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A Feigenbaum fixed point is an even analytic unimodal interval map $g$ with $g(0)=1$, a critical maximum of even order $d$, $a=g(1)\in (-1,0)$ andIn the quadratic class $g^{\prime \prime }(0)<0$. Its spatial scaling uses the Feigenbaum constants convention $\alpha =-1/a$; the values depend on the critical order. The central restrictive intervals are successively reduced by $|a|$ under renormalization. This fixed point describes the limiting shape of repeatedly renormalized maps at a period-doubling accumulation.

// Target: dynamical-systems.bigb

For a normalized even unimodal interval map with $f(0)=1$ and $a=f(1)\in (-1,0)$, defineThis returns the second iterate near the critical point to the original spatial and height normalization. Then $\mathcal{T}(f)(0)=f(1)/a=1$. A restrictive-interval condition is also needed for the composition to remain unimodal and map $[-1,1]$ into itself; normalization alone is not sufficient.

// Target: dynamical-systems.bigb