Mapping schemes realizable by obstructed topological polynomials (Q2888650)

A topological polynomial is a branched cover \(f:\mathbb{S}^2\to\mathbb{S}^2\), having finite postcritical set \(P_f\), such that for some critical point \(\omega\), we have \(f^{-1}(\omega)=\{\omega\}\). Letting \(C_f\) denote the set of critical points, the `mapping scheme of \(f\)' is the pattern of action of \(f|(C_f\cup P_f)\).NEWLINENEWLINETwo topological polynomials \(f\) and \(g\) are (Thurston-)equivalent if there exist two homeomorphisms \(\phi_j:(\mathbb{S}^2,P_f)\to(\mathbb{S}^2.P_g)\) such that \(\phi_1f=g\phi_2\).NEWLINENEWLINEThe paper is about the relationship between the character of the mapping scheme and the existence of an equivalent (ordinary) complex polynomial.NEWLINENEWLINEA cycle in the mapping scheme is called an attractor if it contains a critical point. If all cycles are attractors, the polynomial is called hyperbolic. In 1985, Levy used a theorem of Berstein to show that all hyperbolic topological polynomials are equivalent to complex polynomials.NEWLINENEWLINEThe paper includes a good summary of the history and context, and a list of open problems with comments.