Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Proof of a dynamical Bogomolov conjecture for lines under polynomial actions

Authors: Dragos Ghioca and Thomas J. Tucker
Journal: Proc. Amer. Math. Soc. 138 (2010), 937-942
MSC (2010): Primary 37P05; Secondary 14G25, 11C08
Published electronically: October 20, 2009
MathSciNet review: 2566560
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a dynamical version of the Bogomolov conjecture in the special case of lines in $ \mathbb{A}^m$ under the action of a map $ (f_1,\dots,f_m)$, where each $ f_i$ is a polynomial in $ \overline{\mathbb{Q}}[X]$ of the same degree.

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

  • [AH96] P. Atela and J. Hu, Commuting polynomials and polynomials with same Julia set, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 6 (1996), no. 12A, 2427-2432. MR 1445904 (98c:58133)
  • [BE87] I. N. Baker and A. Erëmenko, A problem on Julia sets, Ann. Acad. Sci. Fenn. Ser. A I Math. 12 (1987), no. 2, 229-236. MR 951972 (89g:30047)
  • [BH05] M. H. Baker and L.-C. Hsia, Canonical heights, transfinite diameters, and polynomial dynamics, J. Reine Angew. Math. 585 (2005), 61-92. MR 2164622 (2006i:11071)
  • [Bea90] A. F. Beardon, Symmetries of Julia sets, Bull. London Math. Soc. 22 (1990), no. 6, 576-582. MR 1099008 (92f:30033)
  • [Bea91] A. F. Beardon, Iteration of rational functions, Springer-Verlag, New York, 1991. MR 1128089 (92j:30026)
  • [Bea92] A. F. Beardon, Polynomials with identical Julia sets, Complex Variables Theory Appl. 17 (1992), no. 3-4, 195-200. MR 1147050 (93k:30033)
  • [BG06] E. Bombieri and W. Gubler, Heights in Diophantine geometry, New Mathematical Monographs, vol. 4, Cambridge University Press, Cambridge, 2006. MR 2216774 (2007a:11092)
  • [Bog91] F. A. Bogomolov, Abelian subgroups of Galois groups, Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 1, 32-67. MR 1130027 (93b:12007)
  • [CS93] G. S. Call and J. Silverman, Canonical heights on varieties with morphism, Compositio Math. 89 (1993), 163-205. MR 1255693 (95d:11077)
  • [DH93] A. Douady and J. H. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math. 171 (1993), no. 2, 263-297. MR 1251582 (94j:58143)
  • [GTZ08] D. Ghioca, T. J. Tucker, and M. E. Zieve, Intersections of polynomials orbits, and a dynamical Mordell-Lang conjecture, Invent. Math. 171 (2008), no. 2, 463-483. MR 2367026 (2008k:37099)
  • [Mil99] J. Milnor, Dynamics in one complex variable, Vieweg, Braunschweig, 1999. MR 1721240 (2002i:37057)
  • [Mim97] A. Mimar, On the preperiodic points of an endomorphism of $ \mathbb{P}^1\times \mathbb{P}^1$ which lie on a curve, Ph.D. thesis, Columbia University, 1997.
  • [Ull98] E. Ullmo, Positivité et discrétion des points algébriques des courbes, Ann. of Math. (2) 147 (1998), no. 1, 167-179. MR 1609514 (99e:14031)
  • [Zha92] S. Zhang, Positive line bundles on arithmetic surfaces, Ann. of Math. (2) 136 (1992), 569-587. MR 1189866 (93j:14024)
  • [Zha95a] S. Zhang, Positive line bundles on arithmetic varieties, J. Amer. Math. Soc. 8 (1995), 187-221. MR 1254133 (95c:14020)
  • [Zha95b] S. Zhang, Small points and adelic metrics, J. Algebraic Geometry 4 (1995), 281-300. MR 1311351 (96e:14025)
  • [Zha98] S. Zhang, Equidistribution of small points on abelian varieties, Ann. of Math. (2) 147 (1998), no. 1, 159-165. MR 1609518 (99e:14032)
  • [Zha06] S. Zhang, Distributions in Algebraic Dynamics, Surveys in Differential Geometry, vol. 10, International Press, Somerville, MA, 2006, pp. 381-430. MR 2408228 (2009k:32016)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37P05, 14G25, 11C08

Retrieve articles in all journals with MSC (2010): 37P05, 14G25, 11C08

Additional Information

Dragos Ghioca
Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB, T1K 3M4, Canada

Thomas J. Tucker
Affiliation: Department of Mathematics, Hylan Building, University of Rochester, Rochester, New York 14627

Received by editor(s): October 22, 2008
Received by editor(s) in revised form: May 1, 2009
Published electronically: October 20, 2009
Additional Notes: The first author was partially supported by NSERC
The second author was partially supported by NSA Grant 06G-067 and NSF Grant DMS-0801072.
Communicated by: Ted Chinburg
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society