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The dynamical Manin–Mumford conjecture and the dynamical Bogomolov conjecture for endomorphisms of  $(\mathbb{P}^{1})^{n}$

  • Dragos Ghioca (a1), Khoa D. Nguyen (a2) and Hexi Ye (a3)

We prove Zhang’s dynamical Manin–Mumford conjecture and dynamical Bogomolov conjecture for dominant endomorphisms $\unicode[STIX]{x1D6F7}$ of $(\mathbb{P}^{1})^{n}$ . We use the equidistribution theorem for points of small height with respect to an algebraic dynamical system, combined with an analysis of the symmetries of the Julia set for a rational function.

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[Bak09] Baker, M., A finiteness theorem for canonical heights attached to rational maps over function fields , J. Reine Angew. Math. 626 (2009), 205233.
[BH05] Baker, M. and Hsia, L.-C., Canonical heights, transfinite diameters, and polynomial dynamics , J. Reine Angew. Math. 585 (2005), 6192.
[BR06] Baker, M. and Rumely, R., Equidistribution of small points, rational dynamics, and potential theory , Ann. Inst. Fourier (Grenoble) 56 (2006), 625688.
[Ben05] Benedetto, R. L., Heights and preperiodic points of polynomials over function fields , Int. Math. Res. Not. IMRN 62 (2005), 38553866.
[BG06] Bombieri, E. and Gubler, W., Heights in diophantine geometry, New Mathematical Monographs, vol. 4 (Cambridge University Press, Cambridge, 2006).
[Bro65] Brolin, H., Invariant sets under iteration of rational functions , Ark. Mat. 6 (1965), 103144.
[CS93] Call, G. and Silverman, J., Canonical heights on varieties with morphisms , Compos. Math. 89 (1993), 163205.
[Cha06] Chambert-Loir, A., Mesures et équidistribution sur les espaces de Berkovich , J. Reine Angew. Math. 595 (2006), 215235.
[DH93] Douady, A. and Hubbard, J., A proof of Thurston’s topological characterization of rational functions , Acta Math. 171 (1993), 263297.
[DF17] Dujardin, R. and Favre, C., The dynamical Manin–Mumford problem for plane polynomial automorphisms , J. Eur. Math. Soc. (JEMS) 19 (2017), 34213465.
[Fab09] Faber, X. W. C., Equidistribution of dynamically small subvarieties over the function field of a curve , Acta Arith. 137 (2009), 345389.
[Fal84] Faltings, G., Calculus on arithmetic surfaces , Ann. of Math. (2) 119 (1984), 387424.
[FR06] Favre, C. and Rivera-Letelier, J., Équidistribution quantitative des points de petite hauteur sur la droite projective , Math. Ann. 355 (2006), 311361.
[FLM83] Freire, A., Lopes, A. and Mañé, R., An invariant measure for rational maps , Bol. Soc. Brasil Mat. 14 (1983), 4562.
[GN16] Ghioca, D. and Nguyen, K. D., Dynamical anomalous subvarieties: structure and bounded height theorems , Adv. Math. 288 (2016), 14331462.
[GNY17] Ghioca, D., Nguyen, K. D. and Ye, H., The Dynamical Manin–Mumford Conjecture and the Dynamical Bogomolov Conjecture for split rational maps , J. Eur. Math. Soc. (JEMS) (2017), 25; to appear.
[GT10] Ghioca, D. and Tucker, T. J., Proof of a Dynamical Bogomolov Conjecture for lines under polynomial actions , Proc. Amer. Math. Soc. 138 (2010), 937942.
[GTZ11] Ghioca, D., Tucker, T. J. and Zhang, S., Towards a Dynamical Manin–Mumford Conjecture , Int. Math. Res. Not. IMRN 2011 (2011), 51095122.
[Gub08] Gubler, W., Equidistribution over function fields , Manuscripta Math. 127 (2008), 485510.
[Ham95] Hamilton, D., Length of Julia curves , Pacific J. Math. 169 (1995), 7593.
[Hri85] Hriljac, P., Heights and Arakelov’s intersection theory , Amer. J. Math. 107 (1985), 2338.
[Lau84] Laurent, M., Equations diophantiennes exponentielles , Invent. Math. 78 (1984), 299327.
[Lev90] Levin, G. M., Symmetries on Julia sets , Mat. Zametki 48 (1990), 72–79, 159.
[Lyu83] Lyubich, M., Entropy properties of rational endomorphisms of the Riemann sphere , Ergod. Th. & Dynam. Sys. 3 (1983), 351385.
[Mañ83] Mañé, R., On the uniqueness of the maximizing measure for rational maps , Bol. Soc. Bras. Math. 14 (1983), 2743.
[McQ95] McQuillan, M., Division points on semi-abelian varieties , Invent. Math. 120 (1995), 143159.
[Med07] Medvedev, A., Minimal sets in ACFA, PhD thesis, University of California, Berkeley (2007), 96 pp.
[MS14] Medvedev, A. and Scanlon, T., Invariant varieties for polynomial dynamical systems , Ann. of Math. (2) 179 (2014), 81177.
[Mil00] Milnor, J., Dynamics in one complex variable (Vieweg, Wiesbaden, Germany, 2000).
[Mil04] Milnor, J., On Lattès maps, Preprint (2004), arXiv:math/0402147 [math.DS].
[Mim97] Mimar, A., On the preperiodic points of an endomorphism of which lie on a curve, PhD thesis, Columbia University (1997).
[Mim13] Mimar, A., On the preperiodic points of an endomorphism of ℙ1 ×ℙ1 which lie on a curve , Trans. Amer. Math. Soc. 365 (2013), 161193.
[Mor96] Moriwaki, A., Hodge index theorem for arithmetic cycles of codimension one , Math. Res. Lett. 3 (1996), 173183.
[Ray83] Raynaud, M., Sous-variétés d’une variété abélienne et points de torsion , in Arithmetic and geometry, Vol. I, Progress in Mathematics, vol. 35 (Birkhäuser, Boston, MA, 1983), 327352.
[Thu85] Thurston, W., On the combinatorics of iterated rational maps, Preprint, 1985.
[Ull98] Ullmo, E., Positivité et discrétion des points algébriques des courbes , Ann. of Math. (2) 147 (1998), 167179.
[Yua08] Yuan, X., Big line bundles over arithmetic varieties , Invent. Math. 173 (2008), 603649.
[YZa] Yuan, X. and Zhang, S., Calabi theorem and algebraic dynamics, Preprint.
[YZb] Yuan, X. and Zhang, S., Small points and Berkovich metrics, Preprint,∼szhang/papers/Preprints.html.
[YZ13] Yuan, X. and Zhang, S., The arithmetic Hodge index theorem for adelic line bundles II, Preprint (2013), arXiv:1304.3539.
[YZ17] Yuan, X. and Zhang, S., The arithmetic Hodge index theorem for adelic line bundles , Math. Ann. 367 (2017), 11231171.
[Zan12] Zannier, U., Some problems of unlikely intersections in arithmetic and geometry, Annals of Mathematics Studies, vol. 181 (Princeton University Press, Princeton, NJ, 2012), with appendixes by David Masser.
[Zdu90] Zdunik, A., Parabolic orbifolds and the dimension of the maximal measure for rational maps , Inv. Math. 99 (1990), 627649.
[Zha95a] Zhang, S., Positive line bundles on arithmetic varieties , J. Amer. Math. Soc. 8 (1995), 187221.
[Zha95b] Zhang, S., Small points and adelic metrics , J. Alg. Geom. 4 (1995), 281300.
[Zha98] Zhang, S., Equidistribution of small points on abelian varieties , Ann. of Math. (2) 147 (1998), 159165.
[Zha06] Zhang, S., Distributions in algebraic dynamics , in Survey in Differential Geometry, Vol. 10 (International Press, 2006), 381430.
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Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
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