Typical orbits of quadratic polynomials with a neutral fixed point: Non-Brjuno type

Cheraghi, Davoud

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 1, pp. 59-138.

On étudie les aspects quantitatifs et analytiques du procédé de renormalisation presque parabolique introduit par Inou et Shishikura en 2006. Ceci fournit des techniques pour étudier la dynamique de certaines applications holomorphes de la forme $f\left(z\right)=e^{2\pi i\alpha }z+𝒪\left(z^2\right)$, dont les polynômes quadratiques $e^{2\pi i\alpha }z+z^2$, pour certaines valeurs irrationnelles de $\alpha$. Les principaux résultats de cet article concernent les propriétés à petite échelle des attracteurs au sens de la théorie de la mesure pour ces applications ainsi que de leur dépendance en fonction des données du problème. On obtient également une borne supérieure optimale sur la taille du domaine maximal de linéarisation en termes de la série de Brjuno-Siegel-Yoccoz de $\alpha$.

We investigate the quantitative and analytic aspects of the near-parabolic renormalization scheme introduced by Inou and Shishikura in 2006. These provide techniques to study the dynamics of some holomorphic maps of the form $f\left(z\right)=e^{2\pi i\alpha }z+𝒪\left(z^2\right)$, including the quadratic polynomials $e^{2\pi i\alpha }z+z^2$, for some irrational values of $\alpha$. The main results of the paper concern fine-scale features of the measure-theoretic attractors of these maps, and their dependence on the data. As a bi-product, we establish an optimal upper bound on the size of the maximal linearization domain in terms of the Siegel-Brjuno-Yoccoz series of $\alpha$.

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