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January  2022, 42(1): 189-210. doi: 10.3934/dcds.2021112

Orbit counting in polarized dynamical systems

Wade Hindes

Mathematics Department, Texas State University, San Marcos, TX 78666

Received  January 2021 Published  January 2022 Early access  August 2021

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.

Keywords: Heights, arithmetic dynamics.
Mathematics Subject Classification: Primary: 37P15, 37P05; Secondary: 11G50, 11D45.
Citation: Wade Hindes. Orbit counting in polarized dynamical systems. Discrete & Continuous Dynamical Systems, 2022, 42 (1) : 189-210. doi: 10.3934/dcds.2021112
References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

W. Bosma, J. 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.  Google Scholar

[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.  Google Scholar

[6]

G. Call and J. Silverman, Canonical heights on varieties with morphisms, Compositio Math., 89 (1993), 163-205.   Google Scholar

[7]

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, 2$^nd$ edition, Springer, New York, 2005.  Google Scholar

[8]

G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., 73 (1983), 349-366.  doi: 10.1007/BF01388432.  Google Scholar

[9] P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[12]

V. Healey and W. Hindes, Stochastic canonical heights, J. Number Theory, 201 (2019), 228-256.  doi: 10.1016/j.jnt.2019.02.020.  Google Scholar

[13]

W. Hindes, Counting points of bounded height in monoid orbits, preprint, arXiv: 2006.08563. Google Scholar

[14]

W. Hindes, Dynamical and arithmetic degrees for random iterations of maps on projective space, Math. Proc. Camb. Philos. Soc, to appear. Google Scholar

[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.  Google Scholar

[16]

S. Kawaguchi, Canonical heights for random iterations in certain varieties, Int. Math. Res. Not., Art. ID rnm 023, 2007 (2007), 33 pp.  Google Scholar

[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.  Google Scholar

[18] R. Mason, Diophantine Equations Over Function Fields, London Mathematical Society Lecture Note Series, 96. Cambridge University Press, Cambridge, 1984.  doi: 10.1017/CBO9780511752490.  Google Scholar
[19]

J. Mello, On quantitative estimates for quasiintegral points in orbits of semigroups of rational maps, New York J. Math., 25 (2019), 1091-1111.   Google Scholar

[20]

S. Schanuel, On heights in number fields, Bull. Amer. Math. Soc., 70 (1964), 262-263.  doi: 10.1090/S0002-9904-1964-11110-1.  Google Scholar

[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.  Google Scholar

[22]

Y. Zhang, Bounded gaps between primes, Ann. of Math., 179 (2014), 1121-1174.  doi: 10.4007/annals.2014.179.3.7.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

W. Bosma, J. 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.  Google Scholar

[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.  Google Scholar

[6]

G. Call and J. Silverman, Canonical heights on varieties with morphisms, Compositio Math., 89 (1993), 163-205.   Google Scholar

[7]

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, 2$^nd$ edition, Springer, New York, 2005.  Google Scholar

[8]

G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., 73 (1983), 349-366.  doi: 10.1007/BF01388432.  Google Scholar

[9] P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[12]

V. Healey and W. Hindes, Stochastic canonical heights, J. Number Theory, 201 (2019), 228-256.  doi: 10.1016/j.jnt.2019.02.020.  Google Scholar

[13]

W. Hindes, Counting points of bounded height in monoid orbits, preprint, arXiv: 2006.08563. Google Scholar

[14]

W. Hindes, Dynamical and arithmetic degrees for random iterations of maps on projective space, Math. Proc. Camb. Philos. Soc, to appear. Google Scholar

[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.  Google Scholar

[16]

S. Kawaguchi, Canonical heights for random iterations in certain varieties, Int. Math. Res. Not., Art. ID rnm 023, 2007 (2007), 33 pp.  Google Scholar

[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.  Google Scholar

[18] R. Mason, Diophantine Equations Over Function Fields, London Mathematical Society Lecture Note Series, 96. Cambridge University Press, Cambridge, 1984.  doi: 10.1017/CBO9780511752490.  Google Scholar
[19]

J. Mello, On quantitative estimates for quasiintegral points in orbits of semigroups of rational maps, New York J. Math., 25 (2019), 1091-1111.   Google Scholar

[20]

S. Schanuel, On heights in number fields, Bull. Amer. Math. Soc., 70 (1964), 262-263.  doi: 10.1090/S0002-9904-1964-11110-1.  Google Scholar

[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.  Google Scholar

[22]

Y. Zhang, Bounded gaps between primes, Ann. of Math., 179 (2014), 1121-1174.  doi: 10.4007/annals.2014.179.3.7.  Google Scholar

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