Connectedness of a space of branched coverings with a periodic cycle

TL;DR

This paper proves that the space of degree-d branched self-coverings of the sphere with specific critical points and a periodic cycle is connected, advancing understanding of the structure of such dynamical systems.

Contribution

It establishes the connectedness of a particular locus of branched coverings with prescribed critical points and periodicity, a novel result in complex dynamics.

Findings

The space of such branched coverings is connected.

The result applies to degree-d coverings with two critical points of order d.

It provides new insights into the topology of dynamical systems on the sphere.

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.