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## Flat limit in the Feigenbaum functional equation

G. SWIATEK explained in Orsay his joint work with G. LEVIN on the functional equation $\tau H^2(x)=H(\tau x)$ introduced by Feigenbaum for the study of the period-doubling “path to chaos” in the quadratic family. For each finite $\ell\geq 2$, they consider $\mathcal H_\ell:=\{|f(x)|^\ell:f\in Diff^2([0,1],[-1,1]), f(0)=0\}$. A result of McMULLEN says that the solution $H_\ell\in\mathcal H_\ell$ is unique for any even integer $\ell$. ECKMANN-WITWER showed the (uniform) convergence towards some $H_\infty$ with a flat critical point using computer-assisted estimates.

G. SWIATEK explained their new proof of the following strengthening:

1. $H_\infty$ extends analytically to the union of two topological disks in the complex plane containing respectively $[0,x_0)$ and $(x_0,1]$ and both mapped to a disk $D(0,R)$. The leading singularity is $e^{-C/|x-x_0|^2}$. Elsewhere the map has negative Schwarzian derivative.
2. $H_\infty$ has a Julia set with the usual characteristic properties (normality, density of periodic points, boundedness of the infinite orbits) which has zero area and full Hausdorff dimension, i.e., 2.
3. In the limit $\ell\to\infty$, the area and dimension of the corresponding Julia sets converge to the above values.

One of the motivations of this work is the well-known problem of Julia sets with positive area.

Question: Can one prove that these Julia sets have zero area (perhaps for $\ell$ large enough)?

Remark: One expects that the dimensions for finite $\ell$ are strictly between $1$ and $2$.

The proof follows McMULLEN’s: compactness (here with a uniform domain of analyticity); existence of uniformly quasi-conformal conjugacies between solutions (here bounded geometry has to be replaced by the presentation functions of LEDRAPPIER and MISIUREWICZ); towers, ie, sequences $H_{n-1}=H_n^2$; rigidity of towers yielding linear conjugacies
.