A model for the parabolic slices $ {\mathrm{Per}}_{1}({e}^{2\pi ip/q})$ in moduli space of quadratic rational maps

Eva Uhre

doi:10.1016/j.crma.2010.10.033

Dynamical Systems

A model for the parabolic slices

{Per}_{1} (e^{2 π i p / q})

in moduli space of quadratic rational maps
[Un modèle pour les sections paraboliques

{Per}_{1} (e^{2 π i p / q})

de l'espace des modules des fractions rationnelles quadratiques]

Présenté par : Étienne Ghys

Eva Uhre ^1,²

¹ Institut de mathématiques de Toulouse, Université Paul-Sabatier, 31062 Toulouse cedex, France
² NSM, Roskilde University, 4000 Roskilde, Denmark

Comptes Rendus. Mathématique, Volume 348 (2010) no. 23-24, pp. 1327-1330.

Résumé
Abstract

Nous introduisons la notion de lieux de parenté dans les sections paraboliques ${Per}_{1} (e^{2 π i p / q})$ de l'espace des modules des fractions rationnelles quadratiques. Ce sont des analogues du lieu de non-connexité dans la section correspondant aux polynômes quadratiques. Nous présentons un modèle pour ces lieux, et donnons une stratégie de preuve de la fidélité de ce modèle.

The notion of relatedness loci in the parabolic slices ${Per}_{1} (e^{2 π i p / q})$ in moduli space of quadratic rational maps is introduced. They are counterparts of the disconnectedness or escape locus in the slice of quadratic polynomials. A model for these loci is presented, and a strategy of proof of the faithfulness of the model is given.

Reçu le : 2009-12-19
Accepté le : 2010-10-26
Publié le : 2010-11-17

DOI : 10.1016/j.crma.2010.10.033

Affiliations des auteurs :

Eva Uhre ^{1, 2}

¹ Institut de mathématiques de Toulouse, Université Paul-Sabatier, 31062 Toulouse cedex, France
² NSM, Roskilde University, 4000 Roskilde, Denmark

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Eva Uhre. A model for the parabolic slices $ {\mathrm{Per}}_{1}({e}^{2\pi ip/q})$ in moduli space of quadratic rational maps. Comptes Rendus. Mathématique, Volume 348 (2010) no. 23-24, pp. 1327-1330. doi : 10.1016/j.crma.2010.10.033. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.10.033/

Bibliographie
Cité par

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[2] J. Milnor, Hyperbolic components in spaces of polynomial maps, IMS Stony Brook, preprint #3(1992).

[3] J. Milnor Geometry and dynamics of quadratic rational maps, Experiment. Math., Volume 2 (1993) no. 1, pp. 37-83

[4] C.L. Petersen No elliptic limits for quadratic maps, Ergod. Th. & Dynam. Sys., Volume 19 (1999), pp. 127-141

[5] C.L. Petersen; L. Tan Analytic coordinates recording cubic dynamics (D. Schleicher, ed.), Complex Dynamics, Families and Friends, A.K. Peters, 2009, pp. 413-450

[6] M. Rees Components of degree two hyperbolic rational maps, Invent. Math., Volume 100 (1990), pp. 357-382

[7] E. Uhre, The structure of parabolic slices ${Per}_{1} (e^{2 π i p / q})$ in moduli space of quadratic rational maps, manuscript, 2009.

[8] B. Wittner, On the bifurcation loci of rational maps of degree two, PhD thesis, Cornell University, 1988.

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