Connectedness of a space of branched coverings with a periodic cycle (Q6594559)

Let \(\mathcal{M}_{d}\) be the space of degree-\(d\) branched self-coverings of the sphere \(S^{2}\) (see [\textit{G. Segal}, Acta Math. 143, 39--72 (1979; Zbl 0427.55006)]). Interesting subspaces arise by imposing dynamical conditions. In the paper under review, the author focuses on \(\mathcal{P}_{d,n}\), the space of degree-\(d\) branched selfcoverings of \(S^{2}\) with two critical points of order \(d\), one of which is \(n\)-periodic.\N\NThe main result is Theorem A: The space \(\mathcal{P}_{d,n}\) is path connected. Equivalently, all branched self-coverings of \(S^{2}\) with two critical points of order \(d\), one of which is \(n\)-periodic, are combinatorially equivalent.