logo Journal de l'École Polytechnique
Rational maps with rational multipliers
[Fractions rationnelles avec multiplicateurs rationnels]
Valentin Huguin1
1 Constructor University Bremen gGmbH Campus Ring 1, 28759 Bremen, Germany
Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 591-599.

Dans cet article, nous montrons que toute fraction rationnelle dont les multiplicateurs sont tous dans un corps de nombres donné est une application puissance, une application de Tchebychev ou un exemple de Lattès. Ceci généralise une conjecture de Milnor concernant les fractions rationnelles avec multiplicateurs entiers, qui a été récemment démontrée par Ji et Xie.

In this article, we show that every rational map whose multipliers all lie in a given number field is a power map, a Chebyshev map or a Lattès map. This strengthens a conjecture by Milnor concerning rational maps with integer multipliers, which was recently proved by Ji and Xie.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.227
Classification : 37P35, 37F10, 37P05
Keywords: Multiplier spectrum, exceptional maps, arithmetic dynamics
Mot clés : Multiplicateur, application exceptionnelle, dynamique arithmétique
Valentin Huguin 1

1 Constructor University Bremen gGmbH Campus Ring 1, 28759 Bremen, Germany
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{JEP_2023__10__591_0,
     author = {Valentin Huguin},
     title = {Rational maps with rational multipliers},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {591--599},
     publisher = {\'Ecole polytechnique},
     volume = {10},
     year = {2023},
     doi = {10.5802/jep.227},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.227/}
}
TY  - JOUR
AU  - Valentin Huguin
TI  - Rational maps with rational multipliers
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2023
SP  - 591
EP  - 599
VL  - 10
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.227/
DO  - 10.5802/jep.227
LA  - en
ID  - JEP_2023__10__591_0
ER  - 
%0 Journal Article
%A Valentin Huguin
%T Rational maps with rational multipliers
%J Journal de l’École polytechnique — Mathématiques
%D 2023
%P 591-599
%V 10
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.227/
%R 10.5802/jep.227
%G en
%F JEP_2023__10__591_0
Valentin Huguin. Rational maps with rational multipliers. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 591-599. doi : 10.5802/jep.227. https://jep.centre-mersenne.org/articles/10.5802/jep.227/

[Aut01] P. Autissier - “Points entiers sur les surfaces arithmétiques”, J. reine angew. Math. 531 (2001), p. 201-235 | DOI | MR | Zbl

[BR10] M. Baker & R. Rumely - Potential theory and dynamics on the Berkovich projective line, Math. Surveys and Monographs, vol. 159, American Mathematical Society, Providence, RI, 2010 | DOI

[EvS11] A. Eremenko & S. van Strien - “Rational maps with real multipliers”, Trans. Amer. Math. Soc. 363 (2011) no. 12, p. 6453-6463 | DOI | MR | Zbl

[FG22] T. Firsova & I. Gorbovickis - “Accumulation set of critical points of the multipliers in the quadratic family”, Ergodic Theory Dynam. Systems (2022), First View (22 pages) | DOI

[FLM83] A. Freire, A. Lopes & R. Mañé - “An invariant measure for rational maps”, Bol. Soc. Brasil. Mat. 14 (1983) no. 1, p. 45-62 | DOI | MR | Zbl

[FRL06] C. Favre & J. Rivera-Letelier - “Équidistribution quantitative des points de petite hauteur sur la droite projective”, Math. Ann. 335 (2006) no. 2, p. 311-361 | DOI | Zbl

[Hug21] V. Huguin - “Unicritical polynomial maps with rational multipliers”, Conform. Geom. Dyn. 25 (2021), p. 79-87 | DOI | MR | Zbl

[Hug22] V. Huguin - “Quadratic rational maps with integer multipliers”, Math. Z. 302 (2022) no. 2, p. 949-969 | DOI | MR | Zbl

[JX22] Z. Ji & J. Xie - “Homoclinic orbits, multiplier spectrum and rigidity theorems in complex dynamics”, 2022 | arXiv

[Lju83] M. J. Ljubich - “Entropy properties of rational endomorphisms of the Riemann sphere”, Ergodic Theory Dynam. Systems 3 (1983) no. 3, p. 351-385 | DOI | MR

[Mañ83] R. Mañé - “On the uniqueness of the maximizing measure for rational maps”, Bol. Soc. Brasil. Mat. 14 (1983) no. 1, p. 27-43 | DOI | MR | Zbl

[McM87] C. McMullen - “Families of rational maps and iterative root-finding algorithms”, Ann. of Math. (2) 125 (1987) no. 3, p. 467-493 | DOI | MR | Zbl

[Mil06] J. Milnor - “On Lattès maps”, in Dynamics on the Riemann sphere, European Mathematical Society, Zürich, 2006, p. 9-43 | DOI | Zbl

[Sil98] J. H. Silverman - “The space of rational maps on ℙ 1 ”, Duke Math. J. 94 (1998) no. 1, p. 41-77 | DOI | Zbl

[Zdu14] A. Zdunik - “Characteristic exponents of rational functions”, Bull. Acad. Polon. Sci. Sér. Sci. Math. 62 (2014) no. 3, p. 257-263 | DOI | Zbl

Cité par Sources :