Duke Mathematical Journal

Preperiodic points and unlikely intersections

Matthew Baker and Laura Demarco

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this article, we combine complex-analytic and arithmetic tools to study the preperiodic points of 1-dimensional complex dynamical systems. We show that for any fixed a,bC and any integer d2, the set of cC for which both a and b are preperiodic for zd+c is infinite if and only if ad=bd. This provides an affirmative answer to a question of Zannier, which itself arose from questions of Masser concerning simultaneous torsion sections on families of elliptic curves. Using similar techniques, we prove that if rational functions ϕ,ψC(z) have infinitely many preperiodic points in common, then all of their preperiodic points coincide (and, in particular, the maps must have the same Julia set). This generalizes a theorem of Mimar, who established the same result assuming that ϕ and ψ are defined over Q¯. The main arithmetic ingredient in the proofs is an adelic equidistribution theorem for preperiodic points over number fields and function fields, with nonarchimedean Berkovich spaces playing an essential role.

Article information

Source
Duke Math. J., Volume 159, Number 1 (2011), 1-29.

Dates
First available in Project Euclid: 11 July 2011

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1310416361

Digital Object Identifier
doi:10.1215/00127094-1384773

Mathematical Reviews number (MathSciNet)
MR2817647

Zentralblatt MATH identifier
1242.37062

Subjects
Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04] 11G50: Heights [See also 14G40, 37P30]
Secondary: 31A99: None of the above, but in this section 11S80: Other analytic theory (analogues of beta and gamma functions, $p$-adic integration, etc.)

Citation

Baker, Matthew; Demarco, Laura. Preperiodic points and unlikely intersections. Duke Math. J. 159 (2011), no. 1, 1--29. doi:10.1215/00127094-1384773. https://projecteuclid.org/euclid.dmj/1310416361


Export citation

References

  • E. Artin, Algebraic Numbers and Algebraic Functions, reprint of the 1967 original, AMS Chelsea, Providence, 2006.
  • M. Baker, “An introduction to Berkovich analytic spaces and non-Archimedean potential theory on curves” in $p$-adic Geometry (Tucson, Ariz., 2007), Univ. Lecture Ser. 45, Amer. Math. Soc., Providence, 2008, 123–174.
  • —, A finiteness theorem for canonical heights attached to rational maps over function fields, J. Reine Angew. Math. 626 (2009), 205–233.
  • M. Baker and L.-C. Hsia, Canonical heights, transfinite diameters, and polynomial dynamics, J. Reine Angew. Math. 585 (2005), 61–92.
  • M. Baker and R. Rumely, Equidistribution of small points, rational dynamics, and potential theory, Ann. Inst. Fourier (Grenoble) 56 (2006), 625–688.
  • —, Potential Theory and Dynamics on the Berkovich Projective Line, Math. Surveys Monogr. 159, Amer. Math. Soc., Providence, 2010.
  • A. F. Beardon, Symmetries of Julia sets, Bull. London Math. Soc. 22 (1990), 576–582.
  • —, Iteration of Rational Functions, Grad. Texts in Math. 132, Springer, New York, 1991.
  • R. L. Benedetto, Heights and preperiodic points of polynomials over function fields, Int. Math. Res. Not. 62 (2005), 3855–3866.
  • —, Non-archimedean dynamics in dimension one, lecture notes, preprint, http://math.arizona.edu/$\sim$swc/aws (accessed 18 April 2011).
  • Y. Bilu, Limit distribution of small points on algebraic tori, Duke Math. J. 89 (1997), 465–476.
  • E. Bombieri and W. Gubler, Heights in Diophantine Geometry, New Math. Monogr. 4, Cambridge Univ. Press, Cambridge, 2006.
  • B. Branner and J. H. Hubbard, The iteration of cubic polynomials, I: The global topology of parameter space, Acta Math. 160 (1988), 143–206.
  • S. Cantat and A. Chambert-Loir, Dynamique p-adique (d'après des exposés de Jean-Christophe Yoccoz), preprint.
  • L. Carleson and T. W. Gamelin, Complex Dynamics, Springer, New York, 1993.
  • A. Chambert-Loir, Mesures et équidistribution sur les espaces de Berkovich, J. Reine Angew. Math. 595 (2006), 215–235.
  • Z. Chatzidakis and E. Hrushovski, Difference fields and descent in algebraic dynamics, I, J. Inst. Math. Jussieu 7 (2008), 653–686.
  • —, Difference fields and descent in algebraic dynamics, II, J. Inst. Math. Jussieu 7 (2008), 687–704.
  • A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes, Publ. Math. Orsay 84, Université de Paris-Sud, Orsay, 1984.
  • R. Dujardin and C. Favre, Distribution of rational maps with a preperiodic critical point, Amer. J. Math. 130 (2008), 979–1032.
  • C. Favre and M. Jonsson, The Valuative Tree, Lecture Notes in Math. 1853, Springer, Berlin, 2004.
  • C. Favre and J. Rivera-Letelier, Théorème d'équidistribution de Brolin en dynamique $p$-adique, C. R. Math. Acad. Sci. Paris 339 (2004), 271–276.
  • —, Équidistribution quantitative des points de petite hauteur sur la droite projective, Math. Ann. 335 (2006), 311–361.
  • A. Freire, A. Lopes, and R. Mañé, An invariant measure for rational maps, Bol. Soc. Brasil. Mat. 14 (1983), 45–62
  • S. Kawaguchi and J. H. Silverman, Dynamics of projective morphisms having identical canonical heights, Proc. Lond. Math. Soc. (3) 95 (2007), 519–544.
  • S. Lang, Fundamentals of Diophantine Geometry, Springer, New York, 1983.
  • G. Levin and F. Przytycki, When do two rational functions have the same Julia set? Proc. Amer. Math. Soc. 125 (1997), 2179–2190.
  • M. Ljubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems 3 (1983), 351–385.
  • D. Masser and U. Zannier, Torsion anomalous points and families of elliptic curves, C. R. Math. Acad. Sci. Paris 346 (2008), 491–494.
  • —, Torsion anomalous points and families of elliptic curves, Amer. J. Math. 132 (2010), 1677–1691.
  • J. Milnor, Dynamics in One Complex Variable, Friedr. Vieweg, Braunschweig, 1999.
  • A. Mimar, On the preperiodic points of an endomorphism of $\mathbb{P}^1 \times \mathbb{P}^1$ which lie on a curve, Ph.D. dissertation, Columbia University, New York, N.Y., 1997.
  • C. Petsche, $S$-integral preperiodic points by dynamical systems over number fields, Bull. Lond. Math. Soc. 40 (2008), 749–758.
  • C. Petsche, L. Szpiro, and M. Tepper, Isotriviality is equivalent to potential good reduction for endomorphisms of $\mathbb{P}^{N}$ over function fields, J. Algebra 322 (2009), 3345–3365.
  • T. Ransford, Potential Theory in the Complex Plane, London Math. Soc. Stud. Texts 28, Cambridge Univ. Press, Cambridge, 1995.
  • J. Rivera-Letelier, Dynamique des fonctions rationnelles sur des corps locaux, Astérisque 287 (2003), 147–230.
  • W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987.
  • A. Thuillier, Théorie du potentiel sur les courbes en géométrie analytique non archimédienne: Applications à la théorie d'Arakelov, Ph.D. dissertation, University of Rennes, Rennes, 2005, http://tel.ccsd.cnrs.fr/documents/ archives0/00/01/09/90/index.html (accessed 18 April 2011).
  • X. Yuan and S. Zhang, Calabi theorem and algebraic dynamics, preprint, 2009.