A new proof of a conjecture of Yoccoz

Buff, Xavier, and Chéritat, Arnaud. "A new proof of a conjecture of Yoccoz." Annales de l’institut Fourier 61.1 (2011): 319-350. <http://eudml.org/doc/219751>.

@article{Buff2011,
abstract = {We give a new proof of the following conjecture of Yoccoz:\[ (\exists C\in \mathbb\{R\})~(\forall \theta \in \mathbb\{R\}\setminus \mathbb\{Q\}) \quad \log \mathrm\{rad\}\,\Delta (Q\_\theta ) \le -Y(\theta ) +C, \]where $Q_\{\theta \}(z)=\mathrm\{e\}^\{2\pi i\theta \}z+z^2$, $\Delta (Q_\theta )$ is its Siegel disk if $Q_\theta $ is linearizable (or $\emptyset$ otherwise), $\mathrm\{rad\}\,\Delta (Q_\theta )$ is the conformal radius of the Siegel disk of $Q_\theta $ (or $0$ if there is none) and $Y(\theta )$ is Yoccoz’s Brjuno function.In a former article we obtained a first proof based on the control of parabolic explosion. Here, we present a more elementary proof based on Yoccoz’s initial methods.We then extend this result to some new families of polynomials such as $z^d+c$ with $d&gt;2$. We also show that the conjecture does not hold for $\mathrm\{e\}^\{2\pi i\theta \} (z + z^d)$ with $d&gt;2$.},
affiliation = {Université Paul Sabatier Institut de Mathématiques de Toulouse 118, route de Narbonne 31062 Toulouse Cedex 9 (France); C.N.R.S Université Paul Sabatier Institut de Mathématiques de Toulouse 118, route de Narbonne 31062 Toulouse Cedex 9},
author = {Buff, Xavier, Chéritat, Arnaud},
journal = {Annales de l’institut Fourier},
keywords = {Siegel disks; quadratic polynomials; harmonic and subharbonic functions; conformal radius; holomorphic motions},
language = {eng},
number = {1},
pages = {319-350},
publisher = {Association des Annales de l’institut Fourier},
title = {A new proof of a conjecture of Yoccoz},
url = {http://eudml.org/doc/219751},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Buff, Xavier
AU - Chéritat, Arnaud
TI - A new proof of a conjecture of Yoccoz
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 1
SP - 319
EP - 350
AB - We give a new proof of the following conjecture of Yoccoz:\[ (\exists C\in \mathbb{R})~(\forall \theta \in \mathbb{R}\setminus \mathbb{Q}) \quad \log \mathrm{rad}\,\Delta (Q_\theta ) \le -Y(\theta ) +C, \]where $Q_{\theta }(z)=\mathrm{e}^{2\pi i\theta }z+z^2$, $\Delta (Q_\theta )$ is its Siegel disk if $Q_\theta $ is linearizable (or $\emptyset$ otherwise), $\mathrm{rad}\,\Delta (Q_\theta )$ is the conformal radius of the Siegel disk of $Q_\theta $ (or $0$ if there is none) and $Y(\theta )$ is Yoccoz’s Brjuno function.In a former article we obtained a first proof based on the control of parabolic explosion. Here, we present a more elementary proof based on Yoccoz’s initial methods.We then extend this result to some new families of polynomials such as $z^d+c$ with $d&gt;2$. We also show that the conjecture does not hold for $\mathrm{e}^{2\pi i\theta } (z + z^d)$ with $d&gt;2$.
LA - eng
KW - Siegel disks; quadratic polynomials; harmonic and subharbonic functions; conformal radius; holomorphic motions
UR - http://eudml.org/doc/219751
ER -