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Bibliography for Arithmetic Dynamics Joseph H. Silverman May 5, 2020 This document lists a wide variety of articles and books in the area of arithmetic dynamics. It also includes some additional material that was referenced in The Arith- metic of Dynamical Systems (Springer-Verlag GTM 241) and some miscellaneous ar- ticles and books that I’ve referenced in my own work. Note that the numbering in this document does not match the numbering of references in GTM 241. Further note that I do not automatically update ArXiv preprints when they appear. Most recent pub- lished articles in arithmetic dynamics can be found by searching the 37P category on MathSciNet. References [1] N. Abarenkova, J.-C. Angles` d’Auriac, S. Boukraa, S. Hassani, and J.-M. Maillard. Topological entropy and Arnold complexity for two-dimensional mappings. Phys. Lett. A, 262(1):44–49, 1999. [2] W. Abram and J. C. Lagarias. Intersections of multiplicative translates of 3-adic cantor sets, 2013. arXiv:1308.3133. [3] R. L. Adler, A. G. Konheim, and M. H. McAndrew. Topological entropy. Trans. Amer. Math. Soc., 114:309–319, 1965. [4] L. V. Ahlfors. Complex Analysis. McGraw-Hill Book Co., New York, 1978. [5] F. Ahmad, R. L. Benedetto, J. Cain, G. Carroll, and L. Fang. The arithmetic basilica: a quadratic PCF arboreal Galois group, 2019. arXiv:1909.00039. [6] O. Ahmadi. A note on stable quadratic polynomials over fields of characteristic two, 2009. arXiv:0910.4556. [7] O. Ahmadi, F. Luca, A. Ostafe, and I. E. Shparlinski. On stable quadratic polynomials. Glasg. Math. J., 54(2):359–369, 2012. [8] O. Ahmadi and K. Monsef-Shokri. A note on the stability of trinomials over finite fields, 2018. arXiv:1810.03142. [9] A. V. Aho and N. J. A. Sloane. Some doubly exponential sequences. Fibonacci Quart., 11(4):429–437, 1973. [10] N. Ailon and Z. Rudnick. Torsion points on curves and common divisors of ak − 1 and bk − 1. Acta Arith., 113(1):31–38, 2004. [11] W. Aitken, F. Hajir, and C. Maire. Finitely ramified iterated extensions. IMRN, 14:855– 880, 2005. [12] A. Akbary and D. Ghioca. Periods of orbits modulo primes. J. Number Theory, 129(11):2831–2842, 2009. [13] S. Akiyama, H. Brunotte, A. Petho,˝ and W. Steiner. Periodicity of certain piecewise affine planar maps. Tsukuba J. Math., 32(1):197–251, 2008. 1 [14] S. Albeverio, M. Gundlach, A. Khrennikov, and K.-O. Lindahl. On the Markovian behavior of p-adic random dynamical systems. Russ. J. Math. Phys., 8(2):135–152, 2001. [15] S. Albeverio, U. A. Rozikov, and I. A. Sattarov. p-adic (2; 1)-rational dynamical sys- tems. J. Math. Anal. Appl., 398(2):553–566, 2013. [16] S. Albeverio, B. Tirotstsi, A. Y. Khrennikov, and S. de Shmedt. p-adic dynamical sys- tems. Teoret. Mat. Fiz., 114(3):349–365, 1998. [17] N. Ali. Stabilite´ des polynomes.ˆ Acta Arith., 119(1):53–63, 2005. [18] I. Aliev and C. Smyth. Power maps and subvarieties of the complex algebraic n-torus, 2008. arXiv:0802.2938. [19] K. Allen, D. DeMark, and C. Petsche. Non-Archimedean Henon´ maps, attractors, and horseshoes. Res. Number Theory, 4(1):Art. 5, 30, 2018. [20] E. Amerik. A computation of invariants of a rational self-map. Ann. Fac. Sci. Toulouse Math. (6), 18(3):445–457, 2009. [21] E. Amerik. Existence of non-preperiodic algebraic points for a rational self-map of infinite order. Math. Res. Lett., 18(2):251–256, 2011. [22] E. Amerik. Some applications of p-adic uniformization to algebraic dynamics. In Ra- tional points, rational curves, and entire holomorphic curves on projective varieties, volume 654 of Contemp. Math., pages 3–21. Amer. Math. Soc., Providence, RI, 2015. [23] E. Amerik, F. Bogomolov, and M. Rovinsky. Remarks on endomorphisms and rational points. Compos. Math., 147(6):1819–1842, 2011. [24] E. Amerik, P. Kurlberg, K. D. Nguyen, A. Towsley, B. Viray, and J. F. Voloch. Evidence for the dynamical Brauer-Manin criterion. Exp. Math., 25(1):54–65, 2016. [25] E. Amerik and M. Verbitsky. Construction of automorphisms of hyperkahler¨ manifolds. Compos. Math., 153(8):1610–1621, 2017. [26] F. Amoroso and R. Dvornicich. A lower bound for the height in abelian extensions. J. Number Theory, 80(2):260–272, 2000. [27] F. Amoroso and U. Zannier. A relative Dobrowolski lower bound over abelian exten- sions. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 29(3):711–727, 2000. [28] V. Anashin. Ergodic transformations in the space of p-adic integers. In p-adic math- ematical physics, volume 826 of AIP Conf. Proc., pages 3–24. Amer. Inst. Phys., Melville, NY, 2006. [29] J. Anderson. Bounds on the radius of the p-adic Mandelbrot set. Acta Arith., 158(3):253–269, 2013. [30] J. Anderson, I. Bouw, O. Ejder, N. Girgin, V. Karemaker, and M. Manes. Dynamical belyi maps, 2017. arXiv:1703.08563. [31] J. Anderson, S. Hamblen, B. Poonen, and L. Walton. Local arboreal representations. Int. Math. Res. Not. IMRN, (19):5974–5994, 2018. [32] J. Anderson, M. Manes, and B. Tobin. Cubic post-critically finite polynomials defined over Q, 2020. arXiv:2001.10471. [33] Y. Andre.´ G-functions and geometry. Aspects of Mathematics, E13. Friedr. Vieweg & Sohn, Braunschweig, 1989. [34] K. Andrei and E. Yurova. Criteria of ergodicity for p-adic dynamical systems in terms of coordinate functions, 2013. arXiv:1303.6472. [35] J. Andrews and C. Petsche. Abelian extensions in dynamical Galois theory, 2020. arXiv:2001.00659. [36] J.-C. Angles` d’Auriac, J.-M. Maillard, and C. M. Viallet. On the complexity of some birational transformations. J. Phys. A, 39(14):3641–3654, 2006. [37] O. Antol´ın-Camarena and S. Koch. On a theorem of Kas and Schlessinger. In Quasicon- formal mappings, Riemann surfaces, and Teichmuller¨ spaces, volume 575 of Contemp. Math., pages 13–22. Amer. Math. Soc., Providence, RI, 2012. 2 [38] T. M. Apostol. Introduction to Analytic Number Theory. Springer-Verlag, New York, 1976. Undergraduate Texts in Mathematics. [39] M. Arfeux. Approximability of dynamical systems between trees of spheres. Indiana Univ. Math. J., 65(6):1945–1977, 2016. [40] M. Arfeux. Compactification and trees of spheres covers. Conform. Geom. Dyn., 21:225–246, 2017. [41] M. Arfeux. Dynamics on trees of spheres. J. Lond. Math. Soc. (2), 95(1):177–202, 2017. [42] M. Arfeux and J. Kiwi. Irreducibility of the set of cubic polynomials with one periodic critical point, 2016. arXiv:1611.09281. [43] J. Arias de Reyna. Dynamical zeta functions and Kummer congruences. Acta Arith., 119(1):39–52, 2005. [44] V. I. Arnol0d. Dynamics of complexity of intersections. Bol. Soc. Brasil. Mat. (N.S.), 21(1):1–10, 1990. [45] V. I. Arnol0d. Dynamics of intersections. In Analysis, et cetera, pages 77–84. Academic Press, Boston, MA, 1990. [46] V. I. Arnol0d. Dynamics, statistics and projective geometry of Galois fields. Cambridge University Press, Cambridge, 2011. Translated from the Russian, With words about Arnold by Maxim Kazarian and Ricardo Uribe-Vargas. [47] D. K. Arrowsmith and F. Vivaldi. Some p-adic representations of the Smale horseshoe. Phys. Lett. A, 176(5):292–294, 1993. [48] D. K. Arrowsmith and F. Vivaldi. Geometry of p-adic Siegel discs. Phys. D, 71(1- 2):222–236, 1994. [49] M. Astorg. Dynamics of post-critically finite maps in higher dimension. Ergodic Theory Dynam. Systems, 2018. DOI: https://doi.org/10.1017/etds.2018.32. [50] P. Autissier. Hauteur des correspondances de Hecke. Bull. Soc. Math. France, 131(3):421–433, 2003. [51] P. Autissier. Dynamique des correspondances algebriques´ et hauteurs. Int. Math. Res. Not., (69):3723–3739, 2004. [52] S. Axler. Linear Algebra Done Right. Undergraduate Texts in Mathematics. Springer- Verlag, New York, second edition, 1997. [53] M. Ayad. Periodicit´ e´ (mod q) des suites elliptiques et points S-entiers sur les courbes elliptiques. Ann. Inst. Fourier (Grenoble), 43(3):585–618, 1993. [54] M. Ayad and D. L. McQuillan. Irreducibility of the iterates of a quadratic polynomial over a field. Acta Arith., 93(1):87–97, 2000. [55] M. Ayad and D. L. McQuillan. Corrections to: “Irreducibility of the iterates of a quadratic polynomial over a field” [Acta Arith. 93 (2000), no. 1, 87–97]. Acta Arith., 99(1):97, 2001. [56] A. Azevedo, M. Carvalho, and A. Machiavelo. Dynamics of a quasi-quadratic map, 2012. arXiv:1210.0042. [57] E. Bach and A. Bridy. On the number of distinct functional graphs of affine-linear transformations over finite fields. Linear Algebra Appl., 439(5):1312–1320, 2013. [58] G. Baier and M. Klein. Maximum hyperchaos in generalized Henon´ maps. Physics Letters A, 151:281–284, 12 1990. [59] S. Baier, S. Jaidee, S. Stevens, and T. Ward. Automorphisms with exotic orbit growth, 2012. arXiv:1201.4503. [60] D. Bajpai, R. Benedetto, R. Chen, E. Kim, O. Marschall, D. Onul, and Y. Xiao. Non- archimedean connected Julia sets with branching, 2014. arXiv:1410.0591. [61] A. Baker. Transcendental number theory. Cambridge Mathematical Library. Cambridge University Press, Cambridge, second edition, 1990. 3 [62] I. N. Baker. Fixpoints of polynomials and rational functions. J. London Math. Soc., 39:615–622, 1964. [63] M. Baker. A lower bound for average values of dynamical Green’s functions. Math. Res. Lett., 13(2-3):245–257, 2006. [64] M. Baker. Uniform structures and Berkovich spaces, 2006. ArXiv:math.NT/ 0606252. [65] M. Baker. A finiteness theorem for canonical heights attached to rational maps over function fields. J. Reine Angew. Math., 626:205–233, 2009. [66] M. Baker and L. De Marco. Special curves and postcritically finite polynomials. Forum Math. Pi, 1:e3, 35, 2013. [67] M. Baker and L.
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  • Annales Scientifiques De L'é.Ns

    Annales Scientifiques De L'é.Ns

    ANNALES SCIENTIFIQUES DE L’É.N.S. J.-B. BOST Potential theory and Lefschetz theorems for arithmetic surfaces Annales scientifiques de l’É.N.S. 4e série, tome 32, no 2 (1999), p. 241-312 <http://www.numdam.org/item?id=ASENS_1999_4_32_2_241_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1999, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systé- matique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. sclent. EC. Norm. Sup., 4° serie, t. 32, 1999, p. 241 a 312. POTENTIAL THEORY AND LEFSCHETZ THEOREMS FOR ARITHMETIC SURFACES BY J.-B. BOST ABSTRACT. - We prove an arithmetic analogue of the so-called Lefschetz theorem which asserts that, if D is an effective divisor in a projective normal surface X which is nef and big, then the inclusion map from the support 1-D) of -D in X induces a surjection from the (algebraic) fondamental group of \D\ onto the one of X. In the arithmetic setting, X is a normal arithmetic surface, quasi-projective over Spec Z, D is an effective divisor in X, proper over Spec Z, and furthermore one is given an open neighbourhood ^ of |-D|(C) on the Riemann surface X(C) such that the inclusion map |D|(C) ^ f^ is a homotopy equivalence.
  • Correspondences in Arakelov Geometry and Applications to the Case of Hecke Operators on Modular Curves

    Correspondences in Arakelov Geometry and Applications to the Case of Hecke Operators on Modular Curves

    CORRESPONDENCES IN ARAKELOV GEOMETRY AND APPLICATIONS TO THE CASE OF HECKE OPERATORS ON MODULAR CURVES RICARDO MENARES Abstract. In the context of arithmetic surfaces, Bost defined a generalized Arithmetic 2 Chow Group (ACG) using the Sobolev space L1. We study the behavior of these groups under pull-back and push-forward and we prove a projection formula. We use these results to define an action of the Hecke operators on the ACG of modular curves and to show that they are self-adjoint with respect to the arithmetic intersection product. The decomposition of the ACG in eigencomponents which follows allows us to define new numerical invariants, which are refined versions of the self-intersection of the dualizing sheaf. Using the Gross-Zagier formula and a calculation due independently to Bost and K¨uhnwe compute these invariants in terms of special values of L series. On the other hand, we obtain a proof of the fact that Hecke correspondences acting on the Jacobian of the modular curves are self-adjoint with respect to the N´eron-Tate height pairing. Contents 1. Introduction1 2 2. Functoriality in L1 5 2 2.1. The space L1 and related spaces5 2 2.2. Pull-back and push-forward in the space L1 6 2 2.3. Pull-back and push-forward of L1-Green functions 12 3. Pull-back and push-forward in Arakelov theory 16 2 3.1. The L1-arithmetic Chow group, the admissible arithmetic Chow group and intersection theory 16 3.2. Functoriality with respect to pull-back and push-forward 18 4.