G. SWIATEK explained in Orsay his joint work with G. LEVIN on the functional equation introduced by Feigenbaum for the study of the period-doubling “path to chaos” in the quadratic family. For each finite , they consider . A result of McMULLEN says that the solution is unique for any even integer . ECKMANN-WITWER showed the (uniform) convergence towards some with a flat critical point using computer-assisted estimates.
G. SWIATEK explained their new proof of the following strengthening:
- extends analytically to the union of two topological disks in the complex plane containing respectively and and both mapped to a disk . The leading singularity is . Elsewhere the map has negative Schwarzian derivative.
- 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.
- In the limit , 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 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 ; rigidity of towers yielding linear conjugacies