= Feigenbaum attractor
{c}
{title2=$F=\bigcap_n\bigcup_{i=0}^{2^n-1}\Delta_i^{(n)}$}
For the normalized <Feigenbaum map>, put $\gamma=-g(1)$ and define
$$
\Delta_0^{(n)}=[-\gamma^n,\gamma^n],\qquad
\Delta_i^{(n)}=g^i(\Delta_0^{(n)}),\quad 0\le i<2^n.
$$
The <Feigenbaum attractor> is the nested intersection $A=\bigcap_n\bigcup_i\Delta_i^{(n)}$. Each level interval has two children, and the largest interval length tends to zero. Consequently every compatible infinite binary address selects one point of this <Cantor set>.<Multifractality of the Feigenbaum attractor and fractional derivatives by Uriel Frisch and others>[Section 2.1, equations (6)–(7) and the following coding paragraph]{ref}
At level $n$, the intervals are visited cyclically: the label changes from $i$ to $i+1$ modulo $2^n$. Expressing $i$ in binary therefore turns iteration on $A$ into the <dyadic adding machine>. One adds one with carries, rather than discarding the first symbol as in a <full shift>. For example, addresses written with the least significant digit first begin $000\ldots\mapsto100\ldots\mapsto010\ldots\mapsto110\ldots$.
There are $2^n$ geometric pieces at depth $n$, but those pieces are visited in a cycle. Counting them does not make the time evolution a two-symbol full shift. The <Feigenbaum geometric potential> uses this depth to study spatial contraction and <Hausdorff dimension>.
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