Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173

 
 

 

Equidistribution of rational functions having a superattracting periodic point towards the activity current and the bifurcation current


Author: Yûsuke Okuyama
Journal: Conform. Geom. Dyn. 18 (2014), 217-228
MSC (2010): Primary 37F45
DOI: https://doi.org/10.1090/S1088-4173-2014-00271-9
Published electronically: November 12, 2014
MathSciNet review: 3276585
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We establish an approximation of the activity current $ T_c$ in the parameter space of a holomorphic family $ f$ of rational functions having a marked critical point $ c$ by parameters for which $ c$ is periodic under $ f$, i.e., is a superattracting periodic point. This partly generalizes a Dujardin-Favre theorem for rational functions having preperiodic points, and refines a Bassanelli-Berteloot theorem on a similar approximation of the bifurcation current $ T_f$ of the holomorphic family $ f$. The proof is based on a dynamical counterpart of this approximation.


References [Enhancements On Off] (What's this?)

  • [1] Tom M. Apostol, Introduction to analytic number theory, Springer-Verlag, New York-Heidelberg, 1976. Undergraduate Texts in Mathematics. MR 0434929 (55 #7892)
  • [2] Giovanni Bassanelli and François Berteloot, Bifurcation currents in holomorphic dynamics on $ \mathbb{P}^k$, J. Reine Angew. Math. 608 (2007), 201-235. MR 2339474 (2008g:32052), https://doi.org/10.1515/CRELLE.2007.058
  • [3] Giovanni Bassanelli and François Berteloot, Lyapunov exponents, bifurcation currents and laminations in bifurcation loci, Math. Ann. 345 (2009), no. 1, 1-23. MR 2520048 (2011i:37063), https://doi.org/10.1007/s00208-008-0325-1
  • [4] Giovanni Bassanelli and François Berteloot, Distribution of polynomials with cycles of a given multiplier, Nagoya Math. J. 201 (2011), 23-43. MR 2772169 (2012h:37098)
  • [5] François Berteloot, Lyapunov exponent of a rational map and multipliers of repelling cycles, Riv. Math. Univ. Parma (N.S.) 1 (2010), no. 2, 263-269. MR 2789444 (2012c:37087)
  • [6] François Berteloot, Bifurcation currents in holomorphic families of rational maps, Pluripotential theory, Lecture Notes in Math., vol. 2075, Springer, Heidelberg, 2013, pp. 1-93. MR 3089068, https://doi.org/10.1007/978-3-642-36421-1_1
  • [7] François Berteloot, Christophe Dupont, and Laura Molino, Normalization of bundle holomorphic contractions and applications to dynamics, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 6, 2137-2168 (English, with English and French summaries). MR 2473632 (2009m:32028)
  • [8] Bodil Branner and John H. Hubbard, The iteration of cubic polynomials. II. Patterns and parapatterns, Acta Math. 169 (1992), no. 3-4, 229-325. MR 1194004 (94d:30044), https://doi.org/10.1007/BF02392761
  • [9] X. Buff and T. Gauthier, Quadratic polynomials, multipliers and equidistribution, Proc. Amer. Math. Soc. (to appear).
  • [10] Demailly, J.-P. Complex analytic and algebraic geometry, available at http://www-fourier.ujf-grenoble.fr/˜demailly/manuscripts/agbook.pdf (2012).
  • [11] Laura DeMarco, Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity, Math. Ann. 326 (2003), no. 1, 43-73. MR 1981611 (2004f:32044), https://doi.org/10.1007/s00208-002-0404-7
  • [12] Tien-Cuong Dinh and Nessim Sibony, Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mappings, Holomorphic dynamical systems, Lecture Notes in Math., vol. 1998, Springer, Berlin, 2010, pp. 165-294. MR 2648690 (2011h:32019), https://doi.org/10.1007/978-3-642-13171-4_4
  • [13] Romain Dujardin, Bifurcation currents and equidistribution in parameter space, Frontiers in Complex Dynamics: in Celebration of John Milnor's 80th birthday, Princeton University Press (2014), 515-566.
  • [14] Romain Dujardin and Charles Favre, Distribution of rational maps with a preperiodic critical point, Amer. J. Math. 130 (2008), no. 4, 979-1032. MR 2427006 (2009f:32035), https://doi.org/10.1353/ajm.0.0009
  • [15] T. Gauthier, Equidistribution towards the bifurcation current I : Mulitpliers and degree d polynomials, ArXiv e-prints (Dec. 2013).
  • [16] Lars Hörmander, The analysis of linear partial differential operators. I: Distribution theory and Fourier analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. MR 717035 (85g:35002a)
  • [17] Mattias Jonsson, Sums of Lyapunov exponents for some polynomial maps of $ {\bf C}^2$, Ergodic Theory Dynam. Systems 18 (1998), no. 3, 613-630. MR 1631728 (99d:32033), https://doi.org/10.1017/S0143385798108209
  • [18] Maciej Klimek, Pluripotential theory, London Mathematical Society Monographs. New Series, vol. 6, The Clarendon Press, Oxford University Press, New York, 1991. Oxford Science Publications. MR 1150978 (93h:32021)
  • [19] G. M. Levin, On the theory of iterations of polynomial families in the complex plane, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 51 (1989), 94-106 (Russian); English transl., J. Soviet Math. 52 (1990), no. 6, 3512-3522. MR 1009151 (90j:30040), https://doi.org/10.1007/BF01095412
  • [20] M. Yu. Lyubich, Some typical properties of the dynamics of rational mappings, Uspekhi Mat. Nauk 38 (1983), no. 5(233), 197-198 (Russian). MR 718838 (85f:58063)
  • [21] R. Mañé, P. Sad, and D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193-217. MR 732343 (85j:58089)
  • [22] Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365 (96b:58097)
  • [23] John Milnor, Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR 2193309 (2006g:37070)
  • [24] Patrick Morton and Franco Vivaldi, Bifurcations and discriminants for polynomial maps, Nonlinearity 8 (1995), no. 4, 571-584. MR 1342504 (96k:11028)
  • [25] Yûsuke Okuyama, Repelling periodic points and logarithmic equidistribution in non-archimedean dynamics, Acta Arith. 152 (2012), no. 3, 267-277. MR 2885787, https://doi.org/10.4064/aa152-3-3
  • [26] N.-m. Pham, Lyapunov exponents and bifurcation current for polynomial-like maps, arXiv preprint math/0512557 (2005).
  • [27] Feliks Przytycki, Lyapunov characteristic exponents are nonnegative, Proc. Amer. Math. Soc. 119 (1993), no. 1, 309-317. MR 1186141 (93k:58193), https://doi.org/10.2307/2159858
  • [28] Joseph H. Silverman, The arithmetic of dynamical systems, Graduate Texts in Mathematics, vol. 241, Springer, New York, 2007. MR 2316407 (2008c:11002)

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2010): 37F45

Retrieve articles in all journals with MSC (2010): 37F45


Additional Information

Yûsuke Okuyama
Affiliation: Division of Mathematics, Kyoto Institute of Technology, Sakyo-ku, Kyoto 606-8585 Japan
Email: okuyama@kit.ac.jp

DOI: https://doi.org/10.1090/S1088-4173-2014-00271-9
Keywords: Holomorphic family, marked critical point, superattracting periodic point, equidistribution, activity current, bifurcation current
Received by editor(s): February 24, 2014
Received by editor(s) in revised form: July 11, 2014, and August 12, 2014
Published electronically: November 12, 2014
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society