On quasi-invariant curves

TL;DR

This paper provides a geometric interpretation of the Denjoy-Yoccoz lemma using hyperbolic metrics, enabling direct construction of quasi-invariant curves in the study of hedgehog dynamics without relying on renormalization techniques.

Contribution

It introduces a new geometric approach to understanding quasi-invariant curves, bypassing the need for complex renormalization methods.

Findings

Geometric interpretation of Denjoy-Yoccoz lemma using hyperbolic metric

Direct construction method for quasi-invariant curves

Enhanced understanding of hedgehog dynamics

Abstract

Quasi-invariant curves are used in the study of hedgehog dynamics. Denjoy-Yoccoz lemma is the preliminary step for Yoccoz's complex renormalization techniques for the study of linearization of analytic circle diffeomorphisms. We give a geometric interpretation of Denjoy-Yoccoz lemma using the hyperbolic metric that gives a direct construction of quasi-invariant curves without renormalization.