On parabolic external maps

TL;DR

This paper proves that certain smooth circle maps with weakly expanding periodic orbits are conjugate to metrically expanding maps, linking parabolic external maps to expanding circle coverings.

Contribution

It establishes a conjugacy result for $C^{1+BV}$ degree $d \\geq 2$ circle maps with weakly expanding periodic orbits, connecting parabolic external maps to metrically expanding maps.

Findings

Weakly expanding periodic orbits imply conjugacy to metrically expanding maps.

Connects parabolic external maps to the space of expanding circle coverings.

Provides a classification framework for certain circle maps.

Abstract

We prove that any $C^{1+BV}$ degree $d \geq 2$ circle covering $h$ having all periodic orbits weakly expanding, is conjugate in the same smoothness class to a metrically expanding map. We use this to connect the space of parabolic external maps (coming from the theory of parabolic-like maps) to metrically expanding circle coverings.