# zbMATH — the first resource for mathematics

Distribution of postcritically finite polynomials. II: Speed of convergence. (English) Zbl 1419.37040
In the moduli space of degree $$d$$ polynomials, the authors prove the equidistribution of postcritically finite polynomials toward the bifurcation measure. More precisely, using complex analytic arguments and pluripotential theory, they prove the exponential speed of convergence for $$\mathcal{C}^2$$-observables.
This improves results obtained with arithmetic methods in the uncritical family and in the space of degree $$d$$ polynomials, see [C. Favre et al., Math. Ann., 335, 311–361 (2006; Zbl 1175.11029)]; Israel Journal of Mathematics, 209, 235–292 (2015; Zbl 1352.37202)].
The techniques allow the deduction from that the equidistribution of hyperbolic parameters with ($$d-1$$) distinct attracting cycles of given multipliers toward the bifurcation measure with exponential speed for $$\mathcal{C}^1$$-observables. As an application, the equidistribution (up to an explicit extraction) of parameters with ($$d-1$$) distinct cycles with prescribed multiplier toward the bifurcation measure for any ($$d-1$$) multipliers outside a pluripolar set, is proved.

##### MSC:
 37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010) 32U40 Currents 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
Full Text:
##### References:
 [1] G. Bassanelli; F. Berteloot, Bifurcation currents in holomorphic dynamics on \begindocument$${\mathbb{P}^k}$$\enddocument, J. Reine Angew. Math., 608, 201-235, (2007) · Zbl 1136.37025 [2] G. Bassanelli; F. Berteloot, Lyapunov exponents, bifurcation currents and laminations in bifurcation loci, Math. Ann., 345, 1-23, (2009) · Zbl 1179.37067 [3] G. Bassanelli; F. Berteloot, Distribution of polynomials with cycles of a given multiplier, Nagoya Math. J., 201, 23-43, (2011) · Zbl 1267.37049 [4] Y. J.-Briend; J. Duval, Exposants de Liapounoff et distribution des points périodiques d’un endomorphisme de CP\^{k}, Acta Math., 182, 143-157, (1999) · Zbl 1144.37436 [5] F. Berteloot; T. Gauthier, On the geometry of bifurcation currents for quadratic rational maps, Ergodic Theory Dynam. Systems, 35, 1369-1379, (2015) · Zbl 1339.37035 [6] X. Buff; T. Gauthier, Quadratic polynomials. multipliers and equidistribution, Proc. Amer. Math. Soc., 143, 3011-3017, (2015) · Zbl 1334.37038 [7] B. Branner; J. H. Hubbard, The iteration of cubic polynomials. Part Ⅰ. The global topology of parameter space, Acta Math., 160, 143-206, (1988) · Zbl 0668.30008 [8] E. Bedford; B. A. Taylor, The Dirichlet problem for a complex Monge-Ampere equation, Bull. Amer. Math. Soc., 82, 102-104, (1976) · Zbl 0322.31008 [9] E. M. Chirka, Complex Analytic Sets, Translated from the Russian by R. A. M. Hoksbergen, Mathematics and its Applications (Soviet Series), 46, Kluwer Academic Publishers Group, Dordrecht, 1989. · Zbl 0683.32002 [10] J. -P. Demailly, Complex Analytic and Differential Geometry, 2011. Free accessible book: http://www-fourier.ujf-grenoble.fr/ demailly/manuscripts/agbook.pdf. [11] L. DeMarco, Dynamics of rational maps: a current on the bifurcation locus, Math. Res. Lett., 8, 57-66, (2001) · Zbl 0991.37030 [12] L. DeMarco, Dynamics of rational maps: Lyapunov exponents. bifurcations, and capacity, Math. Ann., 326, 43-73, (2003) · Zbl 1032.37029 [13] R. Dujardin, The supports of higher bifurcation currents, Ann. Fac. Sci. Toulouse Math. (6), 22, 445-464, (2013) · Zbl 1314.37032 [14] R. Dujardin; C. Favre, Distribution of rational maps with a preperiodic critical point, Amer. J. Math., 130, 979-1032, (2008) · Zbl 1246.37071 [15] A. Douady and J. H. Hubbard, Étude Dynamique des Polynômes Complexes. Partie I, Publications Mathématiques d’Orsay[Mathematical Publications of Orsay], 84, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984. [16] A. Douady and J. H. Hubbard, Étude Dynamique des Polynômes Complexes. Partie II, with the collaboration of P. Lavaurs, Tan Lei and P. Sentenac, Publications Mathématiques d’Orsay [Mathematical Publications of Orsay], 85, Université de Paris-Sud, Département de Mathé-matiques, Orsay, 1985. [17] T.-C. Dinh; N. Sibony, Dynamics of regular birational maps in \begindocument$$\mathbb{P}^k$$\enddocument, J. Funct. Anal., 222, 202-216, (2005) · Zbl 1067.37055 [18] T.-C. Dinh; N. Sibony, Super-potentials of positive closed currents. intersection theory and dynamics, Acta Math., 203, 1-82, (2009) · Zbl 1227.32024 [19] T.-C. Dinh and N. Sibony, Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mappings, in Holomorphic Dynamical Systems, Lecture Notes in Math., 1998, Springer, Berlin, 2010,165-294. · Zbl 1218.37055 [20] H. De Thélin; G. Vigny, Entropy of meromorphic maps and dynamics of birational maps, Mém. Soc. Math. Fr. (N.S.), 122, ⅵ+98, (2010) · Zbl 1214.37004 [21] A. Epstein, Transversality principles in holomorphic dynamics, preprint, 2009. [22] C. Favre; T. Gauthier, Distribution of postcritically finite polynomials, Israel Journal of Mathematics, 209, 235-292, (2015) · Zbl 1352.37202 [23] C. Favre; J. Rivera-Letelier, Equidistribution quantitative des points de petite hauteur sur la droite projective, Math. Ann., 335, 311-361, (2006) · Zbl 1175.11029 [24] T. Gauthier, Strong bifurcation loci of full Hausdorff dimension, Ann. Sci. Éc. Norm. Supér. (4), 45, 947-984, (2012) · Zbl 1326.37036 [25] T. Gauthier, Equidistribution towards the bifurcation current Ⅰ: multipliers and degree d polynomials, Math. Ann., 366, 1-30, (2016) · Zbl 1380.37098 [26] T. Gauthier and G. Vigny, Distribution of postcritically finite polynomials Ⅲ: combinatorial continuity, preprint, arXiv: 1602.00925, 2016. [27] P. Ingram, A finiteness result for post-critically finite polynomials, Int. Math. Res. Not. IMRN, 3, 524-543, (2012) · Zbl 1333.37029 [28] J. Kiwi, Combinatorial continuity in complex polynomial dynamics, Proc. London Math. Soc. (3), 91, 215-248, (2005) · Zbl 1077.37038 [29] G. Levin, Theory of iterations of polynomial families in the complex plane, J. Soviet Math., 52, 3512-3522, (1990) · Zbl 0716.30017 [30] M. Ju. Ljubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems, 3, 351-385, (1983) · Zbl 0537.58035 [31] M. Yu. Lyubich, Investigation of the stability of the dynamics of rational functions, (Russian) Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 42 (1984), 72-91. Translated in Selecta Math. Soviet. 9 (1990), no. 1, 69-90. [32] J. Milnor, Geometry and dynamics of quadratic rational maps. with an appendix by the author and Lei Tan, Experiment. Math., 2, 37-83, (1993) · Zbl 0922.58062 [33] J. Milnor, Cubic polynomial maps with periodic critical orbit. Ⅰ, in Complex Dynamics, A K Peters, Wellesley, MA, 333-411, (2009) · Zbl 1180.37073 [34] R. Mañé; P. Sad; D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4), 16, 193-217, (1983) · Zbl 0524.58025 [35] Y. Okuyama, Equidistribution of rational functions having a superattracting periodic point towards the activity current and the bifurcation current, Conform. Geom. Dyn., 18, 217-228, (2014) · Zbl 1370.37095 [36] Y. Okuyama, Quantitative approximations of the Lyapunov exponent of a rational function over valued fields, Mathematische Zeitschrift, 280, 691-706, (2015) · Zbl 1332.37068 [37] F. Przytycki, Lyapunov characteristic exponents are nonnegative, Proc. Amer. Math. Soc., 119, 309-317, (1993) · Zbl 0787.58037 [38] J. H. Silverman, The Arithmetic of Dynamical Systems, Graduate Texts in Mathematics, 241, Springer, New York, 2007. · Zbl 1130.37001 [39] N. Steinmetz, Rational Iteration, Complex Analytic Dynamical Systems, De Gruyter Studies in Mathematics, 16, Walter de Gruyter & Co., Berlin, 1993. · Zbl 0773.58010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.