Frontière du fractal de Rauzy et système de numération complexe. (Boundary of the Rauzy fractal and complex numeration system). (French) Zbl 0968.28005

Consider the polynomial \(P(x) = x^3 - x^2-x -1\) and denote its real root by \(\beta\) and the two complex roots by \(\alpha\) and \(\overline{\alpha}\), respectively. Then \(\beta > 1\) and the modules of \(\alpha\) and \(\overline{\alpha}\) are strictly less than 1. The Rauzy fractal is the set on the complex plane defined by \[ {\mathcal E} = \left\{ \sum_{i=3}^{\infty} \varepsilon_i \alpha^i \mid \forall i \geq 3, \varepsilon_i \in \{0, 1\},\;\varepsilon_i \varepsilon_{i+1} \varepsilon_{i+2} = 0 \right\}. \] In this paper, the author gives a parametrization for the boundary of the Rauzy fractal \({\mathcal E}\) which makes it possible for him to calculate the Hausdorff dimension of the boundary of \({\mathcal E}\), and to prove that the boundary is a quasi-circle. The author also studies the strictly extreme points and the convex hull of the Rauzy fractal.


28A80 Fractals
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11B85 Automata sequences
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
28A78 Hausdorff and packing measures
37B10 Symbolic dynamics
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