On quasi-invariant curves

Quasi-invariant curves are used in the study of hedgehog dynamics. Denjoy-Yoccoz lemma is the preliminary step for Yoccozs 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.