# 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.

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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). – William Dec 30 '12 at 19:02

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

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Good stuff, thank you! – William Dec 26 '12 at 12:02
[Note that due to updates the #82 above is subject to change; my tiny contribution is presently somewhere in the hundreds...] – Benjamin Dickman Dec 15 '13 at 1:16

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:

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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.

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

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

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