Topics in Arithmetic Dynamics

UN 3951: Undergraduate Seminars, Fall 2021

Welcome to the undergraduate seminar on arithmetic dynamics! In this seminar, we will study a relatively new mathematical subject at the intersection of dynamical systems and number theory. This area of mathematics involves a breadth of mathematical disciplines, including:

Complex analysis
p-adic analysis
Analytic number theory
Algebraic number theory
Algorithmic number theory
Algebraic geometry
Representation theory
Combinatorics

Throughout the semester, students will study and present 30-minute talks on the theory and various problems in arithmetic dynamics. Each assigned presentation will be tailored towards the student's interests. References for the talks include the book The Arithmetic of Dynamical Systems by Joseph Silverman, the survey article "Current trends and open problems in arithmetic dynamics", and various research articles.

Students should be familiar with elementary number theory, groups, rings, fields, and Galois theory. Additional background, such as in the aforementioned branches of mathematics, will be helpful in exploring the discipline.

Grading is based on the quality of presentations as well as active participation during other students' presentations during the semester.

The seminar meets in Math 507 from 4:00PM—6:00PM (ET) on Wednesdays.

A general overview of this semester's undergraduate seminars is available on Courseworks (Canvas).
To contact me, please email me directly using your Columbia UNI email.

Seminar schedule

Date	Speaker	Title	References
September 9 (6pm, Zoom)	Daniela De Silva Raymond Cheng Robin Zhang Davis Lazowski	Organizational meeting (all sections)	—
September 15	Robin Zhang	Organizational meeting and basic introduction to arithmetic dynamics	[Sil1]§Introduction [Sil4]§1 [BIJMST]§1-2
September 22	Ahmed Shaaban	Dynamics in the digits of rational numbers and modular arithmetic	[Viv]§1-2
	Alan Zhao	Classical dynamics I: Geometry, critical points, and Riemann–Hurwitz	[Sil1]§1.1-1.2 [Sil4]§2
	Alan Du	Classical dynamics II: Multipliers, Julia sets, and Fatou sets	[PR]§Frontiers of Chaos; 2; A. Douady [Sil1]§1.3-1.4 [Sil4]§3 [Tip]
September 29	Vivek Anand Yanamadula	Classical dynamics III: Mandelbrot sets, fractals, and computer graphics	[PR]§Frontiers of Chaos; 4; B. B. Mandelbrot; A. Douady [Tip]
	Jacob Weinstein	Classical dynamics IV: Periodic points of complex rational functions and endomorphisms of algebraic groups	[Sil1]§1.5-1.6 [Sil4]§3
	Ahmed Shaaban	Rational periodic points of quadratic polynomials: Periods 1-3	[WR]§1-4
October 6	Alan Zhao	Dynatomic polynomials and dynatomic modular curves	[BL]§1 [Mor1]§1 [Sil1]§4.1-4.2
	Iris Rosenblum-Sellers	Moduli spaces of rational maps and dynamical systems	[Man] [Sil1]§4.3-4.4
	Vivek Anand Yanamadula	Rational periodic points of quadratic polynomials: Periods greater than 3	[FPS]§1 [Pan]§2.1-2.2 [Poo] [Sto2]§1
October 13	No meeting	No meeting	—
October 20	Jacob Weinstein	Arboreal and dynatomic Galois representations	[BJ] [Jon3] [BIJMST]§5-6
	Ahmed Shaaban	Graphs and the Ihara zeta function	[Ter]§1-2, 6
	Alan Du	Artin–Mazur and Ruelle zeta functions	[Bri1]§1 [Bri2]§1 [Fri] [Rue1] [Rue2]§1-7 [Ter]§4 [War]
October 27	Iris Rosenblum-Sellers	Introduction to the p-adic numbers	[Con2] [Gou]§1-3
	Ahmed Shaaban	p-adic periodic points of quadratic polynomials	[BIJMST]§22 [Kru1]§1 [WR]§5-6
	Jacob Weinstein	The dynamics of maps with good reduction	[BIJMST]§12 [Sil1]§2.3, 2.5-2.6 [Sil4]§6
November 3	Alan Du	Asymptotics in dynamics over finite fields	[AG] [BIJMST]§18 [BOSZ] [Gre] [Hea] [Juu]
	Jacob Weinstein	Stability of iterates over finite fields	[BIJMST]§19 [BOSZ] [Fer] [Jon2] [JL]
	Vivek Anand Yanamadula	Dynamical pseudorandom number generators	[BOSZ] [GIGS] [OS]
November 10	Iris Rosenblum-Sellers	Introduction to Diophantine approximation	[Pot]§1 [Sil1]§3.6 [Sil4]§4.2
	Iris Rosenblum-Sellers	Integers in orbits	[BIJMST]§21 [Sil1]§3.7-3.8 [Sil3] [Sil4]§7
	Iris Rosenblum-Sellers	Primes and prime divisors in orbits	[BIJMST]§20 [FG] [GN] [HJM] [Jon1]
November 17	Alan Du	Mazur's torsion theorem and the uniform boundedness conjecture for abelian varieties	[Dem]§1 [KW]§1 [Sil2]§VIII.7 [SS]§Ogg's conjecture [ST]§2.5
	Alan Zhao	Height functions and Northcott's theorem	[BIJMST]§14-15 [Sil1]§3.1-3.2, 3.4 [Sil4]§4.1, 5.1-5.2
	Alan Zhao	The dynamical uniform boundedness conjecture	[BIJMST]§4 [Loo1]§1 [Loo2]§1 [Sil1]§3.3 [Sil4]§5.3
November 24	Thanksgiving Break (no meeting)	Thanksgiving Break (no meeting)	—
December 1	Vivek Anand Yanamadula	Dynatomic Galois groups and dynatomic fields	[Con1] [Sil1]§3.9 [Mor2] [MP] [VH]§1-4
	Jacob Weinstein	Dynatomic Galois groups of quadratic polynomials	[Kru2] [Kru3] [Sto1] [VH]§1, 5-6 [Zha]
	Alan Du	Dynatomic units	[Ben]§3 [DFH]§3.2 [MS]§6-8 [Nar] [PW] [Sil1]§3.11
December 8	Alan Zhao	Galois equidistribution and algebraic dynamics	[BIJMST]§7-8 [Fak] [Sil1]§3.10
	Ahmed Shaaban	Unlikely intersections in dynamics	[BIJMST]§11 [DHY]§1 [Kri] [Zan]§Introduction
	Vivek Anand Yanamadula	Collatz conjecture	[And] [Lag1] [Lag2] [Lag3]

Books and theses

[BL] Xavier Buff and Tan Lei. "The quadratic dynatomic curves are smooth and irreducible". In Araceli Bonifant, Misha Lyubich, and Scott Sutherland (eds.). Frontiers in complex dynamics, 49–72, Princeton Math. Ser., 51, Princeton Univ. Press, 2014.
[Gou] Fernando Gouvêa. p-adic numbers. Springer, 1997.
[Gre] George Grell. Iterated Galois Groups of Quadratic Rational Functions. PhD Thesis – University of Rochester, 2019.
[Pan] Chatchawan Panraksa. Arithmetic dynamics of quadratic polynomials and dynamical units. PhD Thesis – University of Maryland, 2011.
[PR] Heinz-Otto Peitgen and Peter H. Richter. The Beauty of Fractals. Springer, 1986.
[Rue1] David Ruelle. "Dynamical Zeta Functions: Where Do They Come from and What Are They Good for?" In Konrad Schmüdgen (ed.). Mathematical Physics X, 43–51, Springer, 1992.
[Rue2] David Ruelle. Dynamical zeta functions for piecewise monotone maps of the interval. CRM Monograph Series, 4, American Mathematical Society, 1994.
[Sil1] Joseph H. Silverman. The Arithmetic of Dynamical Systems. Springer, 2007.
[Sil2] Joseph H. Silverman. The Arithmetic of Elliptic Curves. Second Edition. Springer, 2009.
[ST] Joseph H. Silverman and John Tate. Rational Points on Elliptic Curves. Second Edition. Springer, 2015.
[Ter] Audrey Terras. Zeta Functions of Graphs: A Stroll through the Garden. Cambridge University Press, 2013.
[Zan] Umberto Zannier, with appendixes by David Masser. Some problems of unlikely intersections in arithmetic and geometry.. Annals of Mathematical Studies, 181, Princeton University Press, 2012.

Articles

[And] Paul J. Andaloro. The 3x + 1 problem and directed graphs. Fibonacci Q., 40(1):43–54, 2002.
[AG] Amir Akbary and Dragos Ghioca. Periods of orbits modulo primes. J. Number Theory, 129(11):2831–2842, 2009.
[Ben] Robert L. Benedetto. An elementary product identity in polynomial dynamics. Amer. Math. Monthly, 108(9):860–864, 2001.
[Bri1] Andrew Bridy. Transcendence of the Artin-Mazur zeta function for polynomial maps of A₁(F_p). Acta Arith., 156(3):293–300, 2012.
[Bri2] Andrew Bridy. The Artin-Mazur zeta function of a dynamically affine rational map in positive characteristic. J. Théor. Nombres Bordeaux 28(2):301–324, 2016.
[BIJMST] Robert L. Benedetto, Patrick Ingram, Rafe Jones, Michelle Manes, Joseph H. Silverman and Thomas J. Tucker. Current trends and open problems in arithmetic dynamics. Bull. Amer. Math. Soc., 56(4):611–685, 2019.
[BJ] Nigel Boston and Rafe Jones. Arboreal Galois representations. Geom. Dedicata, 124:27–35, 2007.
[DFH] John R. Doyle, Paul Fili, and Trevor Hyde. Dynatomic polynomials, necklace operators, and universal relations for dynamical units. arXiv preprint, 2021.
[DFK] John R. Doyle, Xander Faber, and David Krumm. Preperiodic points for quadratic polynomials over quadratic fields. New York J. Math., 20:507–605, 2014.
[DHY] Laura DeMarco, Holly Krieger, and Hexi Ye. Common preperiodic points for quadratic polynomials. arXiv preprint, 2019.
[Fak] Najmuddin Fakhruddin. Questions on self maps of algebraic varieties J. Ramanujan Math. Soc. 18(2):109–122, 2003.
[Fer] Andrea Ferraguti. The set of stable primes for polynomial sequences with large Galois group Proc. Amer. Math. Soc. (4), 146(7):2773–2784, 2018.
[Fri] David Fried. The zeta functions of Ruelle and Selberg. I. Ann. Sci. École Norm. Sup. (4), 19(4):491–517, 1986.
[FG] Xander Faber and Andrew Granville. Prime factors of dynamical sequences. J. reine angew. Math. (4), 2011(661):1–26, 2011.
[FPS] E. V. Flynn, Bjorn Poonen, and Edward F. Schaefer. Cycles of quadratic polynomials and rational points on a genus 2 curve. Duke Math. J., 90(3): 435–463, 1997.
[GIGS] Jaime Gutierrez, Álvar Ibeas, Domingo Gómez-Pérez, and Igor E. Shparlinski. Predicting masked linear pseudorandom number generators over finite fields. Des. Codes Cryptogr., 67:395–402, 2013.
[GN] Richard Guy and Richard Nowakowski. Discovering primes with Euclid. Delta (Waukesha), 5(2):49–63, 1975.
[Hea] D. R. Heath-Brown. Iteration of quadratic polynomials over finite fields. Mathematika, 63(3):1041–1059, 2017.
[HJM] Spencer Hamblen, Rafe Jones, and Kalyani Madhu. The density of primes in orbits of z^d + c. Int. Math. Res. Not., 2015(7):1924–1958, 2015.
[Jon1] Rafe Jones. The density of prime divisors in the arithmetic dynamics of quadratic polynomials. J. London Math. Soc., 78(2):523–544, 2008.
[Jon2] Rafe Jones. An iterative construction of irreducible polynomials reducible modulo every prime. J. Algebra, 369:114–128, 2012.
[Jon3] Rafe Jones. Galois representations from pre-image trees: an arboreal survey. Publications mathématiques de Besançon: Algèbre et théorie des nombres, 107–136, 2013.
[Juu] Jamie Juul. The image size of iterated rational maps over finite fields. Int. Math. Res. Not., 2021(5):3362–3388, 2021.
[JL] Rafe Jones and Alon Levy. Eventually stable rational functions. Int. J. Number Theory, 13(9):2299–2318, 2017.
[Kru1] David Krumm. A local-global principle in the dynamics of quadratic polynomials. Int. J. Number Theory, 12(8):2265–2297, 2016.
[Kru2] David Krumm. Galois groups in a family of dynatomic polynomials. J. Number Theory, 187:469–511, 2018.
[Kru3] David Krumm. A finiteness theorem for specializations of dynatomic polynomials. Algebra Number Theory, 13(4):963–993, 2019.
[KW] Sheldon Kamienny and Joseph L. Wetherell. On torsion in abelian varieties. Comm. Algebra, 26(5):1675–1678, 1998.
[Lag1] Jeffrey C. Lagarias. The 3x + 1 problem and its generalizations. Amer. Math. Monthly, 92(1):3–23, 1985.
[Lag2] Jeffrey C. Lagarias. The set of rational cycles for the 3x + 1 problem. Acta Arith., 56:33–53, 1990.
[Lag3] Jeffrey C. Lagarias. The 3x + 1 problem: an overview. arXiv preprint, 2021.
[Loo1] Nicole R. Looper. Dynamical uniform boundedness and the abc-conjecture. Invent. Math., 225:1–44, 2021.
[Loo2] Nicole R. Looper. The uniform boundedness and dynamical Lang conjectures for polynomials. arXiv preprint, 2021.
[Mor1] Patrick Morton. On certain algebraic curves related to polynomial maps. Compos. Math., 103(3):319–350, 1996.
[Mor2] Patrick Morton. Galois Groups of Periodic Points. J. Algebra, 201(2):401–428, 1998.
[MP] Patrick Morton and Pratishka Patel. The Galois theory of periodic points of polynomial maps. Proc. London Math. Soc., 68(2): 225–263, 1994.
[MS] Patrick Morton and Joseph H. Silverman. Periodic points, multiplicities, and dynamical units. J. Reine Angew. Math., 461:81–122, 1995.
[Nar] Władysław Narkiewicz. Polynomial cycles in algebraic number fields. Colloq. Math., 58(1):151–155, 1989.
[Poo] Bjorn Poonen. The classification of rational preperiodic points of quadratic polynomials over ℚ: a refined conjecture. Math. Z., 228:11–29, 1998.
[PW] Chatchawan Panraksa and Lawrence C. Washington. Arithmetic dynamics and dynamical units. East-West J. Math., 14(2):201–207, 2012.
[OS] Alina Ostafe and Igor E. Shparlinski. On the degree growth in some polynomial dynamical systems and nonlinear pseudorandom number generators. Math. Comp., 79:501–511, 2010.
[Sil3] Joseph H. Silverman. Integer points, Diophantine approximation, and iteration of rational maps. Duke Math J., 71(3): 793–829, 1993.
[Sto1] Michael Stoll. Galois groups over ℚ of some iterated polynomials. Arch. Math., 59:239–244, 1992
[Sto2] Michael Stoll. Rational 6-cycles under iteration of quadratic polynomials. LMS J. Comput. Math., 11:367–380, 2008
[SS] Norbert Schappacher and René Schoof. Beppo Levi and the arithmetic of elliptic curves. Math. Intelligencer, 18(1):57–69, 1996
[VH] Frank Vivaldi and Spyridon J. Hatjispyros. Galois theory of periodic orbits of rational maps. Nonlinearity, 5(4):961–978, 1992.
[WR] Ralph Walde and Paulo Russo. Rational periodic points of the quadratic function Q_c(x) = x² + c. Amer. Math. Monthly, 101(4):318–331, 1994.
[Zha] Robin Zhang. A Galois–dynamics correspondence for unicritical polynomials. Arnold Math. J. 7(3):467–481, 2021

Notes

[BOSZ] Nigel Boston, Alina Ostafe, Igor Shparlinski, and Michael Zieve. The act of iterating rational functions over finite fields. Banff International Research Station, 2013.
[Con1] Keith Conrad. Cyclotomic extensions.
[Con2] Keith Conrad. The p-adic expansion of rational numbers. 2006.
[Dem] Spencer Dembner. Torsion points on elliptic curves and Mazur's theorem. University of Chicago REU, 2019.
[Kri] Holly Krieger. Equidistribution and unlikely intersections in arithmetic dynamics (video). VaNTAGe, 2020.
[Man] Michelle Manes. The moduli space of rational functions of degree d. 2006.
[Pot] Lukas Pottmeyer. Diophantine approximation. Universität Duisburg-Essen, 2021.
[Sil4] Joseph H. Silverman. Lecture notes on arithmetic dynamics. Arizona Winter School, 2010.
[Tip] Brandon Tippings. Julia sets, the Mandelbrot set, and complex dynamics.
[Viv] Franco Vivaldi. An introduction to arithmetic dynamics. The University of London, 2011.
[War] Evan Warner. Artin–Mazur zeta functions in arithmetic dynamics. Stanford University Arithmetic Dynamics Seminar, 2014.

YouTube videos

Inside Dynamical Systems and the Mathematics of Change (2 minutes). Quanta Magazine, 2020.
63 and -7/4 are special (12 minutes). Numberphile, 2014.
What's so special about the Mandelbrot Set? (17 minutes). Numberphile, 2019.
This equation will change how you see the world (the logistic map) (19 minutes). Veritasium, 2020.
Fractals are typically not self-similar (20 minutes). 3Blue1Brown, 2017.
The Simplest Math Problem No One Can Solve - Collatz Conjecture (22 minutes). Veritasium, 2021.
Number Theory and Dynamics, by Joseph Silverman (53 minutes). UConn Mathematics, 2018.
Fractals: The Colours of Infinity (54 minutes). Arthur C. Clarke, 1995.
How to write mathematics badly (56 minutes). Jean-Pierre Serre, 2003. Transcript.