Could somebody please give a big picture description of what exactly is the object of study in the area of Algebraic Dynamics? Is it related to Dynamical Systems? If yes in what sense? Also, what is the main mathematical discipline underpinning Algebraic Dynamics? Is it algebraic geometry, differential geometry e.t.c.?


2 Answers 2


From nLab:

In algebraic dynamics one typically studies discrete dynamical systems on algebraic varieties. Such a system is given by a regular endomorphism $D: X \to X$ of a variety $X$.


The case over number fields is also called arithmetic dynamics...

That said, note also that Joseph Silverman writes in the introduction to The Arithmetic of Dynamical Systems, Springer 2007:

There is no firm line between arithmetic dynamics and algebraic dynamics, and indeed much of the material in this book is quite algebraic.

See also the closing remarks of Alexander and Devaney in A Century of Complex Dynamics:

Another recent topic of interest is algebraic dynamics where questions involving algebraic aspects (rather than the dynamical behavior) of iterated functions arise.

Compare with this excerpt from the abstract for the ICERM Complex and Arithmetic Dynamics program in 2012:

Arithmetic dynamics refers to the study of number theoretic phenomena arising in dynamical systems on algebraic varieties. Many global problems in arithmetic dynamics are analogues of classical problems in the theory of Diophantine equations or arithmetic geometry, including for example uniform bounds for rational periodic points, intersections of orbits with subvarieties, height bounds and/or measure-theoretic distributions of dynamically defined sets of special points, and local-global obstructions.

The same abstract continues:

While global arithmetic dynamics bears a resemblence to arithmetic geometry, the theory of p-adic (nonarchimedean) dynamics draws much of its inspiration from classical complex dynamics. As in complex dynamics, a fundamental question is to characterize orbits by their topological or metric properties. Recent progress in p-adic dynamics, especially in dimension one, has benefited from the introduction of Berkovich space into the subject.

By the way, for a very gentle introduction to a little arithmetic dynamics, see A Glimpse of Arithmetic Dynamics by Grady and Poston. For additional introductory material, see Vivaldi's An introduction to arithmetic dynamics and Chapter 10 of Hutz's An Experimental Introduction to Number Theory.

Beyond that there is the article Current trends and open problems in arithmetic dynamics by Benedetto et al. More resources are available on Silverman's page for his abovementioned book. Also, Benjamin Dickman has recently kindly pointed out to me Benedetto's new book Dynamics in One Non-Archimedean Variable.


The Wiki article states that it is a combination of dynamical systems and number theory. I know it's a redirect, but WP's information on this point is probably reliable enough :)

(Are you checking here because you are not comfortable with WP info? It is a serious question which I'm curious about.)

  • 1
    $\begingroup$ @manos WP=Wikipedia. Yes, I arrived at this page by searching for "Algebraic dynamics", and that page redirected to arithmetic dynamics. Usually it's OK to assume the validity of such redirects. $\endgroup$
    – rschwieb
    Aug 27, 2012 at 18:12
  • $\begingroup$ I am pretty comfortable with Wikipedia. $\endgroup$
    – Manos
    Aug 27, 2012 at 18:47
  • $\begingroup$ It may also refer to rational map actions (quotient of polynomials) on algebraic varieties, or dynamics given by differential equations where the vector fields are polynomial (this last one relates closely to Hilbert's 16th problem). $\endgroup$
    – user2093
    Dec 24, 2012 at 11:38

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