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We consider fixed points of the Feigenbaum (periodic-doubling) operator whoseorders tend to infinity. It is known that the hyperbolic dimension of theirJulia sets go to 2. We prove that the Lebesgue measure of these Julia sets tendto zero. An important part of the proof consists in applying martingale theoryto a stochastic process with non-integrable increments.

Measure of the Julia set of the Feigenbaum map with infinite criticality