order preserving function p : A → S so that for each α ∈ A the ray R X (p(α)) lands on y α . In this case we say that A is an anchor set (of Y ) and p : A → S is an external connecting function (of Y ).
Since p preserves order, p is one-to-one but we do not assume that p is continuous.
Theorem A is our main continuum theory result. It shows that in some cases the fact that a subcontinuum has a non-degenerate finest locally connected model implies that the same can be said about the continuum itself.
Theorem A. Let X be an unshielded plane continuum. If Y is strategically placed in X, and Y has a non-degenerate finest locally connected model, then X has a non-degenerate finest locally connected model.
The main applications of this result are in complex dynamics. Namely, the following theorem holds.
Theorem B. Suppose that P : C → C is a polynomial and J * ⊂ J(P ) is a continuum which is a polynomial-like Julia set of P n for some n > 0. If J * has a non-degenerate finest locally connected model, then so does J(P ).
In Subsection 4.3 we rely upon [BCO13] and prove a version of Theorem B for disconnected Julia sets.

Proof of Theorem A
In the first subsection of this section we give various standard definitions. Then we prove Theorem A.
3.1. Basic definitions. The notion of the principal set is used in studying the limit behavior external rays.
Definition 3.1 (Principal set). Given an external ray R X (α) of an unshielded continuum X we denote by Pr X (α) the set R X (α) \ R X (α) and call it the principal set of the ray R X (α). If Pr X (α) is a single point y we say that the external ray R X (α) lands on y.
More generally, let T ⊂ U ∞ X be an image of R + = (0, ∞) under a continuous map ψ : R + → C such that lim t→∞ |ψ(t)| = ∞ while ∅ = T \ T ⊂ X. Then we say that T = ψ(R + ) accumulates in X, denote T \ T by Pr X (T ) and call it the principal set of the curve T which accumulates in X. If Pr X (T ) is a single point y we say that the curve T which accumulates in X lands on y.
Another important definition is that of a crosscut (see, e.g., [Mil06] for details).
Definition 3.2 (Crosscuts). A crosscut C of X is an open arc C ⊂ U ∞ X so that its closure is a closed arc with two distinct endpoints both of which belong to X. A fundamental chain {C i } (of crosscuts) is a sequence of crosscuts C i of X such that the following holds: ( for each i, C i separates C i+1 from infinity in U ∞ X , and (3) lim diam(C i ) = 0.
For each crosscut C of X its shadow S C is the closure of the bounded complementary domain of C \ [X ∪ C] whose boundary contains C.
Note that every fundamental chain {C i } corresponds to a unique point α ∈ S defined by lim(ϕ X ) −1 (C i ) = α and in this case we say that {C i } is a fundamental chain for α.
It is easy to see that both Pr X (α) and Imp X (α) are continua, that Pr X (α) ⊂ Imp X (α) and that Imp X (α) is independent of the choice of the fundamental chain for α. Moreover, let X be an unshielded continuum. Then, even though α Pr X (α) can be a proper subset of the continuum X, α Imp X (α) = X.

Proof of Theorem A.
Let us recall that the notion of a subcontinuum Y strategically placed in an unshielded continuum X was introduced in Definition 2.2. A part of this definition is a function p (so-called external connecting function) of a dense set A ⊂ S to S which preserves circle order and maps angles such that for each α ∈ A, both the ray R Y (α) and the ray R X (p(α)) land on a point y α ∈ Y . We will show below that the choice of the function p is severely restricted. Moreover, the condition in Lemma 3.4 characterizes the situation in which a subcontinuum is strategically placed in an unshielded continuum (and so this characteristic can be used as an alternative definition of the fact that Y is strategically placed in X).
Lemma 3.4. Suppose that Y ⊂ X are unshielded planar continua. Then the following are equivalent: (1) Y is strategically placed in X with anchor set A, (2) There exists a dense set A ⊂ S so that for α ∈ A, there exists β(α) = β ∈ S so that the ray R X (β) also lands on y α and the rays R Y (α) and R X (β) are homotopic in {y α } ∪ C \ Y under a homotopy which fixes the landing point y α ∈ Y .
Proof. Suppose that Y is strategically placed in X with anchor set A and p : A → S as the external connecting function. Suppose that α ∈ A and the ray R Y (α) lands on y α ∈ Y . Clearly, R X (p(α)) can be viewed as a curve in C \ Y which accumulates in Y ; more precisely, we can say that R X (p(α)) lands on y α . Thus, R X (p(α)) is homotopic to some external ray R Y (β) in C \ Y under a homotopy which fixes y α (so that the ray R Y (β) lands on y α too). If α = β, then both components is an external ray of X which also lands on y γ 2 . Since α < γ 1 < γ 2 , then p(α) < p(γ 1 ) < p(γ 2 ).
Consider the set E of angles in p(A) which belong to (p(α), p(γ 2 )). Consider also the component E of the set containing external rays of X with arguments from E. The external rays of X with argument in E can only land on points from the boundary of E but not on points from other external rays of X; thus, the external rays of X with argument in E can only land on points from [C 2 ∩ Y ] ∪ {y α }. In particular this must be true for the ray R X (p(γ 1 )). However by definition this ray must land on the point y γ 1 ∈ C 1 ∩ Y , a contradiction.
Suppose next that condition (2) holds. It suffices to show that the map p : α → β(α) preserves circular order. Recall that by ψ Y : D ∞ → U ∞ Y we denote the conformal map with derivative converging to a real number as |z| → ∞. Similarly, let ψ X : D ∞ → U ∞ X be the corresponding Riemann map from the complement of the closed unit disk to the unbounded component of X. Assume that α 1 < α 2 < α 3 ∈ S. Let β j = β(α j ); then the rays ψ −1 Y (R X (β j )) = R j land on the points e 2πβ j i = x j ∈ S and x 1 < x 2 < x 3 . Let S r be the circle ψ −1 Y • ψ X ({z ∈ C | |z| = r}) with an induced circular order <. As r ց 1, S r intersects R j in a unique point y j (r) and lim rց1 y j (r) = x j . This implies that y 1 (r) < y 2 (r) < y 3 (r) as required.
Suppose that Y ⊂ X is strategically placed in X with anchor set A. Then Lemma 3.4 implies that for any α ∈ A ⊂ S the ray ψ −1 Y (R X (p(α))) lands on α ∈ S. This visualization is useful in the proof of the next lemma that describes intersections between closures of components of X \ Y and Y . It follows easily from the assumptions that X is unshielded and Y is strategically placed in X.
then there exists a non-degenerate arc [p, q] ⊂ ∂D such that any (not necessarily radial) ray T ∈ C\D which lands at β ∈ (p, q) must intersect To prove the last claim of the lemma choose a fundamental system of crosscuts Then by definition their shadows converge to Imp Y (α). Since all these shadows contain C ∩Y , it follows that C ∩Y ⊂ Imp Y (α) as desired.
Lemma 3.5 motivates the following definition.
Definition 3.6 (Angles associated with components of X \ Y ). Suppose that X is an unshielded continuum and Y ⊂ X is a continuum strategically placed in X. Given a component C of X \ Y we call the angle α such that ψ −1 Y (C) ∩ ∂D = {α} the angle associated with C and denote it by α(C) which defines a map from the family of components of X \ Y to the unit circle. We also define the function C which associates to any point Using the terminology introduced in Definition 3.6 we can restate Lemma 3.5 as follows: if X is an unshielded continuum and Y ⊂ X is a continuum strategically placed in X then for every component C of X \ Y we have C ∩ Y ⊂ Imp Y (α(C)).
We will need the following geometric lemma.
Lemma 3.7. Suppose that X is an unshielded continuum and Y ⊂ X is a continuum strategically placed in X. Let {x i } be a sequence of points of X \ Y such that x i → x and α(x i ) → β. Then either x ∈ Imp Y (β) or x ∈ X \ Y and α(x) = β. In particular, the map α : X \ Y → S is continuous.
Recall, that given a map h : A → B we call point-inverses of h (h-)fibers. The following lemma is proven in [BCO11]. We are ready to prove Theorem A.
Theorem A. Let X be an unshielded plane continuum. If Y is strategically placed in X, and Y has a non-degenerate finest locally connected model, then X has a non-degenerate finest locally connected model.
Proof. By Theorem 1.5 it suffices to show that there exists a monotone map from X to a non-degenerate locally connected continuum L. Since Y has a non-degenerate finest locally connected model, then there exists the finest monotone map m Y : Y → L so that L is a non-degenerate locally connected continuum. We will extend the map m Y to a monotone map m : X → L as follows: for every x ∈ X \ Y set m(x) = m Y (Imp Y (α(x))). Observe that since by Lemma 3.8 the map m Y collapses all Y -impressions to points, then the map m(x) is well-defined. Let us show that this map has the desired properties.
First we show that m is continuous. To see that, we first show that if x i → x then one can find a subsequence x i j such that m(x i j ) → m(x). This is obvious if infinitely many points x i belong to Y because m Y is continuous. Thus we may assume that x i ∈ X \ Y for every i. Choose a subsequence x i j so that α(x i j ) → β. Then by Lemma 3.7 either x ∈ Imp Y (β), or x ∈ X \ Y and α(x) = β. In either case m(x) = m(Imp Y (β)) while m(x i j ) = m(Imp Y (α(x i j ))). Since impressions are upper semi-continuous and m is continuous, then m(Imp Y (α(x i j ))) = m(x i j ) → m(Imp Y (β)) = m(x) as desired.
We claim this implies continuity of m. Indeed, suppose that x i → x but m(x i ) → m(x). Refining our sequence we may assume that m(x i ) → t = m(x). However by the previous paragraph we can find a subsequence x i j of x i such that m(x i j ) → m(x), a contradiction.
Since for y ∈ Y , m −1 (y) is the union of (m Y ) −1 (y) and all components of X \Y whose closure intersects (m Y ) −1 (y), m −1 (y) is connected. Hence m : X → L is the desired monotone map.

Applications
In this section we apply our results to complex dynamics. 4.1. Preliminaries from complex dynamics. We rely upon basic facts discussed, e.g., in [Mil06]. Let us fix a polynomial P of degree at least two.   It is known that the unbounded complementary domain U ∞ J(P ) of J(P ) is in fact the set of all escaping points while its complement K(P ) is in fact the set of all non-escaping points. The set U ∞ J(P ) = U ∞ K(P ) is therefore called the basin of attraction of infinity.
The Julia set J(P ) is a continuum if and only if all critical points of P are non-escaping (in other words, the orbits of all critical points of P are contained in K(P )). We will first assume that J(P ) (equivalently, K(P )) is connected. Then it is known that all repelling and parabolic periodic points of P (and all their pre-images) are the landing points of finitely many rays R J(P ) (α) with α ∈ Q.
In a vast majority of cases the connected Julia set of a polynomial is either locally connected, or at least admits a non-degenerate finest locally connected model. However, this is not always the case. To give an example we need the following alternative definition of a Cremer point.
Definition 4.4. Let P be a polynomial. Suppose that a is a periodic point of P of period n such that (P n ) ′ (a) = e 2πiθ with θ irrational. Moreover, suppose that a belongs to the Julia set J(P ) of P . Then a is a Cremer periodic point of P .
The main result of [BO10] shows that in some cases the finest locally connected model of a connected Julia set is degenerate. In general the existence of a subcontinuum with a non-degenerate finest locally connected model provides no information about such a model for the entire unshielded continuum. However, if the subcontinuum is strategically placed, then Theorem A shows that a nondegenerate model for the entire space does exist. A natural choice of a subcontinuum of J(P ) on which one can hope to have a non-degenerate finest locally model is that of a connected Julia set of a polynomial-like map which is a power of P . This is another application of polynomiallike maps that are a powerful tool in complex dynamics introduced by Douady and Hubbard [DH85].
the compact set of points z ∈ U such that f n (z) is defined and belongs to U for all n ∈ N. The set K f is called the filled Julia set of f . The Julia set J f of f is the boundary of K f . Given a polynomial P , we will often say that P n | K * : K * → K * (or P n | J * : J * → J * ) is a polynomial-like map meaning that there exist open sets U and V as in Definition 4.6 such that K * is the filled Julia set (or J * is the Julia set) of the corresponding polynomial-like map (P n , U, V ).
The term polynomial-like maps is justified by the Straightening Theorem stated below. However first we need one more definition.
Definition 4.8 (Hybrid equivalence [DH85]). Two polynomial-like maps f : U → V and g : U ′ → V ′ are hybrid equivalent if there is a quasiconformal map ϕ : U → U ′ conjugating f to g such that ϕ is conformal almost everywhere on K(f ) (in other words, ϕ is such that ϕ•f = g •ϕ near K f ). The map ϕ is called a straightening map.
An important result of [DH85] is given below; this theorem allows us to talk about finest locally connected models of connected polynomiallike Julia sets.
Straightening Theorem ( [DH85]). Let f : U → V be a polynomiallike map. Then f is hybrid equivalent to a polynomial P . Moreover, if K(f ) is connected, then P is unique up to (global) conjugation by an affine map.

4.2.
Main applications in the connected case. Suppose that the connected Julia set J(P ) of a polynomial P contains a subcontinuum K * so that P n | K * is a polynomial-like map. Then by the Straightening Theorem P n | K * : K * → K * is hybrid equivalent to a polynomial g with connected filled Julia set K(g). In particular, under the hybrid equivalence appropriate arcs contained in external rays of K(g) correspond to arcs inside U which accumulate to the corresponding polynomial-like Julia set J * (the open set U is defined as in Definition 4.6). Slightly abusing the language we will call these arcs polynomial-like rays and will denote them in the same way as we would have denoted external rays of K * (or equivalently, of J * ), i.e. R K * (α) where α is the argument of the external ray of the polynomial g corresponding to R K * (α).
Recall that an external ray R J (α) is said to accumulate in J * if Pr(α) ⊂ J * . Also, it is easy to see that the property of a point being repelling or parabolic is preserved under hybrid equivalence. By Definition 4.2 this allows one to conclude that repelling periodic points of P are dense in J * . Moreover, it follows that if p is a repelling or parabolic periodic point of P , then only finitely many external rays R J * (α) of J * and finitely many external rays R J (β) of J land on p.
Suppose that Y ⊂ X are unshielded plane continua. Above in Lemma 3.4 we considered a map p : A → S; this map associated to a ray R Y (α) the ray R X (p(α)) so that both rays landed on the same point y α ∈ Y and were homotopic outside Y by a homotopy fixing y α . In the case of polynomials f and polynomial-like maps f * it is easier to first consider the "inverse" map which associates rays R J(f ) (β) which land on a point y β ∈ J(f * ) to rays R J(f * ) (ν(β)) which land on y β and are homotopic to R X (β) outside Y by a homotopy which fixes y β . This is accomplished in Lemma 4.9.
In what follows, given a map, we call a point preperiodic if it is not periodic but eventually maps to a periodic point, and (pre)periodic if it is periodic or preperiodic. Recall that if the Julia set J(P ) of a polynomial P is connected and an angle α is (pre)periodic then the external ray R J(P ) lands on a (pre)periodic (in the sense of P ) point in J(P ) [Mil06]. Given a set T ⊂ S we say that a map Ψ : T → S is extendably monotone if Ψ has a monotone (but not necessarily continuous!) extension m : S → S. Lemma 4.9. Suppose that P is a polynomial of degree d with connected Julia set J and J * ⊂ K is a subcontinuum of J such that P n | J * is a polynomial-like map with filled Julia set K * and Julia set J * . Suppose that P n | J * is hybrid equivalent to a polynomial Q of degree k. Let B ⊂ S be the set of all angles β so that R J (β) lands on a point y β ∈ J * . Then there exists a extendably monotone continuous map ν : B → S such that: (1) for each β ∈ B the ray R J * (ν(β)) lands on the same point y β and the rays R J (β) and R J * (ν(β)) are homotopic outside K * under a homotopy which fixes the point y β , (2) if B ′ ⊂ B is the set of all (pre)periodic angles, then ν(B ′ ) is dense in S, Notice that the continuity of ν on B only means that ν is continuous at points of B and does not imply that ν can be extended to a continuous monotone map of the circle to itself.
Proof. Since P n | J * is polynomial-like, there exist Jordan disks U ⊂ U ⊂ V such that J * ⊂ U and P n : U → V is polynomial-like. Denote P n | U by P * .
Let R J (β) be an external ray of J which lands on a point y β ∈ J * . Consider the inverse ξ : U ∞ J * → D ∞ of the corresponding Riemann map from D ∞ to U ∞ J * with derivative converging to a real number at infinity. Then ξ(R J (β)) is a curve which accumulates at a point z ∈ S. Choose the polynomial-like ray R J * (α) of J * whose ξ-image is the radial ray to D ∞ landing at z (the argument of this radial ray and hence the argument of the corresponding polynomial-like ray is denoted by α). Since in the D ∞ -plane the radial ray to z and ξ(R J (α)) are homotopic, it follows that R J (β) and R J * (α) are homotopic outside J * by a homotopy which fixes y (the homotopy carries over to C \ J * under the Riemann map). Define ν(β) = α. Since this construction goes through for all angles β ∈ B, this defines a map ν : B → S.
To see that ν is extendably monotone suppose that ν(β 1 ) = ν(β 2 ). Then ξ(R J (β 1 )) and ξ(R J (β 2 )) are two curves which land on the same point z ∈ S. Denote by T the component of D ∞ \[ξ(R J (β 1 ))∪ξ(R J (β 2 ))] whose closure meets S only in the point z ∈ S (in other words, T is the wedge between ξ(R J (β 1 )) and ξ(R K (β 2 )) which does not contain the unit disk). Then any external ray R J (γ) with ξ(R J (γ)) ⊂ T that lands on a point of J * must land on y so that ξ(R J (γ)) lands on z.
This implies that there exists an arc A z ⊂ S so that ν −1 (z) = A z ∩ B. To see that there exists a monotone extension of ν it remains to observe that circular orientation among points of B is preserved under ν in the following sense: if β 1 < β 1 < β 3 then it is impossible that ν(β 1 ) < ν(β 3 ) < ν(β 2 ) as otherwise some external rays of K will have to intersect. Thus, the arcs A z constructed above for all points z ∈ ν(B) have the same circular order as the points z ∈ S which implies the desired. Now, choose an angle β ∈ B such that y β ∈ J * , the landing point of the external ray R J (β), is preperiodic. Set α = ν(β). Properties of polynomials (and hence of polynomial-like maps) imply that the family of all polynomial-like rays which are preimages of R J * (α) is such that their arguments are dense in S. Each such polynomial-like ray R * with argument α ′ is a unique pullback of R J * (α) under the appropriate branch of the inverse function to P * (recall that y β is not periodic). If we simultaneously pull back R J (β) under the same branch of the inverse function of P * we will obtain an external ray R J (β ′ ) of J with argument β ′ which lands on the same point as R * and is homotopic to R * outside K * . Denote the argument of R by α ′ , then ν(β ′ ) = α ′ . This shows that (2) holds.
To see that ν is continuous consider a sequence β 1 < β 2 < . . . in B so that lim β i = β ∞ ∈ B. Consider the landing points z i of the curves ξ(R J (β i )) and the landing point z ∞ of ξ(R J (β ∞ )). The fact that ν is extendably monotone implies that z 1 ≤ z 2 ≤ · · · ≤ z ∞ . We claim that z ∞ = lim z i . Indeed, otherwise we have that z 1 ≤ lim z i = t < z ∞ . By (2) we can choose a (pre)periodic angleβ ∈ B ′ such that t < ν(β) < z ∞ . Since ν is extendably monotone this contradicts the fact that lim β i = β ∞ . Thus, z ∞ = lim z i as desired. The last claim of the lemma is left to the reader.
The following corollary easily follows from definitions, Lemma 3.4 and Lemma 4.9 Corollary 4.10. Suppose that the connected Julia set J(P ) of a polynomial P contains a subcontinuum J * so that P n | J * is a polynomial-like map for some n ≥ 1. Then J * is strategically placed in J(P ).
Proof. Let us use the notation from Lemma 4.9. Set A = ν(B). Then by Lemma 4.9 the set A is dense in S. Moreover, by Lemma 4.9 conditions listed in Lemma 3.4(2) are satisfied for A ⊂ S and J * ⊂ J(P ). Hence J * is strategically placed in J(P ).
Corollary 4.10 allows one to conclude that connected polynomial-like Julia sets with non-degenerate finest locally connected models force the existence of non-degenerate finest locally connected models of containing them connected polynomial Julia sets.
Theorem B. Suppose that P : C → C is a polynomial and J * ⊂ J(P ) is a continuum which is a polynomial-like Julia set of P n for some n > 0. If J * has a non-degenerate finest locally connected model, then so does J(P ).
Proof. Indeed, by Corollary 4.10 Theorem A implies the desired.
Note that if K * is a filled polynomial-like Julia set of a polynomial P , then K * is a component of P −n (K * ). As it turns out this is almost sufficient (the proof of Theorem 4.11 uses some ideas communicated by M. Lyubich to the third named author). For convenience we state these results in the case that n = 1.
Theorem 4.11 (Theorem B [BOPT15]). Let P : C → C be a polynomial, and Y ⊂ C be a full P -invariant continuum. The following assertions are equivalent: (1) the set Y is the filled Julia set of some polynomial-like map P : U * → V * of degree k, (2) Y is a component of the set P −1 (P (Y )), and, for every attracting or parabolic point y of P in Y , the immediate attracting basin of y or the union of all parabolic domains at y is a subset of Y .
The following corollary is now almost immediate.
Corollary 4.12. Suppose that K * ⊂ K is a subcontinuum of the filled Julia set of a polynomial P such that K * is a component of P −1 •P (K * ) containing all immediate parabolic and attracting basins of all attracting and parabolic points in K * . Then if ∂(U ∞ K * ) has a non-degenerate finest locally connected model, then J(P ) has a non-degenerate finest locally connected model.
Proof. By [BOPT15], P | K * : K * → K * is a polynomial-like map. Hence the result follows from Theorem B. 4.3. Models for non-connected spaces. Models for non-connected spaces were studied in [BCO13]. A compactum is a compact metric space. Since a compactum with infinitely many distinct components is always not locally connected at some point, we need to replace the condition of local connectedness of the model by a suitable notion.
A compactum X is called finitely Suslinian if, for every ε > 0, every collection of disjoint subcontinua of X with diameters at least ε is finite. By Lemma 2.9 [BO04], unshielded planar locally connected continua are finitely Suslinian and vice versa. Thus, in the unshielded case, the notion of finitely Suslinian generalizes the notion of local connectivity. This motivates us to look for good finitely Suslinian models of planar compacta.
Definition 4.13. Let X be a compactum. A finest finitely Suslinian model for X is a finitely Suslinian compactum S and a monotone map m : X → S so that for each monotone map f : X → Y to a finitely Suslinian compactum Y there exists a monotone map g : S → Y with g •m = f . Then the map m : X → S is called a finest finitely Suslinian model map. We say that a compactum X has a non-degenerate finitely Suslinian model S if at least one component of S is non-degenerate.
Observe that by definition of a monotone map it follows that if m is monotone then distinct components of X map to distinct components of m(X). Observe also that the above introduced notion of a degenerate finitely Suslinian model agrees with the notion of a degenerate locally connected model in the case of continua.
By [BCO13] all finest finitely Suslinian models of a compactum X are homeomorphic and we can talk about the finest finitely Suslinian models of compacta. It was shown in [BCO13] that every planar unshielded compactum X has a finest finitely Suslinian model S (which is unique up to homeomorphisms). As previously in the case of continua, the finest finitely Suslinian model S of X may be degenerate (i.e., the finest finitely Suslinian model monotone map m : X → S may well collapse all components of X to points). The following theorem is the main result of [BCO13] concerning finest finitely Suslinian models of polynomial Julia sets (this time including disconnected Julia sets).
Theorem 4.14 (Theorem 6 [BCO13]). The finest finitely Suslinian model monotone map m : J(P ) → S of the Julia set J(P ) of a polynomial P coincides on each component X of J(P ) with the finest monotone map m X of X to a locally connected continuum. In particular, the following holds: (1) the finest finitely Suslinian monotone model of J(P ) is nondegenerate if and only if there exists a periodic component of J(P ) whose finest finitely Suslinian monotone model is nondegenerate; (2) the Julia set J(P ) is finitely Suslinian if and only if all periodic non-degenerate components of J(P ) are locally connected.
Hence, the following theorem immediately follows.
Theorem 4.15. Suppose that J is the Julia set of a polynomial P and J * ⊂ J is a subcontinuum so that, for some integer r, P r | J * : J * → J * is a polynomial-like map and J * has a non-degenerate finest locally connected model. Then J has the finest finitely Suslinian model.
Proof. Suppose that K * is contained in the component C of J. Then C must be periodic of some period n. By a result of [BH88], P n | C : C → C is a polynomial-like map. Hence P n | C is hybrid equivalent to a polynomial g. Since J * ⊂ C it follows from Theorem B that C has a non-degenerate finest locally connected model. Hence, by Theorem 4.14, J has the finest finitely Suslinian model.