Comptes Rendus
no. 7-8
Dynamical Systems
Dimension and measure for semi-hyperbolic rational maps of degree 2
[Dimension et mesure pour les applications rationnelles semi-hyperboliques de degré 2]

Présenté par : Étienne Ghys

Magnus Aspenberg1 ; Jacek Graczyk1
1 Laboratoire de mathématique, Université de Paris-Sud, 91405 Orsay cedex, France
Comptes Rendus. Mathématique, Volume 347 (2009) no. 7-8, pp. 395-400.

Nous démontrons que presque toute application rationnelle semi-hyperbolique de degré 2 a au moins un point critique récurrent. Cette estimation est optimale parce que l'ensemble des applications rationnelles avec tous les points critiques non-récurrents est de pleine dimension de Hausdorff.

We prove that almost every non-hyperbolic rational map of degree 2 has at least one recurrent critical point. This estimate is optimal because the set of rational maps with all critical points non-recurrent is of full Hausdorff dimension.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.02.016
Magnus Aspenberg 1 ; Jacek Graczyk 1

1 Laboratoire de mathématique, Université de Paris-Sud, 91405 Orsay cedex, France 
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Magnus Aspenberg; Jacek Graczyk. Dimension and measure for semi-hyperbolic rational maps of degree 2. Comptes Rendus. Mathématique, Volume 347 (2009) no. 7-8, pp. 395-400. doi : 10.1016/j.crma.2009.02.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.02.016/

[1] M. Aspenberg Rational Misiurewicz maps are rare II (Preprint Comm. Math. Phys., in press) | arXiv

[2] L. Carleson; P.W. Jones; J.-C. Yoccoz Julia and John, Bol. Soc. Brasil. Mat. (N.S.), Volume 25 (1994), pp. 1-30

[3] A. Douady; J. Hubbard On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup., Volume 18 (1985), pp. 287-343

[4] O. Lehto; K.I. Virtanen Quasiconformal Mappings in the Plane, Springer-Verlag, New York, 1973 (viii+258)

[5] C.T. McMullen Complex Dynamics and Renormalization, Annals of Mathematics Studies, vol. 135, Princeton Univ. Press, Princeton, NJ, 1994 (x+214)

[6] R. Mañé On a theorem of Fatou, Bol. Soc. Brasil. Mat. (N.S.), Volume 24 (1993), pp. 1-11

[7] R. Mañé; P. Sad; D. Sullivan On the dynamics of rational maps, Ann. Sci. École Norm. Sup., Volume 16 (1983), pp. 193-217

[8] N. Mihalache La condition de Collet–Eckmann pour les orbites critiques recurrents, Ergod. Th. Dynam. Sys., Volume 27 (2007) no. 4, pp. 1267-1286

[9] M. Shishikura The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, Ann. of Math. (2), Volume 147 (1998), pp. 225-267

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