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 algebraic-dynamics and 28 questions tagged arithmetic-dynamics. 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):
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.