A note on parabolic-like maps

TL;DR

This paper proposes a slight modification to the definition of parabolic-like maps, replacing the smoothness condition with a quasiarc condition on the boundary near the fixed point, broadening the class of such maps.

Contribution

The paper introduces a new, less restrictive boundary condition for parabolic-like maps, expanding the framework for their analysis.

Findings

The modified definition includes quasiarcs instead of $C^1$ dividing arcs.

This change allows for a broader class of parabolic-like maps.

The core properties of parabolic-like maps are preserved under this modification.

Abstract

We show that the definition of parabolic-like map can be slightly modified, by asking $\partial \Delta$ to be a quasiarc out of the parabolic fixed point, instead of the dividing arcs to be $C^1$ on $[-1,0]$ and $[0,1]$.