Points périodiques des fonctions rationnelles dans l'espace hyperbolique $p$-adique

• Autores: Juan Rivera-Letelier
• Localización: Commentarii mathematici helvetici, ISSN 0010-2571, Vol. 80, Nº 3, 2005, págs. 593-629
• Idioma: francés
• DOI: 10.4171/cmh/27
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• Resumen

  • e study the dynamics of rational maps with coefficients in the field ${\Bbb C}_p$ acting on the hyperbolic space ${\Bbb H}_p$. Our main result is that the number of periodic points in ${\Bbb H}_p$ of such a rational map is either $0$, $1$ or $\infty$, and we characterize those rational maps having precisely $0$ or $1$ periodic points. The main property we obtain is a criterion for the existence of infinitely many periodic points (of a special kind) in hyperbolic space. The proof of this criterion is analogous to G. Julia's proof of the density of repelling periodic points in the Julia set of a complex rational map.