## A self similar tiling generated by the minimal Pisot number.(English)Zbl 1024.11066

The paper deals with the study of a fractal boundary for a special case of tiles. The concepts adjacency and vertex are included too. In addition the boundary of tiles are 5 sets of Hausdorff dimension 1.10026.

### MSC:

 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure 11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. 11A63 Radix representation; digital problems

### Keywords:

fractal; plane tiling; Pisot number; beta-expansions
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### References:

 [1] S. Akiyama: Pisot numbers and greedy algorithm. Number Theory, Diophantine, Computational and Algebraic Aspects, edited by K.Gyory, A. Petho and V.T. Sos, 9-21, de Gruyter, Berlin, New York (1998) · Zbl 0919.11063 [2] M. J. Bertin A. Decomps-Guilloux M. Grandet-Hugot M. Pathiaux-Delefosse J. P. Schreiber: Pisot and Salem Numbers. Birkhauser Verlag Basel-Boston-Berlin. (1992) · Zbl 0772.11041 [3] K. J. Falconer: Techniques in Fractal Geometry. John Wiley & Sons.(1997) · Zbl 0869.28003 [4] C. Frougny B. Solomyak: Finite beta-expansions. Ergod. Th. and Dynam. Sys. 12 (1992) 713-723 · Zbl 0814.68065 [5] A. Messaoudi: Frontiere du fractal de Rauzy et systeme de numeration complexe. preprint. · Zbl 0968.28005 [6] W. Parry: On the $$\beta$$-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960) 269-278 · Zbl 0099.28103 [7] G. Rauzy: Nombres Algebriques et substitutions. Bull. Soc. France 110 (1982), 147-178. · Zbl 0522.10032 [8] W. P. Thurston: Groups, Tilings and Finite state automata. Summer 1989 AMS Colloquium lectures, Research Report GCG 1.
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