Locally connected models for Julia sets

Abstract

Let P be a polynomial with a connected Julia set J. We use continuum theory to show that it admits a finest monotone map ϕ onto a locally connected continuum J ∼ P , i.e. a monotone map ϕ : J → J ∼ P such that for any other monotone map ψ : J → J ′ there exists a monotone map h with ψ = h • ϕ. Then we extend ϕ onto the complex plane C (keeping the same notation) and show that ϕ monotonically semiconjugates P | C to a topological polynomial g : C → C. If P does not have Siegel or Cremer periodic points this gives an alternative proof of Kiwi's fundamental results on locally connected models of dynamics on the Julia sets, but the results hold for all polynomials with connected Julia sets. We also give a characterization and a useful sufficient condition for the map ϕ not to collapse all of J into a point.

A core decomposition of compact sets in the plane

“…The studies of Blokh-Curry-Oversteegen [3,2], whose models are generalized by the core decomposition introduced in this paper, already provide very interesting results on the existence of non-degenerate core decompositions. For instance, by [2,Theorem 27], if a continuum X ⊂ K has a "well-slicing family", then the image of X under the natural projection π : K → D P C K is a non-degenerate continuum, hence K has a non-degenerate core decomposition.…”

“…The studies of Blokh-Curry-Oversteegen [3,2], whose models are generalized by the core decomposition introduced in this paper, already provide very interesting results on the existence of non-degenerate core decompositions. For instance, by [2,Theorem 27], if a continuum X ⊂ K has a "well-slicing family", then the image of X under the natural projection π : K → D P C K is a non-degenerate continuum, hence K has a non-degenerate core decomposition. If K is the Julia set of a polynomial, then it is stated in [3,Corollary 24] that K has a nondegenerate core decomposition D P C K if and only if K has a periodic component Q which, as a plane continuum, has a non-degenerate core decomposition D P C Q .…”

“…Here a compactum is finitely Suslinian provided that every collection of pairwise disjoint subcontinua whose diameters are bounded away from zero is finite. Since every finitely Suslinian continuum is locally connected, we see that [2,Theorem 1] is a special case of [3,Theorem 4]. We may wonder about the existence of the core decomposition D F S K of an arbitrary compactum K ⊂ C. However, there are examples of continua K ⊂ C failing to have such a core decomposition (see [3,Example 14] and Section 2 of this paper).…”

“…Moreover, in the last part of [6,Question 5.4], Curry asks: which of these (properties) is the appropriate analogue for the finest locally connected model? Since the core decomposition D P C K in Theorem 7 generalizes the earlier finest decompositions obtained in [2,3], the property of "being a Peano compactum" is a reasonable candidate. What remains is to verify that the decomposition D P C K is "dynamic", in the sense that the rational function R sends every element of D P C K into an element of D P C K .…”

Irreducible Julia sets of rational functions

“…There is more structure on polynomial Julia sets than there are on rational Julia sets, so it is to be expected that more can be concluded in the polynomial case. Locally connected models for polynomial Julia sets have been studied further in joint work with Alexander Blokh and Lex Oversteegen in [BOC08], where we characterized the finest locally connected model for the action of a polynomial on its Julia set.…”