Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds

TL;DR

This paper proves that certain subhyperbolic transcendental entire maps are semiconjugate to hyperbolic maps with connected Julia sets, leading to a detailed topological description of their Julia sets as pinched Cantor bouquets.

Contribution

It establishes a semiconjugacy between subhyperbolic transcendental maps and hyperbolic maps with connected Julia sets, and characterizes the Julia sets as pinched Cantor bouquets.

Findings

Existence of a hyperbolic map g semiconjugate to f on Julia sets.

Semiconjugacy becomes a conjugacy on the escaping set of g.

Julia set of f can be described as a pinched Cantor bouquet.

Abstract

Let f be a transcendental entire map that is subhyperbolic, i.e., the intersection of the Fatou set F(f) and the postsingular set P(f) is compact and the intersection of the Julia set J(f) and P(f) is finite. Assume that no asymptotic value of f belongs to J(f) and that the local degree of f at all points in J(f) is bounded by some finite constant. We prove that there is a hyperbolic map g (of the form g(z)=f(bz) for some complex number b) with connected Fatou set such that f and g are semiconjugate on their Julia sets. Furthermore, we show that this semiconjugacy is a conjugacy when restricted to the escaping set I(g) of g. In the case where f can be written as a finite composition of maps of finite order, our theorem, together with recent results on Julia sets of hyperbolic maps, implies that J(f) is a pinched Cantor bouquet, consisting of dynamic rays and their endpoints. Our result also seems to give the first complete description of topological dynamics of an entire transcendental map whose Julia set is the whole complex plane.