-
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
Distribution of postcritically finite polynomials Ⅱ: Speed of convergence
LAMFA, Universitè de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens Cedex 1, France |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
doi: 10.4310/MRL.2001.v8.n1.a7. |
[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. |
[13] |
R. Dujardin,
The supports of higher bifurcation currents, Ann. Fac. Sci. Toulouse Math. (6), 22 (2013), 445-464.
doi: 10.5802/afst.1378. |
[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. |
[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 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. |
[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. |
[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. |
[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.
|
[21] |
A. Epstein, Transversality principles in holomorphic dynamics, preprint, 2009. |
[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. |
[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. |
[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.
doi: 10.1007/s00208-015-1297-6. |
[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.
doi: 10.1112/S0024611505015248. |
[29] |
G. Levin,
Theory of iterations of polynomial families in the complex plane, J. Soviet Math., 52 (1990), 3512-3522.
doi: 10.1007/BF01095412. |
[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. |
[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.
doi: 10.1080/10586458.1993.10504267. |
[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. |
[34] |
R. Mañé, P. Sad and 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.
doi: 10.1090/S1088-4173-2014-00271-9. |
[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. |
[37] |
F. Przytycki,
Lyapunov characteristic exponents are nonnegative, Proc. Amer. Math. Soc., 119 (1993), 309-317.
doi: 10.1090/S0002-9939-1993-1186141-9. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
doi: 10.4310/MRL.2001.v8.n1.a7. |
[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. |
[13] |
R. Dujardin,
The supports of higher bifurcation currents, Ann. Fac. Sci. Toulouse Math. (6), 22 (2013), 445-464.
doi: 10.5802/afst.1378. |
[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. |
[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 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. |
[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. |
[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. |
[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.
|
[21] |
A. Epstein, Transversality principles in holomorphic dynamics, preprint, 2009. |
[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. |
[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. |
[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.
doi: 10.1007/s00208-015-1297-6. |
[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.
doi: 10.1112/S0024611505015248. |
[29] |
G. Levin,
Theory of iterations of polynomial families in the complex plane, J. Soviet Math., 52 (1990), 3512-3522.
doi: 10.1007/BF01095412. |
[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. |
[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.
doi: 10.1080/10586458.1993.10504267. |
[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. |
[34] |
R. Mañé, P. Sad and 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.
doi: 10.1090/S1088-4173-2014-00271-9. |
[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. |
[37] |
F. Przytycki,
Lyapunov characteristic exponents are nonnegative, Proc. Amer. Math. Soc., 119 (1993), 309-317.
doi: 10.1090/S0002-9939-1993-1186141-9. |
[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. |
[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. |
[1] |
Janos Kollar. Polynomials with integral coefficients, equivalent to a given polynomial. Electronic Research Announcements, 1997, 3: 17-27. |
[2] |
Nathaniel D. Emerson. Dynamics of polynomials with disconnected Julia sets. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 801-834. doi: 10.3934/dcds.2003.9.801 |
[3] |
Hiroki Sumi. Dynamics of postcritically bounded polynomial semigroups I: Connected components of the Julia sets. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1205-1244. doi: 10.3934/dcds.2011.29.1205 |
[4] |
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 |
[5] |
Alexandre Alves, Mostafa Salarinoghabi. On the family of cubic parabolic polynomials. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2051-2064. 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 and 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 and 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] |
Andreas Koutsogiannis. Multiple ergodic averages for variable polynomials. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022067 |
[16] |
Runlin Zhang. Equidistribution of translates of a homogeneous measure on the Borel–Serre compactification. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 2053-2071. doi: 10.3934/dcds.2021183 |
[17] |
Matthieu Arfeux, Jan Kiwi. Topological cubic polynomials with one periodic ramification point. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1799-1811. doi: 10.3934/dcds.2020094 |
[18] |
Nur Fadhilah Ibrahim. An algorithm for the largest eigenvalue of nonhomogeneous nonnegative polynomials. Numerical Algebra, Control and Optimization, 2014, 4 (1) : 75-91. doi: 10.3934/naco.2014.4.75 |
[19] |
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 |
[20] |
Anca Radulescu. The connected Isentropes conjecture in a space of quartic polynomials. Discrete and Continuous Dynamical Systems, 2007, 19 (1) : 139-175. doi: 10.3934/dcds.2007.19.139 |
2020 Impact Factor: 0.848
Tools
Metrics
Other articles
by authors
[Back to Top]