Parameter space for families of Parabolic-like mappings

TL;DR

This paper investigates the parameter space of degree 2 parabolic-like mappings, establishing a continuous map linking parameters to fixed point multipliers and revealing a ramified covering structure over the connectedness locus.

Contribution

It introduces a continuous map from the parameter space to fixed point multipliers and demonstrates a ramified covering property over the connectedness locus.

Findings

The hybrid conjugacy induces a continuous map to fixed point multipliers.

The map restricts to a ramified covering over the connectedness locus.

The structure relates the parameter space to the classical M_1 ackslash {1} locus.

Abstract

In this paper we study analytic families of degree 2 parabolic-like mappings (as we defined in arXiv:1111.7150). We prove that the corresponding family of hybrid conjugacies induces a continuous map, which associates to each parameter the multiplier of the fixed point of the hybrid equivalent rational map P_A. We prove that, under suitable conditions, this map restricts to a ramified covering from the connectedness locus of the family of parabolic-like maps to the connectedness locus M_1 \ {1}.