Sparsity of postcritically finite maps of $\mathbb{P}^k$ and beyond: A complex analytic approach

Thomas Gauthier; Johan Taflin; Gabriel Vigny

doi:10.5802/pmihes.1

Thomas Gauthier ¹ ; Johan Taflin ²; Gabriel Vigny ³

¹Laboratoire de Mathématiques d’Orsay, Bâtiment 307, Université Paris-Saclay, 91405 Orsay Cedex, France
²Institut de Mathématiques de Bourgogne UMR 5584 CNRS, Université Bourgogne Europe, 21000 Dijon, France
³LAMFA, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens Cedex 1, France

Publications Mathématiques de l'IHÉS, Online first, pp. 1-96

Abstract

An endomorphism $f:\mathbb{P}^k\rightarrow \mathbb{P}^k$ of degree $d\ge 2$ is said to be postcritically finite (or PCF) if its critical set $\mathrm{Crit}(f)$ is preperiodic, i.e. if there are integers $m>n\ge 0$ such that $f^m(\mathrm{Crit}(f))\subseteq f^n(\mathrm{Crit}(f))$. When $k\ge 2$, it was conjectured in [61] that, in the space $\mathrm{End}_d^k$ of all endomorphisms of degree $d$ of $\mathbb{P}^k$, such endomorphisms are not Zariski dense. We prove this conjecture. Further, in the space $\mathrm{Poly}_d^2$ of all regular polynomial endomorphisms of degree $d\ge 2$ of the affine plane $\mathbb{A}^2$, we construct a dense and Zariski open subset where we have a uniform bound on the number of preperiodic points lying in the critical set.

The key object in the article are the complex bifurcation measure and its properties. The proofs are a combination of the theory of heights in arithmetic dynamics and methods from real dynamics to produce open subsets with maximal bifurcation.

Received: 2023-05-10
Revised: 2025-10-12
Accepted: 2025-12-18
Online First: 2026-02-05

DOI: 10.5802/pmihes.1

Classification: 37P15, 37P30, 11G50, 37P35, 37F45
Keywords: regular plane polynomial automorphisms, canonical height, algebraic family of rational maps, arithmetic characterizations of stability

Author's affiliations:

Thomas Gauthier ¹ ; Johan Taflin ² ; Gabriel Vigny ³

¹ Laboratoire de Mathématiques d’Orsay, Bâtiment 307, Université Paris-Saclay, 91405 Orsay Cedex, France
² Institut de Mathématiques de Bourgogne UMR 5584 CNRS, Université Bourgogne Europe, 21000 Dijon, France
³ LAMFA, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens Cedex 1, France

@unpublished{PMIHES_0__0__1_0,
     author = {Thomas Gauthier and Johan Taflin and Gabriel Vigny},
     title = {Sparsity of postcritically finite maps of $\mathbb{P}^k$ and beyond: {A} complex analytic approach},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     year = {2026},
     publisher = {IHES},
     doi = {10.5802/pmihes.1},
     language = {en},
     note = {Online first},
}

TY  - UNPB
AU  - Thomas Gauthier
AU  - Johan Taflin
AU  - Gabriel Vigny
TI  - Sparsity of postcritically finite maps of $\mathbb{P}^k$ and beyond: A complex analytic approach
JO  - Publications Mathématiques de l'IHÉS
PY  - 2026
PB  - IHES
N1  - Online first
DO  - 10.5802/pmihes.1
LA  - en
ID  - PMIHES_0__0__1_0
ER  -

%0 Unpublished Work
%A Thomas Gauthier
%A Johan Taflin
%A Gabriel Vigny
%T Sparsity of postcritically finite maps of $\mathbb{P}^k$ and beyond: A complex analytic approach
%J Publications Mathématiques de l'IHÉS
%D 2026
%V 0
%I IHES
%Z Online first
%R 10.5802/pmihes.1
%G en
%F PMIHES_0__0__1_0

Thomas Gauthier; Johan Taflin; Gabriel Vigny. Sparsity of postcritically finite maps of $\mathbb{P}^k$ and beyond: A complex analytic approach. Publications Mathématiques de l'IHÉS, Online first, pp. 1-96

References
Cited by

[1] Matthieu Astorg Dynamics of post-critically finite maps in higher dimension, Ergodic Theory Dyn. Syst., Volume 40 (2020) no. 2, pp. 289-308 | DOI | Zbl | MR

[2] Matthieu Astorg; Fabrizio Bianchi Higher bifurcations for polynomial skew products, J. Mod. Dyn., Volume 18 (2022), pp. 69-99 | Zbl | DOI | MR

[3] Matthieu Astorg; Thomas Gauthier; Nicolae Mihalache; Gabriel Vigny Collet, Eckmann and the bifurcation measure, Invent. Math., Volume 217 (2019) no. 3, pp. 749-797 | Zbl | DOI | MR

[4] Giovanni Bassanelli; François Berteloot Bifurcation currents in holomorphic dynamics on $ℙ^{k}$ , J. Reine Angew. Math., Volume 608 (2007), pp. 201-235 | Zbl | MR

[5] Giovanni Bassanelli; François Berteloot Distribution of polynomials with cycles of a given multiplier, Nagoya Math. J., Volume 201 (2011), pp. 23-43 | Zbl | DOI | MR

[6] Eric Bedford; Mattias Jonsson Dynamics of regular polynomial endomorphisms of $ℂ^{k}$ , Am. J. Math., Volume 122 (2000) no. 1, pp. 153-212 | Zbl | MR | DOI

[7] Eric Bedford; B. Alan Taylor The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math., Volume 37 (1976), pp. 1-44 | Zbl | DOI | MR

[8] Pierre Berger Generic family with robustly infinitely many sinks, Invent. Math., Volume 205 (2016) no. 1, pp. 121-172 correction in ibid. 218 (2) (2019), 649-656 | Zbl | DOI | MR

[9] Pierre Berger; Bernhard Reinke Parametric linearization of skew products (2022) | arXiv | Zbl

[10] François Berteloot; Fabrizio Bianchi; Christophe Dupont Dynamical stability and Lyapunov exponents for holomorphic endomorphisms of $ℙ^{k}$ , Ann. Sci. Éc. Norm. Supér. (4), Volume 51 (2018) no. 1, pp. 215-262 | Zbl | DOI | MR

[11] François Berteloot; Christophe Dupont Une caractérisation des endomorphismes de Lattès par leur mesure de Green, Comment. Math. Helv., Volume 80 (2005) no. 2, pp. 433-454 | Zbl | DOI | MR

[12] François Berteloot; Christophe Dupont A distortion theorem for iterated inverse branches of holomorphic endomorphisms of $ℙ^{k}$ , J. Lond. Math. Soc. (2), Volume 99 (2019) no. 1, pp. 153-172 | Zbl | DOI | MR

[13] François Berteloot; Christophe Dupont; Laura Molino Normalization of bundle holomorphic contractions and applications to dynamics, Ann. Inst. Fourier, Volume 58 (2008) no. 6, pp. 2137-2168 | Zbl | DOI | Numdam | MR

[14] François Berteloot; Jean-Jacques Loeb Une caractérisation géométrique des exemples de Lattès de $ℙ^{k}$ , Bull. Soc. Math. Fr., Volume 129 (2001) no. 2, pp. 175-188 | Zbl | DOI | MR

[15] Fabrizio Bianchi Misiurewicz parameters and dynamical stability of polynomial-like maps of large topological degree, Math. Ann., Volume 373 (2019) no. 3-4, pp. 901-928 | Zbl | DOI | MR

[16] Fabrizio Bianchi; Johan Taflin Bifurcations in the elementary Desboves family, Proc. Am. Math. Soc., Volume 145 (2017) no. 10, pp. 4337-4343 | Zbl | MR | DOI

[17] Sébastien Biebler Lattès maps and the interior of the bifurcation locus, J. Mod. Dyn., Volume 15 (2019), pp. 95-130 | Zbl | DOI | MR

[18] Christian Bonatti; Lorenzo J. Díaz Persistent nonhyperbolic transitive diffeomorphisms, Ann. Math. (2), Volume 143 (1996) no. 2, pp. 357-396 | Zbl | DOI | MR

[19] Christian Bonatti; Lorenzo J. Díaz Robust heterodimensional cycles and $C^{1}$ -generic dynamics, J. Inst. Math. Jussieu, Volume 7 (2008) no. 3, pp. 469-525 | Zbl | DOI | MR

[20] Christian Bonatti; Lorenzo J. Díaz Abundance of $C^{1}$ -robust homoclinic tangencies, Trans. Am. Math. Soc., Volume 364 (2012) no. 10, pp. 5111-5148 | Zbl | DOI | MR

[21] Jean-Yves Briend; Julien Duval Exposants de Liapounoff et distribution des points périodiques d’un endomorphisme de $C P^{k}$ , Acta Math., Volume 182 (1999) no. 2, pp. 143-157 | Zbl | DOI | MR

[22] Xavier Buff; Adam L. Epstein From local to global analytic conjugacies, Ergodic Theory Dyn. Syst., Volume 27 (2007) no. 4, pp. 1073-1094 | Zbl | DOI | MR

[23] Xavier Buff; Adam L. Epstein Bifurcation measure and postcritically finite rational maps, Complex dynamics : families and friends / edited by Dierk Schleicher, A K Peters, 2009, pp. 491-512 | Zbl | DOI

[24] Gregory S. Call; Joseph H. Silverman Canonical heights on varieties with morphisms, Compos. Math., Volume 89 (1993) no. 2, pp. 163-205 | Zbl | Numdam | MR

[25] Serge Cantat; Ziyang Gao; Philipp Habegger; Junyi Xie The geometric Bogomolov conjecture, Duke Math. J., Volume 170 (2021) no. 2, pp. 247-277 | Zbl | MR

[26] Antoine Chambert-Loir Mesures et équidistribution sur les espaces de Berkovich, J. Reine Angew. Math., Volume 595 (2006), pp. 215-235 | Zbl | MR

[27] Evgeniĭ M. Chirka Complex analytic sets, Mathematics and its Applications. Soviet Series, 46, Kluwer Academic Publishers, 1989 (translated from the Russian by R. A. M. Hoksbergen) | Zbl | DOI

[28] John Dalbec; Bernd Sturmfels Introduction to Chow forms, Invariant methods in discrete and computational geometry (Curaçao, 1994), Kluwer Academic Publishers, 1994, pp. 37-58 | Zbl | MR

[29] Jean-Pierre Demailly Mesures de Monge–Ampère et caractérisation géométrique des variétés algébriques affines, Mémoires de la Société Mathématique de France. Nouvelle Série, 19, Société Mathématique de France, 1985, 124 pages | Zbl | Numdam

[30] Laura DeMarco Dynamics of rational maps: a current on the bifurcation locus, Math. Res. Lett., Volume 8 (2001) no. 1-2, pp. 57-66 | Zbl | DOI | MR

[31] Laura DeMarco; Holly Krieger; Hexi Ye Uniform Manin–Mumford for a family of genus 2 curves, Ann. Math. (2), Volume 191 (2020) no. 3, pp. 949-1001 | Zbl

[32] Laura DeMarco; Holly Krieger; Hexi Ye Common preperiodic points for quadratic polynomials, J. Mod. Dyn., Volume 18 (2022), pp. 363-413 | Zbl | MR | DOI

[33] Laura DeMarco; Niki Myrto Mavraki Dynamics on $ℙ^{1}$ : preperiodic points and pairwise stability, Compos. Math., Volume 160 (2024) no. 2, pp. 356-387 | Zbl | MR | DOI

[34] Tien-Cuong Dinh; Nessim Sibony Equidistribution towards the Green current for holomorphic maps, Ann. Sci. Éc. Norm. Supér. (4), Volume 41 (2008) no. 2, pp. 307-336 | Zbl | Numdam | DOI | MR

[35] Tien-Cuong Dinh; Nessim Sibony Super-potentials of positive closed currents, intersection theory and dynamics, Acta Math., Volume 203 (2009) no. 1, pp. 1-82 | Zbl | DOI | MR

[36] Tien-Cuong Dinh; Nessim Sibony Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mappings, Holomorphic dynamical systems (Lecture Notes in Mathematics), Volume 1998, Springer, 2010, pp. 165-294 | Zbl

[37] John R. Doyle; Joseph H. Silverman Moduli spaces for dynamical systems with portraits, Dyn. Syst., Volume 64 (2020) no. 3, pp. 375-465 | Zbl

[38] Romain Dujardin Non-density of stability for holomorphic mappings on $ℙ^{k}$ , J. Éc. Polytech., Math., Volume 4 (2017), pp. 813-843 | Zbl | MR | DOI

[39] Romain Dujardin; Charles Favre Distribution of rational maps with a preperiodic critical point, Am. J. Math., Volume 130 (2008) no. 4, pp. 979-1032 | Zbl | DOI | MR

[40] Christophe Dupont; Johan Taflin Dynamics of fibered endomorphisms of $ℙ^{k}$ , Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), Volume 22 (2021) no. 1, pp. 53-78 | Zbl | MR

[41] Najmuddin Fakhruddin Questions on self maps of algebraic varieties, J. Ramanujan Math. Soc., Volume 18 (2003) no. 2, pp. 109-122 | Zbl | MR

[42] Charles Favre; Thomas Gauthier Distribution of postcritically finite polynomials, Isr. J. Math., Volume 209 (2015) no. 1, pp. 235-292 | Zbl | DOI | MR

[43] Charles Favre; Mattias Jonsson Brolin’s theorem for curves in two complex dimensions, Ann. Inst. Fourier, Volume 53 (2003) no. 5, pp. 1461-1501 | Zbl | Numdam | DOI | MR

[44] John Erik Fornæss; Nessim Sibony Critically finite rational maps on $ℙ^{2}$ , The Madison symposium on complex analysis. Proceedings of the symposium, held June 2-7, 1991, at the University of Wisconsin-Madison, Madison, WI, USA (Contemporary Mathematics), Volume 137, American Mathematical Society, 1991, pp. 245-260 | Zbl | MR

[45] John Erik Fornaess; Nessim Sibony Complex dynamics in higher dimension. II, Modern methods in complex analysis (Princeton, NJ, 1992) (Annals of Mathematics Studies), Volume 137, Princeton University Press, 1992, pp. 135-182 | Zbl

[46] Vesselin Dimitrov anf Ziyang Gao; Philipp Habegger Uniformity in Mordell–Lang for curves, Ann. Math. (2), Volume 194 (2021) no. 1, pp. 237-298 | Zbl | MR

[47] Ziyang Gao; Tangli Ge; Lars Kühne The uniform mordell-lang conjecture (2021) | arXiv | Zbl

[48] Ziyang Gao; Philipp Habegger Heights in families of abelian varieties and the geometric Bogomolov conjecture, Ann. Math. (2), Volume 189 (2019) no. 2, pp. 527-604 | Zbl

[49] Thomas Gauthier Strong bifurcation loci of full Hausdorff dimension, Ann. Sci. Éc. Norm. Supér. (4), Volume 45 (2012) no. 6, pp. 947-984 | Zbl | Numdam | MR | DOI

[50] Thomas Gauthier Dynamical pairs with an absolutely continuous bifurcation measure, Ann. Fac. Sci. Toulouse, Math., Volume 32 (2023) no. 2, pp. 203-230 | Zbl | DOI | Numdam | MR

[51] Thomas Gauthier Good height functions on quasi-projective varieties: equidistribution and applications in dynamics, Ann. Fac. Sci. Toulouse, Math., Volume 35 (2026) no. 1, pp. 57-94 | Zbl | DOI

[52] Thomas Gauthier; Yûsuke Okuyama; Gabriel Vigny Hyperbolic components of rational maps: quantitative equidistribution and counting, Comment. Math. Helv., Volume 94 (2019) no. 2, pp. 347-398 | Zbl | DOI | MR

[53] Thomas Gauthier; Yûsuke Okuyama; Gabriel Vigny Approximation of non-archimedean Lyapunov exponents and applications over global fields, Trans. Am. Math. Soc., Volume 373 (2020) no. 12, pp. 8963-9011 | Zbl | DOI | MR

[54] Thomas Gauthier; Gabriel Vigny The geometric dynamical Northcott and Bogomolov properties, Ann. Sci. Éc. Norm. Supér. (4), Volume 58 (2025) no. 1, pp. 231-273 | Zbl | MR

[55] Igors Gorbovickis Algebraic independence of multipliers of periodic orbits in the space of rational maps of the Riemann sphere, Mosc. Math. J., Volume 15 (2015) no. 1, pp. 73-87 | Zbl | DOI | MR

[56] Marc Hindry; Joseph H. Silverman Diophantine geometry. An introduction, Graduate Texts in Mathematics, 201, Springer, 2000 | Zbl | DOI | MR

[57] Benjamin Hutz Good reduction and canonical heights of subvarieties, Math. Res. Lett., Volume 25 (2018) no. 6, pp. 1837-1863 | Zbl | DOI | MR

[58] Patrick Ingram The critical height is a moduli height, Duke Math. J., Volume 167 (2018) no. 7, pp. 1311-1346 | Zbl | MR

[59] Patrick Ingram Minimally critical endomorphisms of $ℙ^{n}$ (2020) | arXiv | Zbl

[60] Patrick Ingram Minimally critical regular endomorphisms of $𝔸^{N}$ , Ergodic Theory Dyn. Syst., Volume 43 (2023) no. 2, pp. 585-614 | Zbl | DOI | MR

[61] Patrick Ingram; Rohini Ramadas; Joseph H. Silverman Post-critically finite maps on $ℙ^{n}$ for $n \geq 2$ are sparse, Trans. Am. Math. Soc., Volume 376 (2023) no. 5, pp. 3087-3109 | Zbl | DOI | MR

[62] Zhuchao Ji; Junyi Xie Homoclinic orbits, multiplier spectrum and rigidity theorems in complex dynamics, Forum Math. Pi, Volume 11 (2023), e11, 37 pages | Zbl | MR

[63] Mattias Jonsson Dynamics of polynomial skew products on $ℂ^{2}$ , Math. Ann., Volume 314 (1999) no. 3, pp. 403-447 | Zbl | MR | DOI

[64] Sarah Koch Teichmüller theory and critically finite endomorphisms, Adv. Math., Volume 248 (2013), pp. 573-617 | Zbl | DOI | MR

[65] Lars Kühne Equidistribution in families of abelian varieties and uniformity (2021) | arXiv | Zbl

[66] Samuel Lattès Sur les formes réduites des transformations ponctuelles dans le domaine d’un point double, Bull. Soc. Math. Fr., Volume 39 (1911), pp. 309-345 | Zbl | Numdam | MR | DOI

[67] Robert Lazarsfeld Positivity in algebraic geometry. I. Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 48, Springer, 2004, xviii+387 pages | Zbl | MR

[68] Alon Levy The space of morphisms on projective space, Acta Arith., Volume 146 (2011) no. 1, pp. 13-31 | Zbl | DOI | MR

[69] Mikhail Yu. Lyubich Some typical properties of the dynamics of rational mappings, Usp. Mat. Nauk, Volume 38 (1983) no. 5, pp. 197-198 | Zbl | MR

[70] Ricardo Mane; Paulo Sad; Dennis Sullivan On the dynamics of rational maps, Ann. Sci. Éc. Norm. Supér. (4), Volume 16 (1983) no. 2, pp. 193-217 | Zbl | Numdam | DOI | MR

[71] Niki Myrto Mavraki; Harry Schmidt On the dynamical Bogomolov conjecture for families of split rational maps, Duke Math. J., Volume 174 (2025) no. 5, pp. 803-856 | Zbl | MR

[72] Curt McMullen Families of rational maps and iterative root-finding algorithms, Ann. Math. (2), Volume 125 (1987), pp. 467-493 | Zbl | DOI

[73] John Milnor On Lattès maps, Dynamics on the Riemann sphere, European Mathematical Society, 2006, pp. 9-43 | Zbl | DOI

[74] Clayton Petsche; Lucien Szpiro; Michael Tepper Isotriviality is equivalent to potential good reduction for endomorphisms of $ℙ^{N}$ , J. Algebra, Volume 322 (2009) no. 9, pp. 3345-3365 | Zbl | DOI

[75] Federico Rodriguez Hertz; M. A. Rodriguez Hertz; Ali Tahzibi; Raul Ures New criteria for ergodicity and nonuniform hyperbolicity, Duke Math. J., Volume 160 (2011) no. 3, pp. 599-629 | Zbl | MR

[76] Feng Rong The Fatou set for critically finite maps, Proc. Am. Math. Soc., Volume 136 (2008) no. 10, pp. 3621-3625 | Zbl | DOI | MR

[77] George R. Sell Smooth linearization near a fixed point, Am. J. Math., Volume 107 (1985), pp. 1035-1091 | Zbl | DOI | MR

[78] Joseph H. Silverman The space of rational maps on $ℙ^{1}$ , Duke Math. J., Volume 94 (1998) no. 1, pp. 41-77 | Zbl | MR

[79] Joseph H. Silverman The arithmetic of dynamical systems, Graduate Texts in Mathematics, 241, Springer, 2007, ix+511 pages | DOI | Zbl | MR

[80] Bernd Sturmfels; Andrei Zelevinsky Maximal minors and their leading terms, Adv. Math., Volume 98 (1993) no. 1, pp. 65-112 | Zbl | DOI | MR

[81] Johan Taflin Blenders near polynomial product maps of $ℂ^{2}$ , J. Eur. Math. Soc., Volume 23 (2021) no. 11, pp. 3555-3589 | Zbl | DOI | MR

[82] Tetsuo Ueda Complex dynamics on $P^{n}$ and Kobayashi metric, RIMS Kokyuroku, Volume 988 (1997), pp. 188-191 | Zbl

[83] Tetsuo Ueda Critical orbits of holomorphic maps on projective spaces, J. Geom. Anal., Volume 8 (1998) no. 2, pp. 319-334 | Zbl | MR

[84] Emmanuel Ullmo Positivité et discrétion des points algébriques des courbes, Ann. Math. (2), Volume 147 (1998) no. 1, pp. 167-179 | Zbl | DOI | MR

[85] Van Tu Le Periodic points of post-critically algebraic holomorphic endomorphisms, Ergodic Theory Dyn. Syst., Volume 42 (2022) no. 7, pp. 2382-2414 | Zbl | MR

[86] Junyi Xie; Xinyi Yuan Geometric Bogomolov conjecture in arbitrary characteristics, Invent. Math., Volume 229 (2022) no. 2, pp. 607-637 | Zbl | MR

[87] Xinyi Yuan Arithmetic bigness and a uniform bogomolov-type result, Ann. Math. (2), Volume 203 (2026) no. 1, pp. 15-119 | DOI | Zbl | MR

[88] Xinyi Yuan; Shou-Wu Zhang Adelic line bundles over quasi-projective varieties, Annals of Mathematics Studies, 221, Princeton University Press, 2026, 280 pages | DOI | Zbl

[89] Shou-Wu Zhang Positive line bundles on arithmetic varieties, J. Am. Math. Soc., Volume 8 (1995) no. 1, pp. 187-221 | Zbl | DOI | MR

[90] Shou-Wu Zhang Equidistribution of small points on abelian varieties, Ann. Math., Volume 147 (1998) no. 1, pp. 159-165 | Zbl | DOI

Cited by Sources: