Examples of Feigenbaum Julia Sets with Small Hausdorff Dimension
نویسندگان
چکیده
We give examples of infinitely renormalizable quadratic polynomials Fc : z 7→ z + 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. Stony Brook IMS Preprint #2004/04 July 2004
منابع مشابه
Examples of Feigenbaum Julia Sets with Small Hausdorff Dimension Artur Avila and Mikhail Lyubich
We give examples of infinitely renormalizable quadratic polynomials Fc : z 7→ z 2 + 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. Sto...
متن کاملHausdorff Dimension and Conformal Measures of Feigenbaum Julia Sets
1.1. Statement of the results. One of the first questions usually asked about a fractal subset of R is whether it has the maximal possible Hausdorff dimension, n. It certainly happens if the set has positive Lebesgue measure. On the other hand, it is easy to construct fractal sets of zero measure but of dimension n. Moreover, this phenomenon is often observable for fractal sets produced by conf...
متن کامل2 1 A ug 2 00 4 HAUSDORFF DIMENSION AND CONFORMAL MEASURES OF FEIGENBAUM JULIA SETS
We show that contrary to anticipation suggested by the dictionary between rational maps and Kleinian groups and by the " hairiness phenomenon " , there exist many Feigenbaum Julia sets J(f) whose Hausdorff dimension is strictly smaller than two. We also prove that for any Feigen-baum Julia set, the Poincaré critical exponent δcr is equal to the hyperbolic dimension HD hyp (J(f)). Moreover, if a...
متن کاملThe Hausdorff Dimension of Julia Sets of Entire Functions Iv
It is known that, for a transcendental entire function f, the Hausdorff dimension of J( f ) satisfies 1% dim J( f )% 2. For each d ` (1, 2), an example of a transcendental entire function f with dim J( f ) ̄ d is given. It is then indicated how this function can be modified to produce a transcendental meromorphic function F with one pole with dim J(F ) ̄ d. These appear to be the first examples o...
متن کاملTHE HAUSDORFF DIMENSION OF THE BOUNDARY OF THE IMMEDIATE BASIN OF INFINITY OF McMULLEN MAPS
In this paper, we give a formula of the Hausdorff dimension of the boundary of the immediate basin of infinity of McMullen maps fp(z) = z Q + p/z, where Q ≥ 3 and p is small. This gives a lower bound of the Hausdorff dimension of the Julia sets of McMullen maps in the special cases.
متن کامل