• Previous Article
    The critical points of the elastic energy among curves pinned at endpoints
  • DCDS Home
  • This Issue
  • Next Article
    Global existence of solutions to reaction diffusion systems with mass transport type boundary conditions on an evolving domain
doi: 10.3934/dcds.2021112
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Orbit counting in polarized dynamical systems

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

Received  January 2021 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 & Continuous Dynamical Systems, 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. 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.  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. 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.  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

[1]

Vladimir Dragović, Milena Radnović. Pseudo-integrable billiards and arithmetic dynamics. Journal of Modern Dynamics, 2014, 8 (1) : 109-132. doi: 10.3934/jmd.2014.8.109

[2]

Richard Hofer, Arne Winterhof. On the arithmetic autocorrelation of the Legendre sequence. Advances in Mathematics of Communications, 2017, 11 (1) : 237-244. doi: 10.3934/amc.2017015

[3]

Tanja Eisner, Rainer Nagel. Arithmetic progressions -- an operator theoretic view. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 657-667. doi: 10.3934/dcdss.2013.6.657

[4]

Mehdi Pourbarat. On the arithmetic difference of middle Cantor sets. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4259-4278. doi: 10.3934/dcds.2018186

[5]

Gerhard Frey. Relations between arithmetic geometry and public key cryptography. Advances in Mathematics of Communications, 2010, 4 (2) : 281-305. doi: 10.3934/amc.2010.4.281

[6]

Wenzhi Luo, Zeév Rudnick, Peter Sarnak. The variance of arithmetic measures associated to closed geodesics on the modular surface. Journal of Modern Dynamics, 2009, 3 (2) : 271-309. doi: 10.3934/jmd.2009.3.271

[7]

M. D. Todorov, C. I. Christov. Conservative numerical scheme in complex arithmetic for coupled nonlinear Schrödinger equations. Conference Publications, 2007, 2007 (Special) : 982-992. doi: 10.3934/proc.2007.2007.982

[8]

Jiyoung Han. Quantitative oppenheim conjecture for $ S $-arithmetic quadratic forms of rank $ 3 $ and $ 4 $. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2205-2225. doi: 10.3934/dcds.2020359

[9]

Kaushik Nath, Palash Sarkar. Efficient arithmetic in (pseudo-)Mersenne prime order fields. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020113

[10]

Chad Schoen. An arithmetic ball quotient surface whose Albanese variety is not of CM type. Electronic Research Announcements, 2014, 21: 132-136. doi: 10.3934/era.2014.21.132

[11]

Laurent Imbert, Michael J. Jacobson, Jr.. Empirical optimization of divisor arithmetic on hyperelliptic curves over $\mathbb{F}_{2^m}$. Advances in Mathematics of Communications, 2013, 7 (4) : 485-502. doi: 10.3934/amc.2013.7.485

[12]

Bao Qing Hu, Song Wang. A novel approach in uncertain programming part I: new arithmetic and order relation for interval numbers. Journal of Industrial & Management Optimization, 2006, 2 (4) : 351-371. doi: 10.3934/jimo.2006.2.351

[13]

Rodrigo Abarzúa, Nicolas Thériault, Roberto Avanzi, Ismael Soto, Miguel Alfaro. Optimization of the arithmetic of the ideal class group for genus 4 hyperelliptic curves over projective coordinates. Advances in Mathematics of Communications, 2010, 4 (2) : 115-139. doi: 10.3934/amc.2010.4.115

[14]

Brigitte Vallée. Euclidean dynamics. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 281-352. doi: 10.3934/dcds.2006.15.281

[15]

Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225

[16]

N. Romero, A. Rovella, F. Vilamajó. Dynamics of vertical delay endomorphisms. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 409-422. doi: 10.3934/dcdsb.2003.3.409

[17]

Simone Calogero, Juan Calvo, Óscar Sánchez, Juan Soler. Dispersive behavior in galactic dynamics. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 1-16. doi: 10.3934/dcdsb.2010.14.1

[18]

Evelyn Sander, E. Barreto, S.J. Schiff, P. So. Dynamics of noninvertibility in delay equations. Conference Publications, 2005, 2005 (Special) : 768-777. doi: 10.3934/proc.2005.2005.768

[19]

Luis Vega. The dynamics of vortex filaments with corners. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1581-1601. doi: 10.3934/cpaa.2015.14.1581

[20]

Lluís Alsedà, David Juher, Pere Mumbrú. Minimal dynamics for tree maps. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 511-541. doi: 10.3934/dcds.2008.20.511

2020 Impact Factor: 1.392

Article outline

[Back to Top]