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

Orbit counting in polarized dynamical systems

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.

Citation: Wade Hindes. Orbit counting in polarized dynamical systems. Discrete and 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.

[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. Sedgewick, Analytic 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. Mason, Diophantine 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.

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.

[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. Sedgewick, Analytic 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. Mason, Diophantine 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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