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

Alexander Blokh, Lex Oversteegen, Vladlen Timorin
2017 Discrete and Continuous Dynamical Systems. Series A  
Every plane continuum admits a finest locally connected model. The latter is a locally connected continuum onto which the original continuum projects in a monotone fashion. It may so happen that the finest locally connected model is a singleton. For example, this happens if the original continuum is indecomposable. In this paper, we provide sufficient conditions for the existence of a non-degenerate model depending on the existence of subcontinua with certain properties. Applications to complex
more » ... ications to complex polynomial dynamics are discussed. f g ψ We also need to define a concept of a monotone map. In what follows let C be the complex plane and let C be the complex sphere. In his paper [9] 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 one-to-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 [2] it was discovered that an approach based upon continuum theory yields results that extend those of [9] 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 [3] the result was extended to plane compacta). Theorem 1.5 ([2]). 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 LOCALLY CONNECTED MODELS FOR PLANE CONTINUA 5783 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 at 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
doi:10.3934/dcds.2017251 fatcat:tgoywcp5crbuxatmstejuyvex4