Non-density of stability for holomorphic mappings on  k
[Non-densité de la stabilité pour les applications holomorphes sur k ]
Journal de l’École polytechnique — Mathématiques, Tome 4 (2017) , pp. 813-843.

Un théorème célèbre dû à Mañé-Sad-Sullivan et Lyubich affirme que les paramètres J-stables forment un ouvert dense de toute famille holomorphe de systèmes dynamiques rationnels en dimension 1. Dans cet article nous montrons que ce résultat ne subsiste pas en dimension supérieure. Plus précisément nous construisons des ouverts contenus dans le lieu de bifurcation des applications holomorphes de degré d de k () pour tout d2 et k2.

A well-known theorem due to Mañé-Sad-Sullivan and Lyubich asserts that J-stable maps are dense in any holomorphic family of rational maps in dimension 1. In this paper we show that the corresponding result fails in higher dimension. More precisely, we construct open subsets in the bifurcation locus in the space of holomorphic mappings of degree d of k () for every d2 and k2.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/jep.57
Classification : 37F45,  37F10,  37F15
Mots clés : Dynamique holomorphe en dimension supérieure, J-stabilité, bifurcations, mélangeurs
@article{JEP_2017__4__813_0,
     author = {Romain Dujardin},
     title = {Non-density of stability for holomorphic~mappings on~$\protect {\mathbb{P}^k$}},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {813--843},
     publisher = {\'Ecole polytechnique},
     volume = {4},
     year = {2017},
     doi = {10.5802/jep.57},
     mrnumber = {3694096},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.57/}
}
Romain Dujardin. Non-density of stability for holomorphic mappings on $\protect \mathbb{P}^k$. Journal de l’École polytechnique — Mathématiques, Tome 4 (2017) , pp. 813-843. doi : 10.5802/jep.57. https://jep.centre-mersenne.org/articles/10.5802/jep.57/

[BBD17] F. Berteloot, F. Bianchi & C. Dupont - “Dynamical stability and Lyapunov exponents for holomorphic endomorphisms of k ”, Ann. Sci. École Norm. Sup. (4) (2017), to appear | Zbl 06873715

[BD96] C. Bonatti & L. J. Díaz - “Persistent nonhyperbolic transitive diffeomorphisms”, Ann. of Math. (2) 143 (1996) no. 2, p. 357-396 | Article

[BD01] J.-Y. Briend & J. Duval - “Deux caractérisations de la mesure d’équilibre d’un endomorphisme de P k (C)”, Publ. Math. Inst. Hautes Études Sci. 93 (2001), p. 145-159, Erratum: Ibid, 109 (2009), p. 295–296 | Article | MR 1381990 | Zbl 0852.58066

[BDV05] C. Bonatti, L. J. Díaz & M. Viana - Dynamics beyond uniform hyperbolicity, Encyclopaedia of Mathematical Sciences, vol. 102, Springer-Verlag, Berlin, 2005 | MR 2105774 | Zbl 1060.37020

[Bia16] F. Bianchi - Motions of Julia sets and dynamical stability in several complex variables, Université de Toulouse, Sept. 2016, Ph. D. Thesis

[Bie16] S. Biebler - “Persistent homoclinic tangencies and infinitely many sinks for of automorphisms of low degree in 3 ” (2016), arXiv:1611.02011

[BT17] F. Bianchi & J. Taflin - “Bifurcations in the elementary Desboves family”, Proc. Amer. Math. Soc. (2017), to appear | Article | MR 3690617 | Zbl 1375.32037

[Buz97] G. T. Buzzard - “Infinitely many periodic attractors for holomorphic maps of 2 variables”, Ann. of Math. (2) 145 (1997) no. 2, p. 389-417 | Article | MR 1441881 | Zbl 0878.58044

[Chi89] E. M. Chirka - Complex analytic sets, Mathematics and its Applications (Soviet Series), vol. 46, Kluwer Academic Publishers Group, Dordrecht, 1989 | MR 1111477 | Zbl 0683.32002

[DF08] R. Dujardin & C. Favre - “Distribution of rational maps with a preperiodic critical point”, Amer. J. Math. 130 (2008) no. 4, p. 979-1032 | Article | MR 2427006 | Zbl 1246.37071

[DK88] Z. Daróczy & I. Kátai - “Generalized number systems in the complex plane”, Acta Math. Hungar. 51 (1988) no. 3-4, p. 409-416 | Article | MR 956992 | Zbl 0653.10003

[DL15] R. Dujardin & M. Lyubich - “Stability and bifurcations for dissipative polynomial automorphisms of 2 ”, Invent. Math. 200 (2015) no. 2, p. 439-511 | Article | MR 3338008

[DS85] A. Douady & P. Sentenac - “Un tour de valse”, in Étude dynamique des polynômes complexes. Partie II (A. Douady & J. H. Hubbard, eds.), Publications Mathématiques d’Orsay, vol. 85, Université de Paris-Sud, Département de Mathématiques, Orsay, 1985

[Duj14] R. Dujardin - “Bifurcation currents and equidistribution in parameter space”, in Frontiers in complex dynamics (A. Bonifant, M. Lyubich & S. Sutherland, eds.), Princeton Math. Ser., vol. 51, Princeton Univ. Press, Princeton, NJ, 2014, p. 515-566 | MR 3289920

[Gav98] E. A. Gavosto - “Attracting basins in P 2 ”, J. Geom. Anal. 8 (1998) no. 3, p. 433-440 | Article | MR 1707737

[HP94] J. H. Hubbard & P. Papadopol - “Superattractive fixed points in C n ”, Indian J. Math. 43 (1994) no. 1, p. 321-365 | Article | MR 1275463 | Zbl 0858.32023

[HS16] K. G. Hare & N. Sidorov - “Two-dimensional self-affine sets with interior points, and the set of uniqueness”, Nonlinearity 29 (2016) no. 1, p. 1-26 | Article | MR 3460748 | Zbl 1334.28019

[Jon99] M. Jonsson - “Dynamics of polynomial skew products on C 2 ”, Math. Ann. 314 (1999) no. 3, p. 403-447 | Article | MR 1704543

[Lei98] T. Lei - “Hausdorff dimension of subsets of the parameter space for families of rational maps. (A generalization of Shishikura’s result)”, Nonlinearity 11 (1998) no. 2, p. 233-246 | Article | MR 1610752 | Zbl 1019.37502

[Lyu83] M. Lyubich - “Some typical properties of the dynamics of rational mappings”, Uspehi Mat. Nauk 38 (1983) no. 5(233), p. 197-198 | MR 718838 | Zbl 0546.58033

[Lyu84] M. Lyubich - “Investigation of the stability of the dynamics of rational functions”, Teor. Funkciĭ Funkcional. Anal. i Priložen. (1984) no. 42, p. 72-91, Translated in Selecta Math. Soviet. 9 (1990), no. 1, p. 69–90 | MR 751394 | Zbl 0572.30023

[Mañ93] R. Mañé - “On a theorem of Fatou”, Bol. Soc. Brasil. Mat. (N.S.) 24 (1993) no. 1, p. 1-11 | Article | MR 1224298 | Zbl 0781.30023

[McM00] C. T. McMullen - “The Mandelbrot set is universal”, in The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, Cambridge, 2000, p. 1-17 | MR 1765082 | Zbl 1062.37042

[MS98] C. T. McMullen & D. P. Sullivan - “Quasiconformal homeomorphisms and dynamics. III. The Teichmüller space of a holomorphic dynamical system”, Adv. Math. 135 (1998) no. 2, p. 351-395 | Article | Zbl 0926.30028

[MSS83] R. Mañé, P. Sad & D. Sullivan - “On the dynamics of rational maps”, Ann. Sci. École Norm. Sup. (4) 16 (1983) no. 2, p. 193-217 | Article | Numdam | MR 732343 | Zbl 0524.58025

[New70] S. E. Newhouse - “Nondensity of axiom A(a) on S 2 ”, in Global Analysis (Berkeley, Calif., 1968), Proc. Sympos. Pure Math., vol. XIV, American Mathematical Society, Providence, R.I., 1970, p. 191-202 | MR 277005 | Zbl 0206.25801

[PU10] F. Przytycki & M. Urbański - Conformal fractals: ergodic theory methods, London Math. Society Lect. Note Series, vol. 371, Cambridge University Press, Cambridge, 2010 | MR 2656475 | Zbl 1202.37001

[RY08] P. Roesch & Y. Yin - “The boundary of bounded polynomial Fatou components”, Comptes Rendus Mathématique 346 (2008) no. 15-16, p. 877-880 | Article | MR 2441925 | Zbl 1145.37026

[Shi98] M. Shishikura - “The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets”, Ann. of Math. (2) 147 (1998) no. 2, p. 225-267 | Article | MR 1626737 | Zbl 0922.58047

[Taf17] J. Taflin - “Blenders near polynomial product maps of 2 ” (2017), arXiv:1702.02115