Dyadic adding machine ($i\mapsto i+1(\mathrm{mod}{2}^n)$)

The dyadic adding machine is addition by one on the space of infinite binary digit sequences, interpreted with the least significant digit first. Carries propagate toward later digits: $000\dots \mapsto 100\dots \mapsto 010\dots \mapsto 110\dots$. The all-ones sequence maps to the all-zeros sequence.

Equivalently, it is the compatible collection of cyclic maps $i\mapsto i+1(\mathrm{mod}{2}^n)$ for all $n$, or translation by one on the 2-adic integers. Every orbit visits every finite binary cylinder and is therefore dense. At the Feigenbaum attractor, this describes the order of visits to nested intervals; it is not the digit-discarding evolution of a full shift.