American Institute of Mathematical Sciences

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

Distribution of postcritically finite polynomials Ⅱ: Speed of convergence

Thomas Gauthier and  Gabriel Vigny

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.

Keywords: Bifurcation measure, polynomial dynamics, equidistribution, hyperbolic polynomials, speed estimate, postcritically finite polynomials.
Mathematics Subject Classification: Primary: 37F45; Secondary: 32U40, 37F10.
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 and F. Berteloot, Bifurcation currents in holomorphic dynamics on ${\mathbb{P}^k} $, J. Reine Angew. Math., 608 (2007), 201-235.  doi: 10.1515/CRELLE.2007.058.  Google Scholar

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G. Bassanelli and F. Berteloot, Lyapunov exponents, bifurcation currents and laminations in bifurcation loci, Math. Ann., 345 (2009), 1-23.  doi: 10.1007/s00208-008-0325-1.  Google Scholar

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G. Bassanelli and F. Berteloot, Distribution of polynomials with cycles of a given multiplier, Nagoya Math. J., 201 (2011), 23-43.  doi: 10.1017/S0027763000026106.  Google Scholar

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Y. J.-Briend and J. Duval, Exposants de Liapounoff et distribution des points périodiques d'un endomorphisme de CPk, Acta Math., 182 (1999), 143-157.  doi: 10.1007/BF02392572.  Google Scholar

[5]

F. Berteloot and T. Gauthier, On the geometry of bifurcation currents for quadratic rational maps, Ergodic Theory Dynam. Systems, 35 (2015), 1369-1379.  doi: 10.1017/etds.2013.110.  Google Scholar

[6]

X. Buff and T. Gauthier, Quadratic polynomials. multipliers and equidistribution, Proc. Amer. Math. Soc., 143 (2015), 3011-3017.  doi: 10.1090/S0002-9939-2015-12506-3.  Google Scholar

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B. Branner and J. H. Hubbard, The iteration of cubic polynomials. Part Ⅰ. The global topology of parameter space, Acta Math., 160 (1988), 143-206.  doi: 10.1007/BF02392275.  Google Scholar

[8]

E. Bedford and B. A. Taylor, The Dirichlet problem for a complex Monge-Ampere equation, Bull. Amer. Math. Soc., 82 (1976), 102-104.  doi: 10.1090/S0002-9904-1976-13977-8.  Google Scholar

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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. doi: 10.1007/978-94-009-2366-9.  Google Scholar

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J. -P. Demailly, Complex Analytic and Differential Geometry, 2011. Free accessible book: http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf. Google Scholar

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L. DeMarco, Dynamics of rational maps: a current on the bifurcation locus, Math. Res. Lett., 8 (2001), 57-66.  doi: 10.4310/MRL.2001.v8.n1.a7.  Google Scholar

[12]

L. DeMarco, Dynamics of rational maps: Lyapunov exponents. bifurcations, and capacity, Math. Ann., 326 (2003), 43-73.  doi: 10.1007/s00208-002-0404-7.  Google Scholar

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R. Dujardin, The supports of higher bifurcation currents, Ann. Fac. Sci. Toulouse Math. (6), 22 (2013), 445-464.  doi: 10.5802/afst.1378.  Google Scholar

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R. Dujardin and C. Favre, Distribution of rational maps with a preperiodic critical point, Amer. J. Math., 130 (2008), 979-1032.  doi: 10.1353/ajm.0.0009.  Google Scholar

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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.  Google Scholar

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T.-C. Dinh and N. Sibony, Dynamics of regular birational maps in $\mathbb{P}^k $, J. Funct. Anal., 222 (2005), 202-216.  doi: 10.1016/j.jfa.2004.07.018.  Google Scholar

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T.-C. Dinh and N. Sibony, Super-potentials of positive closed currents. intersection theory and dynamics, Acta Math., 203 (2009), 1-82.  doi: 10.1007/s11511-009-0038-7.  Google Scholar

[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. doi: 10.1007/978-3-642-13171-4_4.  Google Scholar

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H. De Thélin and G. Vigny, Entropy of meromorphic maps and dynamics of birational maps, Mém. Soc. Math. Fr. (N.S.), 122 (2010), ⅵ+98.   Google Scholar

[21]

A. Epstein, Transversality principles in holomorphic dynamics, preprint, 2009. Google Scholar

[22]

C. Favre and T. Gauthier, Distribution of postcritically finite polynomials, Israel Journal of Mathematics, 209 (2015), 235-292.  doi: 10.1007/s11856-015-1218-0.  Google Scholar

[23]

C. Favre and J. Rivera-Letelier, Equidistribution quantitative des points de petite hauteur sur la droite projective, Math. Ann., 335 (2006), 311-361.  doi: 10.1007/s00208-006-0751-x.  Google Scholar

[24]

T. Gauthier, Strong bifurcation loci of full Hausdorff dimension, Ann. Sci. Éc. Norm. Supér. (4), 45 (2012), 947-984.   Google Scholar

[25]

T. Gauthier, Equidistribution towards the bifurcation current Ⅰ: multipliers and degree d polynomials, Math. Ann., 366 (2016), 1-30.  doi: 10.1007/s00208-015-1297-6.  Google Scholar

[26]

T. Gauthier and G. Vigny, Distribution of postcritically finite polynomials Ⅲ: combinatorial continuity, preprint, arXiv: 1602.00925, 2016. Google Scholar

[27]

P. Ingram, A finiteness result for post-critically finite polynomials, Int. Math. Res. Not. IMRN, 3 (2012), 524-543.   Google Scholar

[28]

J. Kiwi, Combinatorial continuity in complex polynomial dynamics, Proc. London Math. Soc. (3), 91 (2005), 215-248.  doi: 10.1112/S0024611505015248.  Google Scholar

[29]

G. Levin, Theory of iterations of polynomial families in the complex plane, J. Soviet Math., 52 (1990), 3512-3522.  doi: 10.1007/BF01095412.  Google Scholar

[30]

M. Ju. Ljubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems, 3 (1983), 351-385.  doi: 10.1017/S0143385700002030.  Google Scholar

[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.  Google Scholar

[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.  doi: 10.1080/10586458.1993.10504267.  Google Scholar

[33]

J. Milnor, Cubic polynomial maps with periodic critical orbit. Ⅰ, in Complex Dynamics, A K Peters, Wellesley, MA, (2009), 333-411.  doi: 10.1201/b10617-13.  Google Scholar

[34]

R. Mañé, P. Sad and D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4), 16 (1983), 193-217.   Google Scholar

[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.  doi: 10.1090/S1088-4173-2014-00271-9.  Google Scholar

[36]

Y. Okuyama, Quantitative approximations of the Lyapunov exponent of a rational function over valued fields, Mathematische Zeitschrift, 280 (2015), 691-706.  doi: 10.1007/s00209-015-1443-6.  Google Scholar

[37]

F. Przytycki, Lyapunov characteristic exponents are nonnegative, Proc. Amer. Math. Soc., 119 (1993), 309-317.  doi: 10.1090/S0002-9939-1993-1186141-9.  Google Scholar

[38]

J. H. Silverman, The Arithmetic of Dynamical Systems, Graduate Texts in Mathematics, 241, Springer, New York, 2007. doi: 10.1007/978-0-387-69904-2.  Google Scholar

[39]

N. Steinmetz, Rational Iteration, Complex Analytic Dynamical Systems, De Gruyter Studies in Mathematics, 16, Walter de Gruyter & Co., Berlin, 1993. doi: 10.1515/9783110889314.  Google Scholar

show all references

References:
[1]

G. Bassanelli and F. Berteloot, Bifurcation currents in holomorphic dynamics on ${\mathbb{P}^k} $, J. Reine Angew. Math., 608 (2007), 201-235.  doi: 10.1515/CRELLE.2007.058.  Google Scholar

[2]

G. Bassanelli and F. Berteloot, Lyapunov exponents, bifurcation currents and laminations in bifurcation loci, Math. Ann., 345 (2009), 1-23.  doi: 10.1007/s00208-008-0325-1.  Google Scholar

[3]

G. Bassanelli and F. Berteloot, Distribution of polynomials with cycles of a given multiplier, Nagoya Math. J., 201 (2011), 23-43.  doi: 10.1017/S0027763000026106.  Google Scholar

[4]

Y. J.-Briend and J. Duval, Exposants de Liapounoff et distribution des points périodiques d'un endomorphisme de CPk, Acta Math., 182 (1999), 143-157.  doi: 10.1007/BF02392572.  Google Scholar

[5]

F. Berteloot and T. Gauthier, On the geometry of bifurcation currents for quadratic rational maps, Ergodic Theory Dynam. Systems, 35 (2015), 1369-1379.  doi: 10.1017/etds.2013.110.  Google Scholar

[6]

X. Buff and T. Gauthier, Quadratic polynomials. multipliers and equidistribution, Proc. Amer. Math. Soc., 143 (2015), 3011-3017.  doi: 10.1090/S0002-9939-2015-12506-3.  Google Scholar

[7]

B. Branner and J. H. Hubbard, The iteration of cubic polynomials. Part Ⅰ. The global topology of parameter space, Acta Math., 160 (1988), 143-206.  doi: 10.1007/BF02392275.  Google Scholar

[8]

E. Bedford and B. A. Taylor, The Dirichlet problem for a complex Monge-Ampere equation, Bull. Amer. Math. Soc., 82 (1976), 102-104.  doi: 10.1090/S0002-9904-1976-13977-8.  Google Scholar

[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. doi: 10.1007/978-94-009-2366-9.  Google Scholar

[10]

J. -P. Demailly, Complex Analytic and Differential Geometry, 2011. Free accessible book: http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf. Google Scholar

[11]

L. DeMarco, Dynamics of rational maps: a current on the bifurcation locus, Math. Res. Lett., 8 (2001), 57-66.  doi: 10.4310/MRL.2001.v8.n1.a7.  Google Scholar

[12]

L. DeMarco, Dynamics of rational maps: Lyapunov exponents. bifurcations, and capacity, Math. Ann., 326 (2003), 43-73.  doi: 10.1007/s00208-002-0404-7.  Google Scholar

[13]

R. Dujardin, The supports of higher bifurcation currents, Ann. Fac. Sci. Toulouse Math. (6), 22 (2013), 445-464.  doi: 10.5802/afst.1378.  Google Scholar

[14]

R. Dujardin and C. Favre, Distribution of rational maps with a preperiodic critical point, Amer. J. Math., 130 (2008), 979-1032.  doi: 10.1353/ajm.0.0009.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

T.-C. Dinh and N. Sibony, Dynamics of regular birational maps in $\mathbb{P}^k $, J. Funct. Anal., 222 (2005), 202-216.  doi: 10.1016/j.jfa.2004.07.018.  Google Scholar

[18]

T.-C. Dinh and N. Sibony, Super-potentials of positive closed currents. intersection theory and dynamics, Acta Math., 203 (2009), 1-82.  doi: 10.1007/s11511-009-0038-7.  Google Scholar

[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. doi: 10.1007/978-3-642-13171-4_4.  Google Scholar

[20]

H. De Thélin and G. Vigny, Entropy of meromorphic maps and dynamics of birational maps, Mém. Soc. Math. Fr. (N.S.), 122 (2010), ⅵ+98.   Google Scholar

[21]

A. Epstein, Transversality principles in holomorphic dynamics, preprint, 2009. Google Scholar

[22]

C. Favre and T. Gauthier, Distribution of postcritically finite polynomials, Israel Journal of Mathematics, 209 (2015), 235-292.  doi: 10.1007/s11856-015-1218-0.  Google Scholar

[23]

C. Favre and J. Rivera-Letelier, Equidistribution quantitative des points de petite hauteur sur la droite projective, Math. Ann., 335 (2006), 311-361.  doi: 10.1007/s00208-006-0751-x.  Google Scholar

[24]

T. Gauthier, Strong bifurcation loci of full Hausdorff dimension, Ann. Sci. Éc. Norm. Supér. (4), 45 (2012), 947-984.   Google Scholar

[25]

T. Gauthier, Equidistribution towards the bifurcation current Ⅰ: multipliers and degree d polynomials, Math. Ann., 366 (2016), 1-30.  doi: 10.1007/s00208-015-1297-6.  Google Scholar

[26]

T. Gauthier and G. Vigny, Distribution of postcritically finite polynomials Ⅲ: combinatorial continuity, preprint, arXiv: 1602.00925, 2016. Google Scholar

[27]

P. Ingram, A finiteness result for post-critically finite polynomials, Int. Math. Res. Not. IMRN, 3 (2012), 524-543.   Google Scholar

[28]

J. Kiwi, Combinatorial continuity in complex polynomial dynamics, Proc. London Math. Soc. (3), 91 (2005), 215-248.  doi: 10.1112/S0024611505015248.  Google Scholar

[29]

G. Levin, Theory of iterations of polynomial families in the complex plane, J. Soviet Math., 52 (1990), 3512-3522.  doi: 10.1007/BF01095412.  Google Scholar

[30]

M. Ju. Ljubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems, 3 (1983), 351-385.  doi: 10.1017/S0143385700002030.  Google Scholar

[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.  Google Scholar

[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.  doi: 10.1080/10586458.1993.10504267.  Google Scholar

[33]

J. Milnor, Cubic polynomial maps with periodic critical orbit. Ⅰ, in Complex Dynamics, A K Peters, Wellesley, MA, (2009), 333-411.  doi: 10.1201/b10617-13.  Google Scholar

[34]

R. Mañé, P. Sad and D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4), 16 (1983), 193-217.   Google Scholar

[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.  doi: 10.1090/S1088-4173-2014-00271-9.  Google Scholar

[36]

Y. Okuyama, Quantitative approximations of the Lyapunov exponent of a rational function over valued fields, Mathematische Zeitschrift, 280 (2015), 691-706.  doi: 10.1007/s00209-015-1443-6.  Google Scholar

[37]

F. Przytycki, Lyapunov characteristic exponents are nonnegative, Proc. Amer. Math. Soc., 119 (1993), 309-317.  doi: 10.1090/S0002-9939-1993-1186141-9.  Google Scholar

[38]

J. H. Silverman, The Arithmetic of Dynamical Systems, Graduate Texts in Mathematics, 241, Springer, New York, 2007. doi: 10.1007/978-0-387-69904-2.  Google Scholar

[39]

N. Steinmetz, Rational Iteration, Complex Analytic Dynamical Systems, De Gruyter Studies in Mathematics, 16, Walter de Gruyter & Co., Berlin, 1993. doi: 10.1515/9783110889314.  Google Scholar

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