Distribution of postcritically finite polynomials Ⅱ: Speed of convergence

Thomas Gauthier; Gabriel Vigny

doi:10.3934/jmd.2017004

Previous Article
On spectra of Koopman, groupoid and quasi-regular representations
JMD Home
This Volume
Next Article
Positive metric entropy in nondegenerate nearly integrable systems

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
[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 CP^k, 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

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 CP^k, 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

[1]	Nathaniel D. Emerson. Dynamics of polynomials with disconnected Julia sets. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 801-834. doi: 10.3934/dcds.2003.9.801
[2]	Janos Kollar. Polynomials with integral coefficients, equivalent to a given polynomial. Electronic Research Announcements, 1997, 3: 17-27.
[3]	Jean-François Biasse, Michael J. Jacobson, Jr.. Smoothness testing of polynomials over finite fields. Advances in Mathematics of Communications, 2014, 8 (4) : 459-477. doi: 10.3934/amc.2014.8.459
[4]	Hiroki Sumi. Dynamics of postcritically bounded polynomial semigroups I: Connected components of the Julia sets. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 1205-1244. doi: 10.3934/dcds.2011.29.1205
[5]	Alexandre Alves, Mostafa Salarinoghabi. On the family of cubic parabolic polynomials. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021121
[6]	Yuzhou Tian, Yulin Zhao. Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2941-2956. doi: 10.3934/dcdsb.2020214
[7]	Susanne Pumplün. Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes. Advances in Mathematics of Communications, 2017, 11 (3) : 615-634. doi: 10.3934/amc.2017046
[8]	Takao Komatsu, Bijan Kumar Patel, Claudio Pita-Ruiz. Several formulas for Bernoulli numbers and polynomials. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021006
[9]	Michael Boshernitzan, Máté Wierdl. Almost-everywhere convergence and polynomials. Journal of Modern Dynamics, 2008, 2 (3) : 465-470. doi: 10.3934/jmd.2008.2.465
[10]	Elisavet Konstantinou, Aristides Kontogeorgis. Some remarks on the construction of class polynomials. Advances in Mathematics of Communications, 2011, 5 (1) : 109-118. doi: 10.3934/amc.2011.5.109
[11]	Shrihari Sridharan, Atma Ram Tiwari. The dependence of Lyapunov exponents of polynomials on their coefficients. Journal of Computational Dynamics, 2019, 6 (1) : 95-109. doi: 10.3934/jcd.2019004
[12]	Bin Han. Some multivariate polynomials for doubled permutations. Electronic Research Archive, 2021, 29 (2) : 1925-1944. doi: 10.3934/era.2020098
[13]	Abdon E. Choque-Rivero, Iván Area. A Favard type theorem for Hurwitz polynomials. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 529-544. doi: 10.3934/dcdsb.2019252
[14]	Jayadev S. Athreya, Gregory A. Margulis. Values of random polynomials at integer points. Journal of Modern Dynamics, 2018, 12: 9-16. doi: 10.3934/jmd.2018002
[15]	Runlin Zhang. Equidistribution of translates of a homogeneous measure on the Borel–Serre compactification. Discrete & Continuous Dynamical Systems, 2021 doi: 10.3934/dcds.2021183
[16]	Matthieu Arfeux, Jan Kiwi. Topological cubic polynomials with one periodic ramification point. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1799-1811. doi: 10.3934/dcds.2020094
[17]	Nur Fadhilah Ibrahim. An algorithm for the largest eigenvalue of nonhomogeneous nonnegative polynomials. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 75-91. doi: 10.3934/naco.2014.4.75
[18]	Peter H. van der Kamp, D. I. McLaren, G. R. W. Quispel. Homogeneous darboux polynomials and generalising integrable ODE systems. Journal of Computational Dynamics, 2021, 8 (1) : 1-8. doi: 10.3934/jcd.2021001
[19]	Anca Radulescu. The connected Isentropes conjecture in a space of quartic polynomials. Discrete & Continuous Dynamical Systems, 2007, 19 (1) : 139-175. doi: 10.3934/dcds.2007.19.139
[20]	Ricardo García López. A note on L-series and Hodge spectrum of polynomials. Electronic Research Announcements, 2009, 16: 56-62. doi: 10.3934/era.2009.16.56

American Institute of Mathematical Sciences

Distribution of postcritically finite polynomials Ⅱ: Speed of convergence

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Distribution of postcritically finite polynomials Ⅱ: Speed of convergence

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors