# Should algebraic-dynamics tag be made a synonym for arithmetic-dynamics tag?

Disclosure: On Meta at Mathematics Stack Exchange, I have asked whether an algebraic-dynamics tag should be created and made a synonym of the existing arithmetic-dynamics tag there, given the apparent overlap or lack of clear boundary between the two areas. (See https://math.meta.stackexchange.com/q/31109/118539.)

In checking the situation on Math Overflow, I have found 8 questions tagged and 28 questions tagged . I wonder whether it would be helpful on MO to make one of the tags a synonym of the other. (If I understand correctly, tag synonyms are not symmetric, as one tag remains the primary tag.)

Note the following (also used in my answer to a Mathematics Stack Exchange question asking about algebraic dynamics):

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.

• Perhaps also the tag (tags) would be suitable for this question? (Personally I do not think that adding that tag would do any harm, but I'll leave the decision on this to the OP and other users.) Jan 8, 2020 at 10:41
• @MartinSleziak I think your tag suggestions are good and not likely to be polemical, so I'd suggest you directly edit (in the unlikely case OP doesn't like it, OP can revert anyway).
– YCor
Jan 8, 2020 at 17:09
• I think it's a good suggestion. Next one should choose the common name. I think "arithmetic-dynamics" would exclude algebraic dynamics that are not of arithmetic flavour. So I'd suggest algebraic-arithmetic-dynamics, or just algebraic-dynamics, with tag info saying explicitly that both algebraic dynamics and arithmetic dynamics are concerned.
– YCor
Jan 8, 2020 at 17:13
• @YCor: Good points. That said, arithmetic dynamics might be the more common term catch-all term these days. In any case, there doesn't seem to be as clear a split in practice as there is between say algebraic geometry and arithmetic geometry. Of course, if I'm wrong about this I'd welcome clarification from the experts.
– J W
Jan 13, 2020 at 18:17
• @JW I'm not sure the arithmetic side is broader. Actually, I see a lot of questions in complex dynamics that are about iterations of polynomial or rational functions. They belong on algebraic dynamics but it doesn't appear as a tag. Actually, the number of questions tagged algebraic-dynamics is ridiculously small compared to the number of questions pertaining to it (also maybe arithmetic-dynamics).
– YCor
Jan 13, 2020 at 20:32
• My 2 cents: The analogy is "algebraic geometry" is work over an algebraically closed base field, "arithmetic geometry" is work where the base field need not be algebraically closed, with an implication that the fact that the base field is not algebraically closed is intrinsic to the problem being studied. So to me, arithmetic dynamics has the connotation that the base field is not closed and the problems have to do with that fact. Algebraic dynamics suggests working over an algebraically closed field like $\mathbb C$ and studying dynamics of endomorphisms of algebraic varieties. ... Jan 14, 2020 at 17:47
• But it's all kind of amorphous. Part of the problem: there are now three fields interacting, arithmetic (number theory), algebraic geometry, and dynamical systems. The first two are algebraic in nature, but algebra is too broad, which is why I prefer arithmetic dynamics to algebraic dynamics. The latter should be "algebraic geometry dynamics", but that's more words than most people want to use. As a practical matter, the MSC category 37P is "Arithmetic and non-Archimedean dynamical systems", which leads to arithmetic dynamics being used more than algebraic dynamics. Jan 14, 2020 at 17:53
• @JoeSilverman I disagree. I think "algebraic geometry" in its modern sense (as notably formalized by Grothendieck) encompasses work over arbitrary fields, and arithmetic geometry is a subdomain of it, such as the use of algebraic geometry concepts in arithmetics, typically proving the Weil conjectures, and possibly by extension, concepts that emerged from such studies, where the context is no longer arithmetic in its traditional sense; still I view it as strictly contained in algebraic geometry (which doesn't mean arithmetic geometry is narrow, but that algebraic geometry is broad).
– YCor
Jan 15, 2020 at 2:04