La cascade de doublements de période de Coullet-Tresser Feigenbaum et la dimension de Hausdorff des ensembles de Julia quadratiques (Michel Zinsmeister-Université d'Orléans)

Séminaire « Analyse complexe et équations différentielles »
Salle Kampé de Fériet

When c decreases from 0 to cF eig = −1, 4011.., the family (fc(x) = x2 + c) undergoes a period-doubling
cascade of bifurcations which was introduced and studied by Coullet-Tresser and Feigenbaum to serve
as a model of transition to chaos. Going to the dynamical plane the quadratic polynomials fc(z) = z2 +c
generate Julia sets Jc which are, except for c = 0, −2, fractal sets with Hausdorff dimension d(c) ≥ 1
if c ∈ [−2, 0].
The aim of the talk is to prove that on the interval (cF eig, 0] the function d(c) (which is known (Mc-
Mullen) to be continuous), is actually C1 except at the first bifurcation point c = −3/4.
The proof is by combining a paper by Jaksztas and Z. (Adv. Math., 2020) with a recent preprint by
Dutko, Gorobovickis, and Tucker (arXiv :22040788v1)

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