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DYNAMIQUE ANALYTIQUE SUR Z. I: MESURES D’ÉQUILIBRE SUR UNE DROITE PROJECTIVE RELATIVE

Part of: Arithmetic problems. Diophantine geometry Arithmetic and non-Archimedean dynamical systems

Published online by Cambridge University Press:  24 November 2025

Jérôme Poineau Open the ORCID record for Jérôme Poineau [Opens in a new window]
Jérôme Poineau*
Affiliation:
LMNO, Université de Caen Normandie , France
*
(jerome.poineau@unicaen.fr)
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Résumé

Considérons un espace de Berkovich sur un bon anneau de Banach et la droite projective relative sur celui-ci. (C’est un espace dont les fibres sont des droites projectives sur différents corps valués complets.) Pour tout endomorphisme polarisé de cette droite, nous montrons que la famille des mesures d’équilibre associées aux restrictions de l’endomorphisme aux fibres est continue. Le résultat vaut, par exemple, lorsque l’anneau de Banach est un corps valué complet, un corps hybride, un anneau de valuation discrète complet ou un anneau d’entiers de corps de nombres.

Abstract

Abstract

(Analytic dynamics over Z. I: Equilibrium measures on a relative projective line.) Consider a Berkovich space over a good Banach ring and the relative projective line over it. (It is a space whose fibers are projective lines over different complete valued fields.) For each polarized endomorphism of this line, we prove that the family of equilibrium measures associated to the restrictions of the endomorphism to the fibers is continuous. The result holds, in particular, when the Banach ring is a complete valued field, a hybrid field, a complete discrete valuation ring, or the ring of integers of a number field.

Keywords

Espaces de Berkovich sur Zespaces hybridesmesures d’équilibrethéorie du potentiel

MSC classification

Primary: 37P50: Dynamical systems on Berkovich spaces 37P15: Global ground fields
Secondary: 14G22: Rigid analytic geometry

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , First View , pp. 1 - 43
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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