Examples of Feigenbaum Julia sets with small Hausdorff dimension
Artur Avila
Université Pierre et Marie Curie, Paris, FranceMikhail Lyubich
Stony Brook University, USA
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Abstract
We give examples of infinitely renormalizable quadratic polynomials Fc : z ↦ z2 + c with stationary combinatorics whose Julia sets have Hausdorff dimension arbitrary close to 1. The combinatorics of the renormalization involved is close to the Chebyshev one. The argument is based upon a new tool, a "Recursive Quadratic Estimate" for the Poincaré series of an infinitely renormalizable map.