Tilings, Quasicrystals, Discrete Planes, Generalized Substitutions, and Multidimensional Continued Fractions

The aim of this paper is to give an overview of recent results about tilings, discrete approximations of lines and planes, and Markov partitions for toral automorphisms.The main tool is a generalization of the notion of substitution. The simplest examples which correspond to algebraic parameters, are related to the iteration of one substitution, but we show that it is possible to treat arbitrary irrationalexamples by using multidimensional continued fractions.We give some non-trivial applications to Diophantine approximation, numeration systems and tilings, and we expose the main unsolved questions.

[1]  Benjamin Weiss,et al.  SIMILARITY OF AUTOMORPHISMS OF THE TORUS , 1970 .

[2]  Laurent Vuillon,et al.  Combinatoire des motifs d'une suite sturmienne bidimensionnelle , 1998, Theor. Comput. Sci..

[3]  M. S. Keane,et al.  On almost everywhere exponential convergence of the modified Jacobi-Perron algorithm: a corrected proof , 1996, Ergodic Theory and Dynamical Systems.

[4]  Laurent Vuillon.  Combinatorics of patterns of a bidimensional Sturmian sequence. , 1998 .

[5]  S. Akiyama.  Self affine tiling and Pisot numeration system , 1999 .

[6]  Shunji Ito.  ON PERIODIC EXPANSIONS OF CUBIC NUMBERS AND RAUZY FRACTALS , 2000 .

[7]  Y. Takahashi.  Algorithms, Fractals and Dynamics , 1996 .

[8]  Shunji Ito,et al.  Fractal Domains of Quasi-Periodic Motions on T 2 , 1995 .

[9]  de Ng Dick Bruijn.  Sequences of zeros and ones generated by special production rules , 1981 .

[10]  Rufus Bowen,et al.  Markov partitions are not smooth , 1978 .

[11]  Makoto Ohtsuki,et al.  Modified Jacobi-Perron Algorithm and Generating Markov Partitions for Special Hyperbolic Toral Automorphisms , 1993 .

[12]  Laurent Vuillon,et al.  Tilings and rotations on the torus: a two-dimensional generalization of Sturmian sequences , 2000, Discret. Math..

[13]  Valérie Berthé,et al.  Balance properties of multi-dimensional words , 2002, Theor. Comput. Sci..

[14]  Makoto Ohtsuki,et al.  Parallelogram Tilings and Jacobi-Perron Algorithm , 1994 .

[15]  Pierre Arnoux,et al.  Higher dimensional extensions of substitutions and their dual maps , 2001 .

[16]  Shigeki Akiyama,et al.  Pisot numbers and greedy algorithm , 1998 .

[17]  P. Arnoux,et al.  Pisot substitutions and Rauzy fractals , 2001 .

[18]  京都大学数理解析研究所,et al.  Number Theory and Its Applications , 1999 .

[19]  Sébastien Ferenczi,et al.  Imbalances in Arnoux-Rauzy sequences , 2000 .

[20]  P. Arnoux,et al.  Mesures de Gauss pour des algorithmes de fractions continues multidimensionnelles , 1993 .

[21]  Sébastien Ferenczi.  Bounded remainder sets , 1992 .

[22]  M. Senechal.  Quasicrystals and geometry , 1995 .

[23]  Rémy Mosseri,et al.  Generalized Rauzy tilings: construction and electronic properties , 2000 .

[24]  A. Messaoudi,et al.  Frontiere du fractal de Rauzy et systeme de numeration complexe , 2000 .

[25]  J. Vidal,et al.  Generalized quasiperiodic Rauzy tilings , 2000 .

[26]  G. A. Hedlund,et al.  Symbolic Dynamics II. Sturmian Trajectories , 1940 .

[27]  佐野 友紀,et al.  On purely periodic beta-expansions of Pisot numbers , 2001 .

[28]  Pierre Arnoux,et al.  THE SCENERY FLOW FOR GEOMETRIC STRUCTURES ON THE TORUS： THE LINEAR SETTING , 2001 .