# American Institue of Mathematical Sciences

2017, 11: 57-98. doi: 10.3934/jmd.2017004

## Distribution of postcritically finite polynomials Ⅱ:speed of convergence

 LAMFA, Universitè de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens Cedex 1, France

Received  March 28, 2016 Revised  October 18, 2016 Published  January 2017

In the moduli space of degree $d$ polynomials, we prove the equidistribution of postcritically finite polynomials toward the bifurcation measure. More precisely, using complex analytic arguments and pluripotential theory, we prove the exponential speed of convergence for $\mathscr{C}^2$-observables. This improves results obtained with arithmetic methods by Favre and Rivera-Letellier in the unicritical family and Favre and the first author in the space of degree $d$ polynomials.

We deduce from that the equidistribution of hyperbolic parameters with $(d-1)$ distinct attracting cycles of given multipliers toward the bifurcation measure with exponential speed for $\mathscr{C}^1$-observables. As an application, we prove 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.

Citation: Thomas Gauthier, Gabriel Vigny. Distribution of postcritically finite polynomials Ⅱ:speed of convergence. Journal of Modern Dynamics, 2017, 11: 57-98. doi: 10.3934/jmd.2017004
##### References:
 [1] G. Bassanelli, F. Berteloot, Bifurcation currents in holomorphic dynamics on Pk, J. Reine Angew. Math., 608 (2007), 201-235. [2] G. Bassanelli, F. Berteloot, Lyapunov exponents, bifurcation currents and laminations in bifurcation loci, Math. Ann., 345 (2009), 1-23. [3] G. Bassanelli, F. Berteloot, Distribution of polynomials with cycles of a given multiplier, Nagoya Math. J., 201 (2011), 23-43. [4] Y. J.-Briend, J. Duval, Exposants de Liapounoff et distribution des points périodiques d'un endomorphisme de CPk, Acta Math., 182 (1999), 143-157. [5] F. Berteloot, T. Gauthier, On the geometry of bifurcation currents for quadratic rational maps, Ergodic Theory Dynam. Systems, 35 (2015), 1369-1379. [6] X. Buff, T. Gauthier, Quadratic polynomials. multipliers and equidistribution, Proc. Amer. Math. Soc., 143 (2015), 3011-3017. [7] B. Branner, J. H. Hubbard, The iteration of cubic polynomials. Part Ⅰ. The global topology of parameter space, Acta Math., 160 (1988), 143-206. [8] E. Bedford, B. A. Taylor, The Dirichlet problem for a complex Monge-Ampere equation, Bull. Amer. Math. Soc., 82 (1976), 102-104. [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. [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 (2001), 57-66. [12] L. DeMarco, Dynamics of rational maps: Lyapunov exponents. bifurcations, and capacity, Math. Ann., 326 (2003), 43-73. [13] R. Dujardin, The supports of higher bifurcation currents, Ann. Fac. Sci. Toulouse Math. (6), 22 (2013), 445-464. [14] R. Dujardin, C. Favre, Distribution of rational maps with a preperiodic critical point, Amer. J. Math., 130 (2008), 979-1032. [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] C. T.-Dinh, N. Sibony, Dynamics of regular birational maps in Pk, J. Funct. Anal., 222 (2005), 202-216. [18] C. T.-Dinh, N. Sibony, Super-potentials of positive closed currents. intersection theory and dynamics, Acta Math., 203 (2009), 1-82. [19] C. T.-Dinh, 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. [20] H. De Thélin, G. Vigny, Entropy of meromorphic maps and dynamics of birational maps, Mém. Soc. Math. Fr. (N.S.), 122 (2010), vi+98. [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 (2015), 235-292. [23] C. Favre, J. Rivera-Letelier, Equidistribution quantitative des points de petite hauteur sur la droite projective, Math. Ann., 335 (2006), 311-361. [24] T. Gauthier, Strong bifurcation loci of full Hausdorff dimension, Ann. Sci. Éc. Norm. Supér. (4), 45 (2012), 947-984. [25] T. Gauthier, Equidistribution towards the bifurcation current Ⅰ: multipliers and degree d polynomials, Math. Ann., 366 (2016), 1-30. [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 (2012), 524-543. [28] J. Kiwi, Combinatorial continuity in complex polynomial dynamics, Proc. London Math. Soc. (3), 91 (2005), 215-248. [29] G. Levin, Theory of iterations of polynomial families in the complex plane, J. Soviet Math., 52 (1990), 3512-3522. [30] M. Ju. Ljubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems, 3 (1983), 351-385. [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 (1993), 37-83. [33] J. Milnor, Cubic polynomial maps with periodic critical orbit. Ⅰ, in Complex Dynamics, A K Peters, Wellesley, MA, (2009), 333-411. [34] R. Mañé, P. Sad, D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4), 16 (1983), 193-217. [35] Y. Okuyama, Equidistribution of rational functions having a superattracting periodic point towards the activity current and the bifurcation current, Conform. Geom. Dyn., 18 (2014), 217-228. [36] Y. Okuyama, Quantitative approximations of the Lyapunov exponent of a rational function over valued fields, Mathematische Zeitschrift, 280 (2015), 691-706. [37] F. Przytycki, Lyapunov characteristic exponents are nonnegative, Proc. Amer. Math. Soc., 119 (1993), 309-317. [38] J. H. Silverman, The Arithmetic of Dynamical Systems, Graduate Texts in Mathematics, 241, Springer, New York, 2007. [39] N. Steinmetz, Rational Iteration, Complex Analytic Dynamical Systems, De Gruyter Studies in Mathematics, 16, Walter de Gruyter & Co. , Berlin, 1993.

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##### References:
 [1] G. Bassanelli, F. Berteloot, Bifurcation currents in holomorphic dynamics on Pk, J. Reine Angew. Math., 608 (2007), 201-235. [2] G. Bassanelli, F. Berteloot, Lyapunov exponents, bifurcation currents and laminations in bifurcation loci, Math. Ann., 345 (2009), 1-23. [3] G. Bassanelli, F. Berteloot, Distribution of polynomials with cycles of a given multiplier, Nagoya Math. J., 201 (2011), 23-43. [4] Y. J.-Briend, J. Duval, Exposants de Liapounoff et distribution des points périodiques d'un endomorphisme de CPk, Acta Math., 182 (1999), 143-157. [5] F. Berteloot, T. Gauthier, On the geometry of bifurcation currents for quadratic rational maps, Ergodic Theory Dynam. Systems, 35 (2015), 1369-1379. [6] X. Buff, T. Gauthier, Quadratic polynomials. multipliers and equidistribution, Proc. Amer. Math. Soc., 143 (2015), 3011-3017. [7] B. Branner, J. H. Hubbard, The iteration of cubic polynomials. Part Ⅰ. The global topology of parameter space, Acta Math., 160 (1988), 143-206. [8] E. Bedford, B. A. Taylor, The Dirichlet problem for a complex Monge-Ampere equation, Bull. Amer. Math. Soc., 82 (1976), 102-104. [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. [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 (2001), 57-66. [12] L. DeMarco, Dynamics of rational maps: Lyapunov exponents. bifurcations, and capacity, Math. Ann., 326 (2003), 43-73. [13] R. Dujardin, The supports of higher bifurcation currents, Ann. Fac. Sci. Toulouse Math. (6), 22 (2013), 445-464. [14] R. Dujardin, C. Favre, Distribution of rational maps with a preperiodic critical point, Amer. J. Math., 130 (2008), 979-1032. [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] C. T.-Dinh, N. Sibony, Dynamics of regular birational maps in Pk, J. Funct. Anal., 222 (2005), 202-216. [18] C. T.-Dinh, N. Sibony, Super-potentials of positive closed currents. intersection theory and dynamics, Acta Math., 203 (2009), 1-82. [19] C. T.-Dinh, 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. [20] H. De Thélin, G. Vigny, Entropy of meromorphic maps and dynamics of birational maps, Mém. Soc. Math. Fr. (N.S.), 122 (2010), vi+98. [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 (2015), 235-292. [23] C. Favre, J. Rivera-Letelier, Equidistribution quantitative des points de petite hauteur sur la droite projective, Math. Ann., 335 (2006), 311-361. [24] T. Gauthier, Strong bifurcation loci of full Hausdorff dimension, Ann. Sci. Éc. Norm. Supér. (4), 45 (2012), 947-984. [25] T. Gauthier, Equidistribution towards the bifurcation current Ⅰ: multipliers and degree d polynomials, Math. Ann., 366 (2016), 1-30. [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 (2012), 524-543. [28] J. Kiwi, Combinatorial continuity in complex polynomial dynamics, Proc. London Math. Soc. (3), 91 (2005), 215-248. [29] G. Levin, Theory of iterations of polynomial families in the complex plane, J. Soviet Math., 52 (1990), 3512-3522. [30] M. Ju. Ljubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems, 3 (1983), 351-385. [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 (1993), 37-83. [33] J. Milnor, Cubic polynomial maps with periodic critical orbit. Ⅰ, in Complex Dynamics, A K Peters, Wellesley, MA, (2009), 333-411. [34] R. Mañé, P. Sad, D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4), 16 (1983), 193-217. [35] Y. Okuyama, Equidistribution of rational functions having a superattracting periodic point towards the activity current and the bifurcation current, Conform. Geom. Dyn., 18 (2014), 217-228. [36] Y. Okuyama, Quantitative approximations of the Lyapunov exponent of a rational function over valued fields, Mathematische Zeitschrift, 280 (2015), 691-706. [37] F. Przytycki, Lyapunov characteristic exponents are nonnegative, Proc. Amer. Math. Soc., 119 (1993), 309-317. [38] J. H. Silverman, The Arithmetic of Dynamical Systems, Graduate Texts in Mathematics, 241, Springer, New York, 2007. [39] N. Steinmetz, Rational Iteration, Complex Analytic Dynamical Systems, De Gruyter Studies in Mathematics, 16, Walter de Gruyter & Co. , Berlin, 1993.
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