Feigenbaum period-doubling map ($g(g(\beta x))=\beta g(x)$)

The smooth quadratic-critical fixed point of period-doubling renormalization is normalized by $g(0)=1$ and $\beta =g(1)<0$. It has $g^{\prime }(0)=0$ and $g^{\prime \prime }(0)\ne 0$. Differentiating its displayed functional equation twice at zero gives $g^{\prime }(1)=1/\beta$. Its restrictive intervals generate a Cantor attractor described by Feigenbaum geometric potential weights.

The geometric symbolic potential for the Feigenbaum period-doubling map has uniformly negative bounded values and exponentially decreasing variations. Potential sums describe logarithmic lengths of odd-address period-doubling intervals up to a uniform additive error. This is a geometric scaling potential, distinct from an unbounded derivative potential at a critical point. Its pressure is continuous, strictly decreasing and convex by the almost-additive partition-function limit and finite-volume inequalities.

Odd-address intervals lie in the first image of the central restrictive interval, away from the critical point. The derivative of the Feigenbaum map is bounded above and away from zero there, so the next even interval has comparable length by the mean value theorem. For the wraparound image, $g^{2^n}({\beta }^{n}x)={\beta }^{n}g(x)$ makes its length a fixed fraction $(1-\beta )/2$ of the central interval. Thus the odd and full partition length-power sums differ only by constant factors for each fixed real exponent, and have the same exponential growth rate.

The Feigenbaum attractor is the compact Cantor set obtained from the nested restrictive-interval partitions of the Feigenbaum map. Every level interval has two descendant intervals. Their lengths vary with symbolic address, so a geometric topological pressure is useful for estimating its Hausdorff dimension.