Locally Connected Models for Julia Sets of Polynomials
by
Clinton Curry
University of Alabama at Birmingham
Coauthors: Alexander Blokh, Lex Oversteegen

Let P:C → C be a polynomial of degree d with connected Julia set J. A locally connected model of P|_J is a dynamical system P _~ :J _~ → J _~ on a locally connected continuum J _~ to which P|_J is monotonically semiconjugate. Jan Kiwi (2004) constructed non-degenerate (i.e., not the identity map on a point) locally connected models for polynomials P without irrationally neutral periodic points, and showed in that case that the locally connected model J _~ comes from a finite-to-one map F:S¹ → J _~ which semiconjugates z→ z^d to P _~ .

We extend Kiwi's work to all polynomials with connected Julia set. We prove that there is a finest locally connected model of P|_J, and that the semiconjugacy is the finest monotone map of J to any locally connected continuum. (By a finest monotone map of J to any locally connected continuum, we mean a map m of J onto a locally connected continuum such that any other monotone map m' of J onto a locally connected continuum is a composition of m with another monotone map.) We characterize the models and their associated laminations, and characterize dynamically when the finest locally connected model is non-degenerate.

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Date received: February 19, 2009