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Orbit counting in polarized dynamical systems

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  • We extend recent orbit counts for finitely generated semigroups acting on $ \mathbb{P}^N $ to certain infinitely generated, polarized semigroups acting on projective varieties. We then apply these results to semigroup orbits generated by some infinite sets of unicritical polynomials.

    Mathematics Subject Classification: Primary: 37P15, 37P05; Secondary: 11G50, 11D45.

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  • [1] A. Baragar, Rational points on $K3$ surfaces in $\mathbb{P}^1\times\mathbb{P}^1\times \mathbb{P}^1$, Math. Ann., 305 (1996), 541-558.  doi: 10.1007/BF01444236.
    [2] E. Bombieri and J. Pila, The number of integral points on arcs and ovals, Duke Math. J., 59 (1989), 337-357.  doi: 10.1215/S0012-7094-89-05915-2.
    [3] Y. Bilu and R. Tichy, The Diophantine equation $f(x) = g (y)$, Acta Arithmetica, 95 (2000), 261-288.  doi: 10.4064/aa-95-3-261-288.
    [4] W. BosmaJ. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput., 24 (1997), 235-265.  doi: 10.1006/jsco.1996.0125.
    [5] B. Brindza, On $S$-integral solutions of the Catalan equation, Acta Arith., 48 (1987), 397-412.  doi: 10.4064/aa-48-4-397-412.
    [6] G. Call and J. Silverman, Canonical heights on varieties with morphisms, Compositio Math., 89 (1993), 163-205. 
    [7] R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, 2$^nd$ edition, Springer, New York, 2005.
    [8] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., 73 (1983), 349-366.  doi: 10.1007/BF01388432.
    [9] P. Flajolet and  R. SedgewickAnalytic Combinatorics, Cambridge University Press, 2009. 
    [10] D. R. Heath-Brown, Counting rational points on algebraic varieties, Analytic Number Theory, Lecture Notes in Math. Springer, Berlin, Heidelberg, (2006), 51–95. doi: 10.1007/978-3-540-36364-4_2.
    [11] D. R. Heath-Brown, The density of rational points on curves and surfaces, Ann. of Math., 155 (2002), 553-598.  doi: 10.2307/3062125.
    [12] V. Healey and W. Hindes, Stochastic canonical heights, J. Number Theory, 201 (2019), 228-256.  doi: 10.1016/j.jnt.2019.02.020.
    [13] W. Hindes, Counting points of bounded height in monoid orbits, preprint, arXiv: 2006.08563.
    [14] W. Hindes, Dynamical and arithmetic degrees for random iterations of maps on projective space, Math. Proc. Camb. Philos. Soc, to appear.
    [15] P. Ingram, Lower bounds on the canonical height associated to the morphism, $\phi(z) = z^ d+ c$, Monatsh. Math., 157 (2009), 69-89.  doi: 10.1007/s00605-008-0018-6.
    [16] S. Kawaguchi, Canonical heights for random iterations in certain varieties, Int. Math. Res. Not., Art. ID rnm 023, 2007 (2007), 33 pp.
    [17] S. Kawaguchi and Joseph H. Silverman, On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties, J. Reine Angew. Math., 713 (2016), 21-48.  doi: 10.1515/crelle-2014-0020.
    [18] R. MasonDiophantine Equations Over Function Fields, London Mathematical Society Lecture Note Series, 96. Cambridge University Press, Cambridge, 1984.  doi: 10.1017/CBO9780511752490.
    [19] J. Mello, On quantitative estimates for quasiintegral points in orbits of semigroups of rational maps, New York J. Math., 25 (2019), 1091-1111. 
    [20] S. Schanuel, On heights in number fields, Bull. Amer. Math. Soc., 70 (1964), 262-263.  doi: 10.1090/S0002-9904-1964-11110-1.
    [21] D. Zagier, On the number of Markoff numbers below a given bound, Math. Comp., 39 (1982), 709-723.  doi: 10.1090/S0025-5718-1982-0669663-7.
    [22] Y. Zhang, Bounded gaps between primes, Ann. of Math., 179 (2014), 1121-1174.  doi: 10.4007/annals.2014.179.3.7.
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