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Distribution of postcritically finite polynomials. (English) Zbl 1352.37202
Summary: We prove that Misiurewicz parameters with prescribed combinatorics and hyperbolic parameters with \((d - 1)\) distinct attracting cycles with given multipliers are equidistributed with respect to the bifurcation measure in the moduli space of degree \(d\) complex polynomials. Our proof relies on Yuan’s equidistribution results of points of small heights, and uses in a crucial way Epstein’s transversality results.

37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
37P30 Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems
Full Text: DOI
[1] Berteloot, F., Bifurcation currents in holomorphic families of rational maps, No. 2075, 1-93, (2013), Berlin · Zbl 1280.37039
[2] Bassanelli, G.; Berteloot, F., bifurcation currents in holomorphic dynamics on ℙ\^{k}, Journal für die Reine und Angewante Mathematik, 608, 201-235, (2007) · Zbl 1136.37025
[3] Bassanelli, G.; Berteloot, F., Lyapunov exponents, bifurcation currents and laminations in bifurcation loci, Mathematische Annalen, 345, 1-23, (2009) · Zbl 1179.37067
[4] Bassanelli, G.; Berteloot, F., Distribution of polynomials with cycles of a given multiplier, Nagoya Mathematical Journal, 201, 23-43, (2011) · Zbl 1267.37049
[5] Briend, J.-Y.; Duval, J., deux caractérisations de la mesure d’équilibre d’un endomorphisme de P\^{k}(C), Publications Mathématiques. Institut de Hautes Études Scientifiques, 93, 145-159, (2001) · Zbl 1010.37004
[6] Baker, M.; Marco, L., Special curves and postcritically finite polynomials, Forum of Mathematics. Pi, 1, e3, (2013) · Zbl 1320.37022
[7] Buff, X.; Epstein, A. L., Bifurcation measure and postcritically finite rational maps, 491-512, (2009), Wellesley, Massachussets · Zbl 1180.37056
[8] Bielefeld, B.; Fisher, Y.; Hubbard, J., The classification of critically preperiodic polynomials as dynamical systems, Journal of the American Mathematical Society, 5, 721-762, (1992) · Zbl 0784.58031
[9] F. Berteloot and T. Gauthier, On the geometry of bifurcation currents for quadratic rational maps, Egodic Theory and Dynamical Systems, appeared online, March 2014, 11 pages. · Zbl 1333.37029
[10] Buff, X.; Gauthier, T., Pertubations of flexible Lattès maps, Bulletin de la Société Mathématique de France, 141, 603-614, (2013) · Zbl 1326.37035
[11] Baker, M.; H’sia, L. H., Canonical heights, transfinite diameters, and polynomial dynamics, Journal für die Reine und Angewandte Mathematik, 585, 61-92, (2005) · Zbl 1071.11040
[12] Branner, B.; Hubbard, J. H., The iteration of cubic polynomials. I. the global topology of parameter space, Acta Mathematics, 160, 143-206, (1988) · Zbl 0668.30008
[13] Bedford, E.; Taylor, B. A., The Dirichlet problem for a complex Monge-ampere equation, Bulletin of the American Mathematical Society, 82, 102-104, (1976) · Zbl 0322.31008
[14] L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993.
[15] A. Chambert-Loir, Mesures et équidistribution sur les espaces de Berkovich, Journal für die Reine und Angewandte Mathematik 595 (2006), 215-235. · Zbl 1112.14022
[16] Chambert-Loir, A., Heights and measures on analytic spaces. A survey of recent results, and some remarks, No. 384, 1-50, (2011), Cambridge · Zbl 1279.14027
[17] DeMarco, L., Dynamics of rational maps: a current on the bifurcation locus, Mathematical Research Letters, 8, 57-66, (2001) · Zbl 0991.37030
[18] L. DeMarco, X. Wang and H. Ye, Bifurcation measures and quadratic rational maps, 2014, preprint. · Zbl 1112.14022
[19] Douady, A., Systèmes dynamiques holomorphes, No. 105, 39-63, (1983), Paris · Zbl 0532.30019
[20] R. Dujardin, Bifurcation currents and equidistribution on parameter space, in Frontiers in Complex Dynamics: In Celebration of John Milnor’s 80th Birthday, Princeton Mathematical Series, edited by Araceli Bonifant, Misha Lyubich and Scott Sutherland, 2014, pp. 515-566. · Zbl 1405.37003
[21] Dujardin, R.; Favre, C., Distribution of rational maps with a preperiodic critical point, American Journal of Mathematics, 130, 979-1032, (2008) · Zbl 1246.37071
[22] Douady, A.; Hubbard, J. H., A proof of thurston’s topological characterization of rational functions, Acta Mathematica, 171, 263-297, (1993) · Zbl 0806.30027
[23] A. Epstein, Infinitesimal Thurston rigidity and the Fatou-Shishikura inequality, preprint, arXiv DS.9902158, 1999. · Zbl 1180.37056
[24] A. Epstein, Transversality principles in holomorphic dynamics, 2009, preprint. · Zbl 1320.37022
[25] Y. Fisher, The classification of critically preperiodic polynomials, PhD Thesis, Cornell University, 1989. · Zbl 1136.37025
[26] C. Favre and J. Rivera-Letelier, Equidistribution quantitative des points de petite hauteur sur la droite projective, Mathematische Annalen 335 (2006), 311-361. · Zbl 1175.11029
[27] Gauthier, T., Strong bifurcation loci of full Hausdorff dimension, Annales Scientifiques de l’École Normale Supérieure, 45, 947-984, (2012) · Zbl 1326.37036
[28] D. Ghioca, L.-C. H’sia and T. Tucker, Preperiodic points for families of rational maps, Proceedings of the London Mathematical Society, to appear. · Zbl 1317.37112
[29] M. Hindry and J. H. Silverman, Diophantine Geometry. An Introduction, Graduate Texts in Mathematics, Vol. 201, Springer-Verlag, New York, 2000. · Zbl 0948.11023
[30] Ingram, P., A finiteness result for post-critically finite polynomials, International Mathematics Research Notices, 3, 524-543, (2012) · Zbl 1333.37029
[31] Kiwi, J., Combinatorial continuity in complex polynomial dynamics, Proceedings of the London Mathematical Society, 91, 215-248, (2005) · Zbl 1077.37038
[32] G. Levin, Perturbations of weakly expanding critical orbits, in Frontiers in Complex Dynamics: In Celebration of John Milnor’s 80th Birthday Princeton Mathematical Series, edited by Araceli Bonifant, Misha Lyubich and Scott Sutherland, 2014, pp. 163-196.
[33] Levin, G. M., Bifurcation set of parameters of a family of quadratic mappings, 103-109, (1982), Kuybyshev
[34] Levin, G., On the theory of iterations of polynomial families in the complex plane, Journal of Soviet Mathematics, 52, 3512-3522, (1990) · Zbl 0716.30017
[35] Levin, G., Multipliers of periodic orbits in spaces of rational maps, Ergodic Theory and Dynamical Systems, 31, 197-243, (2011) · Zbl 1209.37052
[36] A. Levy, An algebraic proof of thurston’s rigidity for a polynomial, 2012, preprint arXiv:math.AG/1201.1969.
[37] C. T. McMullen, Complex Dynamics and Renormalization, Annals of Mathematics Studies, Vol. 135, Princeton University Press, Princeton, NJ, 1994. · Zbl 0822.30002
[38] A. Poirier, On postcritically finite polynomials, part 2: Hubbard trees, 1993. IMS - ims93-7, preprint. · Zbl 1146.14016
[39] J. H. Silverman, The Arithmetic of Dynamical Systems, Graduate Texts in Mathematics, Vol. 241, Springer, New York, 2007. · Zbl 1130.37001
[40] Yuan, X., Big line bundles over arithmetic varieties, Inventiones Mathematicae, 173, 603-649, (2008) · Zbl 1146.14016
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