# Arithmetic dynamics and dynamics on moduli spaces

The following question is more of a request for pointers to suitable literature on introductory material for arithmetic dynamics and dynamics on moduli spaces.

In my dissertation, I have been working mostly with smooth dynamical systems, and a lot with a class of dynamical systems given by iteration of certain polynomial maps on smooth two-dimensional (in fact algebraic) submanifolds of $\mathbb{R}^3$. Naturally I have also been looking at holomorphic dynamics. At some point I started to ask questions (not related to my current work) about those dynamical systems, which seem to be better formulated in the algebraic context, rather than analytic, measure-theoretic or differential-geometric (all of which I am familiar with). As a result I discovered a field (which seems to be actively developing) of arithmetic dynamics and (somewhat related but not entirely, I guess) dynamics on moduli spaces. I have searched the internet for some introductory material, only to find that literature is rather scarce, and I haven't been able to find a description of major problems or conjectures which are driving the field.

Questions:

1) What would be good references for some of the fundamental results in arithmetic dynamics?

2) What are questions of interest in arithmetic dynamics? Are there any major actively researched conjectures? What is driving the field? Are there any strong connections with well-known problems in other fields?

3) Is there any introductory literature of expository nature?

4) Questions (1) - (3) applied to dynamics on moduli spaces (I really don't know much about this field, other than the phrase "dynamics on moduli spaces" that I seem to come across often lately).

Note: Not sure whether I should make this a community wiki; please advise.

• A quick answer to (2) is that there are a number of conjectures that are analogues from arithmetic geometry that are helping to drive the field of arithmetic dynamics. These include dynamical analogues of (1) the uniform boundedness conjecture for torsion on abelian varieties; (2) Raynaud's theorem on torsion points on subvarieties of abelian varieties; (3) Faltings' theorem (Mordell-Lang conjecture) on rational points on subvarieties of abelian varieties. Replace torsion points with preperiodic points and Mordell-Weil groups with orbits to get rough analogues, although there are subtleties. – Joe Silverman Dec 26 '12 at 21:17
• I'd like to thank everyone for comments and answers - all the answers are very helpful and informative! I find it difficult to accept any one answer (since all are acceptable), so perhaps I will keep this question open (hope this doesn't stand too sharply at odds with the MO rules). – user39719 Dec 30 '12 at 19:02

Concerning dynamics on moduli spaces (and work of Avila, Eskin, Forni, Gouëzel, Hubert, Kontsevich, McMullen, Yoccoz, Eskin,...), I would first study three Bourbaki Seminars talks:

The Arithmetic of Dynamical Systems, Springer-Verlag, GTM 241, 2007,

there's also the following monograph that discusses dynamical-related moduli spaces from an algebraic and arithmetic viewpoint:

Moduli Spaces and Arithmetic Dynamics, CRM Monograph Series 30, AMS, 2012.

As for the list of references http://www.math.brown.edu/~jhs/ADSBIB.pdf that Benjamin Dickman mentioned, I update it once or twice a year, so it's reasonably up-to-date, but I make no claim to its being complete. You might also try searching MathSciNet and the ArXiv for articles whose classification number is 37Pxx, which will catch most recent articles. In particular, there's

37P45 = Dynamical systems and ergodic theory / Arithmetic and non-Archimedean dynamical systems / Families and moduli spaces.

• @Joe Silverman: Thank you! This is very helpful indeed. – user39719 Dec 26 '12 at 21:29

Joseph H. Silverman has compiled a long list of articles and books on arithmetic dynamical systems.

Edit: You may wish to check out a book published in March 2019:

Benedetto, R. L. (2019). Dynamics in one non-archimedean variable (Vol. 198). American Mathematical Society. AMS Link.

The theory of complex dynamics in one variable, initiated by Fatou and Julia in the early twentieth century, concerns the iteration of a rational function acting on the Riemann sphere. Building on foundational investigations of $$p$$-adic dynamics in the late twentieth century, dynamics in one non-archimedean variable is the analogous theory over non-archimedean fields rather than over the complex numbers. It is also an essential component of the number-theoretic study of arithmetic dynamics.

This textbook presents the fundamentals of non-archimedean dynamics, including a unified exposition of Rivera-Letelier's classification theorem, as well as results on wandering domains, repelling periodic points, and equilibrium measures. The Berkovich projective line, which is the appropriate setting for the associated Fatou and Julia sets, is developed from the ground up, as are relevant results in non-archimedean analysis. The presentation is accessible to graduate students with only first-year courses in algebra and analysis under their belts, although some previous exposure to non-archimedean fields, such as the $$p$$-adic numbers, is recommended. The book should also be a useful reference for more advanced students and researchers in arithmetic and non-archimedean dynamics.

• Another 2019 update is the article "Current Trends and Open Problems in Arithmetic Dynamics" (arxiv.org/abs/1806.04980) which gives a survey of many current research directions. – user47305 Mar 14 at 5:28
• @Mark Great; thanks! PS: There I am in Reference 35 at the arXiv piece you linked (: – Benjamin Dickman Mar 26 at 21:33

Although the following is a book review, it is also an expositionary introduction:

http://www.ams.org/bull/2009-46-01/S0273-0979-08-01216-0/S0273-0979-08-01216-0.pdf

BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 46, Number 1, January 2009, Pages 157–164 Robert L. Benedetto reviews The arithmetic of dynamical systems, by Joseph H. Silverman