= Dyadic adding machine
{title2=$i\mapsto i+1\pmod{2^n}$}
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: $000\ldots\mapsto100\ldots\mapsto010\ldots\mapsto110\ldots$. The all-ones sequence maps to the all-zeros sequence.
Equivalently, it is the compatible collection of cyclic maps $i\mapsto i+1\pmod{2^n}$ 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>.
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