[Paper Review] Dynamics of certain smooth one-dimensional mappings II: geometrically finite one-dimensional mappings

Yunping Jiang|ArXiv.org|Jan 11, 1991

Mathematical Dynamics and Fractals8 references3 citations

TL;DR

This paper studies geometrically finite one-dimensional mappings in the $C^{1+\alpha}$ class with finitely many critically finite critical points. It establishes that this class is closed under quasisymmetrical conjugacy and proves that topological conjugacy implies quasisymmetric conjugacy, providing a rigidity result for such systems.

ABSTRACT

We study geometrically finite one-dimensional mappings. These are a subspace of $C^{1+α}$ 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.

Motivation & Objective

• To analyze the geometric and dynamical properties of geometrically finite one-dimensional mappings in the $C^{1+\alpha}$ class.
• To investigate the closure properties of this class under quasisymmetrical conjugacy.
• To establish a rigidity result linking topological and quasisymmetric conjugacy for mappings in this class.
• To provide illustrative examples of geometrically finite mappings to clarify their structure and behavior.
• To contribute to the understanding of one-dimensional dynamical systems with finite critical structure and smoothness constraints.

Proposed method

• Define geometrically finite one-dimensional mappings as $C^{1+\alpha}$ mappings with finitely many critically finite critical points.
• Use quasisymmetry to analyze conjugacy relations between mappings in the defined class.
• Apply topological conjugacy arguments to show that topological equivalence implies quasisymmetric equivalence.
• Employ techniques from smooth dynamics and quasiconformal mappings to prove closure under quasisymmetrical conjugacy.
• Construct explicit examples of geometrically finite mappings to demonstrate the theoretical framework.
• Leverage the $C^{1+\alpha}$ regularity to control distortion and ensure structural stability under conjugacy.

Experimental results

Research questions

• RQ1 Is the class of geometrically finite one-dimensional mappings closed under quasisymmetrical conjugacy?
• RQ2 Under what conditions does topological conjugacy between two such mappings imply quasisymmetric conjugacy?
• RQ3 How do the geometric and dynamical properties of these mappings constrain their conjugacy classes?
• RQ4 What role does the finite critical point structure play in the rigidity of the system?
• RQ5 Can explicit examples of geometrically finite mappings be constructed that illustrate the theoretical results?

Key findings

• The class of geometrically finite one-dimensional mappings is closed under quasisymmetrical conjugacy.
• Topological conjugacy between two mappings in this class implies quasisymmetric conjugacy.
• The rigidity result establishes a strong link between topological and quasisymmetric conjugacy in the $C^{1+\alpha}$ setting.
• The finite number of critically finite critical points ensures structural control necessary for the rigidity theorems.
• Examples of geometrically finite mappings are constructed to illustrate the theoretical framework and verify the results.
• The $C^{1+\alpha}$ regularity condition is essential for controlling distortion and enabling the quasisymmetry arguments.