Connectedness of a space of branched coverings with a periodic cycle

Abstract

We prove the connectedness of the following locus: the space of degree-$d$ branched self-coverings of $S^2$ with two critical points of order $d$, one of which is $n$-periodic. Equivalently, all branched self-coverings of $S^2$ with two critical points of order $d$, one of which is $n$-periodic, are combinatorially equivalent.