On Rauzy fractal. (English) Zbl 0734.28010

Ito, Shunji; Kimura, Minako

Japan J. Ind. Appl. Math. 8, No. 3, 461-486 (1991).

Summary: The boundary of a space tiling set, called the Rauzy fractal, can be constructed by means of the endomorphism \(\theta\) on a free group of rank 3 according to Dekking’s fractal generating method. Using this method, the Hausdorff dimension of the Rauzy fractal is calculated by \(\frac{\log \lambda_ E}{\log (1/\zeta)}\Doteq 1.09338...\), where \(\lambda_ E\) is a maximal solution of \(\lambda^ 4-2\lambda -1=0,\) and \(\zeta\) is a positive solution of \(x^ 3+x^ 2+x-1=0.\)

MSC:

28A80	Fractals
11K06	General theory of distribution modulo \(1\)
28A78	Hausdorff and packing measures

Keywords:

Weil automorphism; fractal curve; domain exchange transformation; space tiling set; Dekking’s fractal generating method; Hausdorff dimension; Rauzy fractal

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