Parabolic-like maps

TL;DR

This paper introduces parabolic-like maps, a new class similar to polynomial-like maps but with a parabolic external class, and proves a straightening theorem linking them to a specific family of quadratic rational maps.

Contribution

It defines parabolic-like maps and establishes a straightening theorem showing their hybrid conjugacy to Per_1(1) maps, with uniqueness for connected Julia sets.

Findings

Parabolic-like maps are introduced as a new class of dynamical systems.

A straightening theorem for degree 2 parabolic-like maps is proved.

Any such map is hybrid conjugate to a unique member of Per_1(1) if the Julia set is connected.

Abstract

In this paper we introduce the notion of parabolic-like mapping, which is an object similar to a polynomial-like mapping, but with a parabolic external class, i.e. an external map with a parabolic fixed point. We prove a straightening theorem for parabolic-like maps, which states that any parabolic-like map of degree 2 is hybrid conjugate to a member of the family Per_1(1), and this member is unique (up to holomorphic conjugacy) if the filled Julia set of the parabolic-like map is connected.