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On Rauzy fractal. (English) Zbl 0734.28010

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
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References:

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