Dynamics on the Riemann Sphere


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pp: 71–87

DOI: 10.4171/011-1/3

Examples of Feigenbaum Julia sets with small Hausdorff dimension

Artur Avila[1] and Mikhail Lyubich[2]

(1) Université Pierre et Marie Curie, Paris, France
(2) Stony Brook University, USA

We give examples of infinitely renormalizable quadratic polynomials Fc : zz2 + 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.

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