Current research on MMGD focuses on three related questions concerning finite dynamics and geometric realization.

Fig.2  Reversal Preservation

What is preserved when the order of numerical interactions changes?

Multiplication is commutative, but inserting a last-nonzero-digit observation after each step can make the resulting dynamics dependent on the order of the reference sequence. The question is which periodic structures remain invariant under changes of order and under what conditions.

Order　(2,3,5) ↔ (5,3,2)　Spec={3,12}

How far does reversal preserve periodic structure?

Computational results show that, for decimal words of length three, reversal preserves cycle type in all 9^3=729 cases. At length four, counterexamples first appear. The current problem is to explain the mechanism underlying this transition.

What can geometric realization distinguish?

Different systems may share the same cycle spectrum while differing in their individual orbits or functional graphs. The question is whether geometric realization retains information that coarse dynamical invariants do not distinguish.