A Fundamental Domain for $V_{3}$ - Tome (2010) no. 121

A Fundamental Domain for

V_{3}

[Un domaine fondamental pour

V_{3}

]

Rees, Mary

Mémoires de la Société Mathématique de France, no. 121 (2010) , 141 p.

Résumé
Abstract

Nous décrivons un domaine fondamental pour la surface de Riemann $V_{3, m}$ qui paramétrise (à conjugaison près) l’ensemble des fonctions rationelles par le biais des points critiques énumérés, de manière à ce que le premier point critique ait une période de 3, et que le deuxième point critique ne soit pas envoyé sur le premier après $m$ itérations ou moins. Cela nous fournit une description, à conjugaison topologique près, des dynamiques de toutes les composantes de type III en $V_{3}$ , et nous donne des indications sur un modèle topologique de $V_{3}$ , au même temps que l’ensemble des composantes hyperboliques qui y sont contenues.

We describe a fundamental domain for the punctured Riemann surface $V_{3, m}$ which parametrises (up to Möbius conjugacy) the set of quadratic rational maps with numbered critical points, such that the first critical point has period three, and such that the second critical point is not mapped in $m$ iterates or less to the periodic orbit of the first. This gives, in turn, a description, up to topological conjugacy, of all dynamics in all type III hyperbolic components in $V_{3}$ , and gives indications of a topological model for $V_{3}$ , together with the hyperbolic components contained in it.

MR Zbl | 1 citation dans Numdam

DOI : 10.24033/msmf.433

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     title = {A {Fundamental} {Domain} for $V_{3}$},
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     publisher = {Soci\'et\'e math\'ematique de France},
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     year = {2010},
     doi = {10.24033/msmf.433},
     mrnumber = {2768577},
     zbl = {1222.30001},
     language = {en},
     url = {http://www.numdam.org/item/MSMF_2010_2_121__1_0/}
}

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Rees, Mary. A Fundamental Domain for $V_{3}$. Mémoires de la Société Mathématique de France, Série 2, no. 121 (2010), 141 p. doi : 10.24033/msmf.433. http://numdam.org/item/MSMF_2010_2_121__1_0/

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