Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds (Q2841344)

A transcendental entire function \(f:\mathbb{C} \to \mathbb{C}\) is called ``strongly subhyperbolic'' if the following conditions are met:NEWLINENEWLINE(i) the intersection of the Fatou set \(F(f)\) and the post-singular set \(P(f)\) is compact,NEWLINENEWLINE(ii) the intersection of the Julia set \(J(f)\) and \(P(f)\) is finite,NEWLINENEWLINE(iii) \(J(f)\) contains no asymptotic values of \(f\),NEWLINENEWLINE(iv) the local degree of points in \(J(f)\) is uniformly bounded.NEWLINENEWLINEThe main result of the paper under review is that if \(f\) is strongly subhyperbolic and \(\lambda \in \mathbb{C}\) is such that \(g(z) = f(\lambda z)\) is hyperbolic and has connected Fatou set, then there is a semiconjugacy between \(J(f)\) and \(J(g)\), that is, there exists a continuous surjection \(\phi : J(g) \to J(f)\) such that \(f(\phi(z)) = \phi(g(z))\) for \(z\in J(g)\), and further, \(\phi\) restricts to a homeomorphism between the escaping sets \(I(f)\) and \(I(g)\).NEWLINENEWLINEThis result extends the understanding of the dynamics of hyperbolic functions with connected set as studied in [\textit{L. Rempe}, Acta Math. 203, No. 2, 235--267 (2009; Zbl 1226.37027)]. Note that the hypothesis of this theorem is always satisfied for small enough \(\lambda\). As a corollary to the main result, if \(f\) is a strongly subhyperbolic function which can be written as a finite composition of entire maps with bounded singular values and finite order of growth, then \(J(f)\) must be a pinched Cantor bouquet, that is, a Cantor bouquet with an equivalence relation on the endpoints.NEWLINENEWLINEThe definition of strong subhyperbolicity allows the application of the theory of Riemann orbifolds. The question of whether such results are true in the absence of a condition on the local degree of points in \(J(f)\) is left open, with a remark that such functions arise in the class of Poincaré functions.