MMGD defines a deterministic dynamics on a finite state space.

Let

A=(A₁,…,Aₘ), Aᵢ∈{1,…,9},

be a finite reference sequence. Its elements are referenced periodically.

At each step, the current state rₙ₋₁ is multiplied by the referenced element Aₚ₍ₙ₎:

Rₙ = rₙ₋₁Aₚ₍ₙ₎.

The next state is obtained by taking the last nonzero decimal digit of the product:

rₙ = O(Rₙ).

The numerical state is determined by the observed digit together with the current reference phase. Since this state space is finite and the update rule is deterministic, every orbit eventually enters a periodic cycle.

Although multiplication itself is commutative, inserting the observation O after each multiplication makes the resulting dynamics dependent on the order of the reference sequence.

日本語

MMGDは、有限状態空間上の決定論的な力学を定める。

有限参照列を

A=(A₁,…,Aₘ), Aᵢ∈{1,…,9}

とし、その要素を周期的に参照する。

各ステップでは、現在状態 rₙ₋₁ と参照要素 Aₚ₍ₙ₎ を乗算する。

Rₙ = rₙ₋₁Aₚ₍ₙ₎