[Paper Review] One-dimensional maps and Poincaré metric

Grzegorz Świątek|arXiv (Cornell University)|Aug 12, 1990

Analytic and geometric function theory4 references4 citations

TL;DR

This paper develops distortion control techniques for compositions of one-dimensional maps using the Poincaré metric, proving bounded joint distortion under Koebe principle conditions and showing that nearly linear-fractional maps with small total nonlinearity can be well-approximated by affine maps. The framework is applied to establish a rigidity result for critical circle maps.

ABSTRACT

Invertible compositions of one-dimensional maps are studied which are assumed to include maps with non-positive Schwarzian derivative and others whose sum of distortions is bounded. If the assumptions of the Koebe principle hold, we show that the joint distortion of the composition is bounded. On the other hand, if all maps with possibly non-negative Schwarzian derivative are almost linear-fractional and their nonlinearities tend to cancel leaving only a small total, then they can all be replaced with affine maps with the same domains and images and the resulting composition is a very good approximation of the original one. These technical tools are then applied to prove a theorem about critical circle maps.

Motivation & Objective

• To analyze the distortion properties of compositions of one-dimensional maps with non-positive Schwarzian derivative.
• To extend distortion control to maps with possibly non-negative Schwarzian derivatives under bounded total nonlinearity.
• To develop tools for approximating complex map compositions with affine maps when nonlinearities nearly cancel.
• To apply these techniques to prove a rigidity theorem for critical circle maps.
• To establish conditions under which the Koebe principle ensures uniform distortion bounds in compositions.

Proposed method

• Applies the Koebe principle to control distortion in compositions of maps with non-positive Schwarzian derivative.
• Uses the Poincaré metric to measure distortion and establish bounds on the joint distortion of map compositions.
• Introduces a notion of total nonlinearity for maps with possibly non-negative Schwarzian derivatives.
• Shows that if total nonlinearity is small, the maps can be replaced by affine maps with negligible error in the composition.
• Employs distortion estimates based on the sum of distortions and the Poincaré metric to control the behavior of iterated maps.
• Applies the developed tools to critical circle maps, proving rigidity under specific dynamical conditions.

Experimental results

Research questions

• RQ1 Under what conditions is the joint distortion of a composition of one-dimensional maps uniformly bounded?
• RQ2 How can maps with non-negative Schwarzian derivatives be approximated by affine maps while preserving dynamical properties?
• RQ3 What role does the total nonlinearity play in determining the quality of affine approximation for map compositions?
• RQ4 Can the Koebe principle be extended to compositions involving maps with non-positive Schwarzian derivatives?
• RQ5 What dynamical rigidity results follow from distortion control in critical circle maps?

Key findings

• Compositions of maps with non-positive Schwarzian derivative satisfy uniform distortion bounds under the Koebe principle.
• When the total nonlinearity of maps with possibly non-negative Schwarzian derivatives is small, they can be replaced by affine maps with negligible error in the composition.
• The Poincaré metric provides an effective tool for measuring and controlling distortion in one-dimensional map compositions.
• The distortion of the composition is bounded if the sum of distortions is bounded and the Koebe principle applies.
• The developed techniques yield a rigidity theorem for critical circle maps, showing that such maps are uniquely determined by their combinatorial and metric data.
• Affine approximations preserve the essential dynamical structure of the original map composition when nonlinearity is sufficiently small.