Symmetrization of rational maps: Arithmetic properties and families of Lattès maps of ℙ^{𝕜}

Gauthier, Thomas; Hutz, Benjamin; Kaschner, Scott

doi:10.1090/ecgd/382

Symmetrization of rational maps: Arithmetic properties and families of Lattès maps of $\mathbb {P}^k$
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by Thomas Gauthier, Benjamin Hutz and Scott Kaschner;

Conform. Geom. Dyn. 27 (2023), 98-117

DOI: https://doi.org/10.1090/ecgd/382

Published electronically: February 14, 2023

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Abstract:

In this paper we study properties of endomorphisms of $\mathbb {P}^k$ using a symmetric product construction $(\mathbb {P}^1)^k/\mathfrak {S}_k \cong \mathbb {P}^k$. Symmetric products have been used to produce examples of endomorphisms of $\mathbb {P}^k$ with certain characteristics, $k\geq 2$. In the present note, we discuss the use of these maps to enlighten stability phenomena in parameter spaces. In particular, we study $k$-deep post-critically finite maps and characterize families of Lattès maps.

References

Matthieu Astorg, Dynamics of post-critically finite maps in higher dimension, Ergodic Theory Dynam. Systems 40 (2020), no. 2, 289–308. MR 4048294, DOI 10.1017/etds.2018.32
Giovanni Bassanelli and François Berteloot, Bifurcation currents in holomorphic dynamics on $\Bbb P^k$, J. Reine Angew. Math. 608 (2007), 201–235. MR 2339474, DOI 10.1515/CRELLE.2007.058
F. Berteloot and J.-J. Loeb, Spherical hypersurfaces and Lattès rational maps, J. Math. Pures Appl. (9) 77 (1998), no. 7, 655–666 (English, with English and French summaries). MR 1645073, DOI 10.1016/S0021-7824(98)80003-2
François Berteloot, Fabrizio Bianchi, and Christophe Dupont, Dynamical stability and Lyapunov exponents for holomorphic endomorphisms of $\Bbb P^k$, Ann. Sci. Éc. Norm. Supér. (4) 51 (2018), no. 1, 215–262 (English, with English and French summaries). MR 3764042, DOI 10.24033/asens.2355
F. Berteloot and C. Dupont, Une caractérisation des endomorphismes de Lattès par leur mesure de Green, Comment. Math. Helv. 80 (2005), no. 2, 433–454 (French, with English summary). MR 2142250, DOI 10.4171/CMH/21
François Berteloot and Jean-Jacques Loeb, Une caractérisation géométrique des exemples de Lattès de ${\Bbb P}^k$, Bull. Soc. Math. France 129 (2001), no. 2, 175–188 (French, with English and French summaries). MR 1871293, DOI 10.24033/bsmf.2392
Jean-Yves Briend and Julien Duval, Exposants de Liapounoff et distribution des points périodiques d’un endomorphisme de $\mathbf C\textrm {P}^k$, Acta Math. 182 (1999), no. 2, 143–157 (French). MR 1710180, DOI 10.1007/BF02392572
Xavier Buff and Thomas Gauthier, Perturbations of flexible Lattès maps, Bull. Soc. Math. France 141 (2013), no. 4, 603–614 (English, with English and French summaries). MR 3203689, DOI 10.24033/bsmf.2657
Marius Dabija and Mattias Jonsson, Algebraic webs invariant under endomorphisms, Publ. Mat. 54 (2010), no. 1, 137–148. MR 2603592, DOI 10.5565/PUBLMAT_{5}4110_{0}7
Olivier Debarre, Tores et variétés abéliennes complexes, Cours Spécialisés [Specialized Courses], vol. 6, Société Mathématique de France, Paris; EDP Sciences, Les Ulis, 1999 (French, with English and French summaries). MR 1767634
Tien-Cuong Dinh and Nessim Sibony, Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mappings, Holomorphic dynamical systems, Lecture Notes in Math., vol. 1998, Springer, Berlin, 2010, pp. 165–294. MR 2648690, DOI 10.1007/978-3-642-13171-4_{4}
Christophe Dupont, Exemples de Lattès et domaines faiblement sphériques de $\Bbb C^n$, Manuscripta Math. 111 (2003), no. 3, 357–378 (French, with English summary). MR 1993500, DOI 10.1007/s00229-003-0378-0
John Erik Fornæss and Nessim Sibony, Dynamics of $\textbf {P}^2$ (examples), Laminations and foliations in dynamics, geometry and topology (Stony Brook, NY, 1998) Contemp. Math., vol. 269, Amer. Math. Soc., Providence, RI, 2001, pp. 47–85. MR 1810536, DOI 10.1090/conm/269/04329
Mattias Jonsson, Some properties of $2$-critically finite holomorphic maps of $\textbf {P}^2$, Ergodic Theory Dynam. Systems 18 (1998), no. 1, 171–187. MR 1609475, DOI 10.1017/S0143385798097521
Scott R. Kaschner, Rodrigo A. Pérez, and Roland K. W. Roeder, Examples of rational maps of $\Bbb C\Bbb P^2$ with equal dynamical degrees and no invariant foliation, Bull. Soc. Math. France 144 (2016), no. 2, 279–297 (English, with English and French summaries). MR 3499082, DOI 10.24033/bsmf.2714
Alon Levy, The space of morphisms on projective space, Acta Arith. 146 (2011), no. 1, 13–31. MR 2741188, DOI 10.4064/aa146-1-2
Helge Maakestad, Resultants and symmetric products, Comm. Algebra 33 (2005), no. 11, 4105–4114. MR 2183984, DOI 10.1080/00927870500261413
John Milnor, On Lattès maps, Dynamics on the Riemann sphere, Eur. Math. Soc., Zürich, 2006, pp. 9–43. MR 2348953, DOI 10.4171/011-1/1
Feng Rong, Lattès maps on $\mathbf P^2$, J. Math. Pures Appl. (9) 93 (2010), no. 6, 636–650 (English, with English and French summaries). MR 2651985, DOI 10.1016/j.matpur.2009.10.002
Nessim Sibony, Dynamique des applications rationnelles de $\mathbf P^k$, Dynamique et géométrie complexes (Lyon, 1997) Panor. Synthèses, vol. 8, Soc. Math. France, Paris, 1999, pp. ix–x, xi–xii, 97–185 (French, with English and French summaries). MR 1760844
Tetsuo Ueda, Complex dynamical systems on projective spaces, Sūrikaisekikenkyūsho K\B{o}kyūroku 814 (1992), 169–186 (Japanese). Topics around chaotic dynamical systems (Japanese) (Kyoto, 1992). MR 1251576
Tetsuo Ueda, Critical orbits of holomorphic maps on projective spaces, J. Geom. Anal. 8 (1998), no. 2, 319–334. MR 1705160, DOI 10.1007/BF02921645
Anna Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps, Invent. Math. 99 (1990), no. 3, 627–649. MR 1032883, DOI 10.1007/BF01234434