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On the Lebesgue measure of the Feigenbaum Julia set. (English) Zbl 1454.37045

The quadratic Feigenbaum polynomial is given by \(f_\text{Feig}(z)=z^2+c_\text{Feig}\), where \(c_\text{Feig}\approx -1.4011551890\) is the limit of the sequence of parameters real period doubling parameters. It is shown that the Hausdorff dimension of the Julia set of \(f_\text{Feig}\) is strictly less than \(2\). In particular, this implies that the Julia set has Lebesgue measure zero. This answers a well-known question in complex dynamics. The paper contains many results and some proofs are computer-assisted.

MSC:

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F25 Renormalization of holomorphic dynamical systems
37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
28A78 Hausdorff and packing measures
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable

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Online Encyclopedia of Integer Sequences:

Decimal expansion of Feigenbaum’s constant 0.399535...

References:

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