Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds

Abstract:

Let $f:\mathbb {C}\rightarrow \mathbb {C}$ be a transcendental entire map that is subhyperbolic, i.e., the intersection of the Fatou set $\mathcal {F}(f)$ and the postsingular set $P(f)$ is compact and the intersection of the Julia set $\mathcal {J}(f)$ and $P(f)$ is finite. Assume that no asymptotic value of $f$ belongs to $\mathcal {J}(f)$ and that the local degree of $f$ at all points in $\mathcal {J}(f)$ is bounded by some finite constant. We prove that there is a hyperbolic map $g\in \{z\mapsto f(\lambda z):\; \lambda \in \mathbb {C}\}$ 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 $\mathcal {J}(f)$ 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.