Coullet-Tresser Cascade of Bifurcations in the logistic Family and Hausdorff Dimension of real quadratic Julia Sets

Abstract: In a paper with L. Jacksztas (Adv Math 2020) we have proven that if $c_0$ is a parabolic parameter (i.e. with a parabolic cycle) in $(c_{Feig},1/4)$ ($c_{Feig}$ being the limit point of the cascade of bifurcations) then the function $d(c)=$ Hausdorff dimension of the Julia set $J_c$ of $z^2+c$ has an infinite derivative at $c_0$ if $d(c_0)\leq 4/3$, while it is $C^1$ across $c_0$ if $d(c_0)>4/3$.

Recently A. Dudko, I. Gorobovickis and W. Tucker have proven that $d(c)>4/3$ on $[-1.53,-1.23]$ (arXiv:2204.07880). The combination of these two results implies that $d$ is $C^1$ on $(c_{Feig},-3/4)$ while $d'(-3/4)=-\infty$ (a former result of L. Jacksztas).\\ After some description (including a history) of the Coullet-Tresser Feigenbaum phenomenon, I will outline the proof of J-Z theorem and briefly describe D-G-T's result.

(Joint work with L. Jacksztas)

complex variablesdynamical systems

Audience: researchers in the topic

CAvid: Complex Analysis video seminar

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