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: Joseph H. Silverman has compiled a long list of articles and books on arithmetic dynamical systems.
It is available for free here.

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.

From the aforelinked:

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.

A: 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:

*

*Raphaël Krikorian, Déviations de moyennes ergodiques, Exp. 927, 2003-2004

*Jean-Christophe Yoccoz, Échanges d'intervalles et surfaces de translation, Exp. 996, 2008-2009.

*Julien Grivaux & Pascal Hubert, Exposants de Lyapunov du ﬂot de Teichmüller, 2012.

A: In addition to the graduate textbook that Felipe already mentioned, 
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.
A: 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
A: To start with, there is Silverman's book: http://www.math.brown.edu/~jhs/ADSHome.html
There was a semester at ICERM last Spring devoted to arithmetic dynamics and the lectures at the workshops (including surveys, open problems, etc) were recorded as videos. Here is a link to the videos of one of the workshops (there is more at the ICERM website). http://icerm.brown.edu/html/videos/sp_s12_w2/
