On the Continuity of Hausdorff Dimension of Julia Sets Concerning Potts Models

Continuity of Hausdorff Dimension of Julia-lavaurs Sets as a Function of the Phase

Let f0(z) = z2 + 1/4 and E0 the set of phases σ such that the critical point 0 escapes in one step by the Lavaurs map gσ ; it is a topological strip in the cylinder of phases whose boundary consists of two Jordan curves symmetric wrt R/Z. We prove that if σn ∈ E0 converges to σ ∈ ∂E0 in such a way that gσn (0) converges to gσ(0) along an external ray, then the Hausdorff dimension of the Julia-L...

Calculating Hausdorff Dimension of Julia Sets and Kleinian Limit Sets

The 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...

Area and Hausdorff Dimension of Julia Sets of Entire Functions

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Hausdorff 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...