Proof of a dynamical Bogomolov conjecture for lines under polynomial actions

Ghioca, Dragos; Tucker, Thomas

doi:10.1090/S0002-9939-09-10182-X

Proof of a dynamical Bogomolov conjecture for lines under polynomial actions
HTML articles powered by AMS MathViewer

by Dragos Ghioca and Thomas J. Tucker PDF

Proc. Amer. Math. Soc. 138 (2010), 937-942 Request permission

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

Pau Atela and Jun Hu, Commuting polynomials and polynomials with same Julia set, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 6 (1996), no. 12A, 2427–2432. MR 1445904, DOI 10.1142/S0218127496001570
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, DOI 10.5186/aasfm.1987.1205
Matthew H. Baker and Liang-Chung Hsia, Canonical heights, transfinite diameters, and polynomial dynamics, J. Reine Angew. Math. 585 (2005), 61–92. MR 2164622, DOI 10.1515/crll.2005.2005.585.61
A. F. Beardon, Symmetries of Julia sets, Bull. London Math. Soc. 22 (1990), no. 6, 576–582. MR 1099008, DOI 10.1112/blms/22.6.576
Alan F. Beardon, Iteration of rational functions, Graduate Texts in Mathematics, vol. 132, Springer-Verlag, New York, 1991. Complex analytic dynamical systems. MR 1128089, DOI 10.1007/978-1-4612-4422-6
A. F. Beardon, Polynomials with identical Julia sets, Complex Variables Theory Appl. 17 (1992), no. 3-4, 195–200. MR 1147050, DOI 10.1080/17476939208814512
Enrico Bombieri and Walter Gubler, Heights in Diophantine geometry, New Mathematical Monographs, vol. 4, Cambridge University Press, Cambridge, 2006. MR 2216774, DOI 10.1017/CBO9780511542879
F. A. Bogomolov, Abelian subgroups of Galois groups, Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 1, 32–67 (Russian); English transl., Math. USSR-Izv. 38 (1992), no. 1, 27–67. MR 1130027, DOI 10.1070/IM1992v038n01ABEH002186
Gregory S. Call and Joseph H. Silverman, Canonical heights on varieties with morphisms, Compositio Math. 89 (1993), no. 2, 163–205. MR 1255693
Adrien Douady and John H. Hubbard, A proof of Thurston’s topological characterization of rational functions, Acta Math. 171 (1993), no. 2, 263–297. MR 1251582, DOI 10.1007/BF02392534
Dragos Ghioca, Thomas J. Tucker, and Michael E. Zieve, Intersections of polynomials orbits, and a dynamical Mordell-Lang conjecture, Invent. Math. 171 (2008), no. 2, 463–483. MR 2367026, DOI 10.1007/s00222-007-0087-5
John Milnor, Dynamics in one complex variable, Friedr. Vieweg & Sohn, Braunschweig, 1999. Introductory lectures. MR 1721240
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.
Emmanuel Ullmo, Positivité et discrétion des points algébriques des courbes, Ann. of Math. (2) 147 (1998), no. 1, 167–179 (French). MR 1609514, DOI 10.2307/120987
Shouwu Zhang, Positive line bundles on arithmetic surfaces, Ann. of Math. (2) 136 (1992), no. 3, 569–587. MR 1189866, DOI 10.2307/2946601
Shouwu Zhang, Positive line bundles on arithmetic varieties, J. Amer. Math. Soc. 8 (1995), no. 1, 187–221. MR 1254133, DOI 10.1090/S0894-0347-1995-1254133-7
Shouwu Zhang, Small points and adelic metrics, J. Algebraic Geom. 4 (1995), no. 2, 281–300. MR 1311351
Shou-Wu Zhang, Equidistribution of small points on abelian varieties, Ann. of Math. (2) 147 (1998), no. 1, 159–165. MR 1609518, DOI 10.2307/120986
Shou-Wu Zhang, Distributions in algebraic dynamics, Surveys in differential geometry. Vol. X, Surv. Differ. Geom., vol. 10, Int. Press, Somerville, MA, 2006, pp. 381–430. MR 2408228, DOI 10.4310/SDG.2005.v10.n1.a9