Dynamics of certain smooth one-dimensional mappings II: geometrically finite one-dimensional mappings

TL;DR

This paper investigates geometrically finite one-dimensional mappings, demonstrating their stability under quasisymmetric conjugacy and establishing conditions under which topological conjugacy implies quasisymmetric conjugacy.

Contribution

It proves that the subspace of geometrically finite mappings is closed under quasisymmetric conjugacy and that topological conjugacy implies quasisymmetric conjugacy within this class.

Findings

The subspace of geometrically finite mappings is closed under quasisymmetric conjugacy.

Topological conjugacy implies quasisymmetric conjugacy for these mappings.

Examples illustrating the properties of geometrically finite mappings.

Abstract

We study geometrically finite one-dimensional mappings. These are a subspace of $C^{1+\alpha}$ one-dimensional mappings with finitely many, critically finite critical points. We study some geometric properties of a mapping in this subspace. We prove that this subspace is closed under quasisymmetrical conjugacy. We also prove that if two mappings in this subspace are topologically conjugate, they are then quasisymmetrically conjugate. We show some examples of geometrically finite one-dimensional mappings.