Dynamics of certain smooth one-dimensional mappings I: The $C^{1+α}$-Denjoy-Koebe distortion lemma

On quasi-invariant curves

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.

刺猬的唯一遍历性

Unique ergodicity of hedgehogs

本文证明了非线性化刺猬上的动力学具有唯一遍历性，解决了作者1995年的猜想，其唯一不变测度为惰性不动点处的狄拉克质量，证明基于拟不变曲线和Denjoy-Yoccoz引理的双曲形式。

Dynamics of certain non-conformal semigroups

A semigroup generated by two dimensional $C^{1+α}$ contracting maps is considered. We call a such semigroup regular if the maximum $K$ of the conformal dilatations of generators, the maximum $l$ of the norms of the derivatives of generators and the smoothness $α$ of the generators satisfy a compatibility condition $K< 1/l^α$. We prove that the shape of the image of the core of a ball under any element of a regular semigroup is good (bounded geometric distortion like the Koebe $1/4$-lemma \cite{a}). And we use it to show a lower and a upper bounds of the Hausdorff dimension of the limit set of …