Period-Doubling Systematics

Symbol names for nodes belonging to a period doubling cascade are built up easily and systematically by the following algorithm, which is illustrated for the cascade based on the period three node $001$. The node symbol sequence of length $p$ is repeated (this produces an even parity word of length $2p$) and the penultimate symbol is changed. This step is repeated forever.

$$001 \rightarrow 001~0\underline{0}1=001~011 \rightarrow 001~011~~001~0\underline{1}1= 001~011~~001~001 \rightarrow \cdots$$