Algebra & Number Theory

The algebraic dynamics of generic endomorphisms of $\mathbb{P}^n$

Najmuddin Fakhruddin

Full-text: Open access

Abstract

We investigate some general questions in algebraic dynamics in the case of generic endomorphisms of projective spaces over a field of characteristic zero. The main results that we prove are that a generic endomorphism has no nontrivial preperiodic subvarieties, any infinite set of preperiodic points is Zariski-dense and any infinite subset of a single orbit is also Zariski-dense, thereby verifying the dynamical “Manin–Mumford” conjecture of Zhang and the dynamical “Mordell–Lang” conjecture of Denis and Ghioca and Tucker in this case.

Article information

Source
Algebra Number Theory, Volume 8, Number 3 (2014), 587-608.

Dates
Received: 30 November 2012
Revised: 25 May 2013
Accepted: 4 July 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513730168

Digital Object Identifier
doi:10.2140/ant.2014.8.587

Mathematical Reviews number (MathSciNet)
MR3218803

Zentralblatt MATH identifier
1317.37116

Subjects
Primary: 37P55: Arithmetic dynamics on general algebraic varieties
Secondary: 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04]

Keywords
generic endomorphisms projective space

Citation

Fakhruddin, Najmuddin. The algebraic dynamics of generic endomorphisms of $\mathbb{P}^n$. Algebra Number Theory 8 (2014), no. 3, 587--608. doi:10.2140/ant.2014.8.587. https://projecteuclid.org/euclid.ant/1513730168


Export citation

References

  • E. Amerik, “Existence of non-preperiodic algebraic points for a rational self-map of infinite order”, Math. Res. Lett. 18:2 (2011), 251–256.
  • J. P. Bell, D. Ghioca, and T. J. Tucker, “The dynamical Mordell–Lang problem for étale maps”, Amer. J. Math. 132:6 (2010), 1655–1675.
  • T. Bousch, Sur quelques problemes de dynamique holomorphe, Ph.D. thesis, Université de Paris-Sud, 1992.
  • L. Denis, “Géométrie et suites récurrentes”, Bull. Soc. Math. France 122:1 (1994), 13–27.
  • A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes, I, Publications Mathématiques d'Orsay 84, Université de Paris-Sud, 1984.
  • A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes, II, Publications Mathématiques d'Orsay 85, Université de Paris-Sud, 1985.
  • D. Eberlein, Rational parameter rays of multibrot sets, Master's thesis, Technische Universität München, 1999.
  • N. Fakhruddin, “Questions on self maps of algebraic varieties”, J. Ramanujan Math. Soc. 18:2 (2003), 109–122.
  • D. Ghioca and T. J. Tucker, “Periodic points, linearizing maps, and the dynamical Mordell–Lang problem”, J. Number Theory 129:6 (2009), 1392–1403.
  • D. Ghioca, T. J. Tucker, and S. Zhang, “Towards a dynamical Manin–Mumford conjecture”, Int. Math. Res. Not. 2011:22 (2011), 5109–5122.
  • E. Lau and D. Schleicher, “Internal addresses in the Mandelbrot set and irreducibility of polynomials”, preprint, Stony Brook University, 1994.
  • P. Morton, “Galois groups of periodic points”, J. Algebra 201:2 (1998), 401–428.
  • T. Tao and V. Vu, Additive combinatorics, Cambridge Studies in Advanced Mathematics 105, Cambridge University Press, 2006.