= Feigenbaum period-doubling map
{c}
{title2=$g(g(\beta x))=\beta g(x)$}
= Feigenbaum map
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{synonym}
The <Feigenbaum period-doubling map> is the universal, even, quadratic-critical fixed map associated with repeated <period-doubling bifurcations>. One normalization is
$$
g(0)=1,\qquad \beta=g(1)<0,\qquad g(g(\beta x))=\beta g(x).
$$
Composing twice and rescaling therefore reproduces the same graph. This self-similarity is the mechanism behind a common spatial scaling in otherwise different one-dimensional maps with a quadratic maximum.<Quantitative Universality for a Class of Nonlinear Transformations by Mitchell Feigenbaum>[1978: universal spatial scaling and the period-doubling functional equation]{ref}
The quadratic maximum means $g'(0)=0$ and $g''(0)\ne0$. Twice applying the <chain rule> at zero gives
$$
g'(g(0))g''(0)\beta^2=\beta g''(0),\qquad g'(1)=\frac1\beta.
$$
The term involving $g'(0)^2$ vanishes, and cancellation is legitimate because the maximum is nondegenerate. Thus the endpoint derivative is determined by the spatial scaling.
Writing $\gamma=-\beta>0$ makes the successive central intervals $[-\gamma^n,\gamma^n]$ shrink by the factor $\gamma$. Their iterates generate the <Feigenbaum attractor>; nonuniform interval lengths are encoded by the <Feigenbaum geometric potential>. The constant $\beta$ here describes spatial scaling, rather than the spacing of successive bifurcation parameter values.
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