One could ask whether there exist dynamical systems satisfying (2), (3), (4), and a weaker version of axiom (1) other than the obvious ones. Obviously a full understanding of the “obvious ones” requires some deep number theory, but it might still be possible to reach a conjectural understanding.
In particular, can we construct any “exotic” dynamical systems satisfying (2), (3), (4) at all?

PS You left some extra dots in Equation (3), where they are unneeded.

]]>