Connectedness of number theoretic tilings

We also study the connectedness of Pisot dual tilings which play an important role in the study of β-expansion, substitution and symbolic dynamical system. It is shown that each tile generated by a Pisot unit of degree 3 is arcwise connected. This is naturally expected since the digit set consists of consecutive integers as above. However surprisingly, we found families of disconnected Pisot dual tiles of degree 4. Also we give a simple necessary and sufficient condition for the connectedness of the Pisot dual tiles of degree 4. As a byproduct, a complete classification of the β-expansion of 1 for quartic Pisot units is given.

[1]  Michael Baake,et al.  Digit tiling of euclidean space , 2000 .

[2]  M. Marden.  Geometry of Polynomials , 1970 .

[3]  Shigeki Akiyama,et al.  Cubic Pisot units with finite beta expansions , 2004 .

[4]  Shigeki Akiyama,et al.  The topological structure of fractal tilings generated by quadratic number systems , 2005 .

[5]  J. J. P. Veerman,et al.  Hausdorff Dimension of Boundaries of Self-Affine Tiles In R N , 1997, math/9701215.

[6]  W. Parry.  On theβ-expansions of real numbers , 1960 .

[7]  Shigeki Akiyama,et al.  Topological properties of two-dimensional number systems , 2000 .

[8]  Frédérique Bassino.  Beta-Expansions for Cubic Pisot Numbers , 2002, LATIN.

[9]  Randolph B. Tarrier,et al.  Groups , 1973, Algebra.

[10]  J. Allouche.  Algebraic Combinatorics on Words , 2005 .

[11]  Jun Luo,et al.  On the boundary connectedness of connected tiles , 2004, Mathematical Proceedings of the Cambridge Philosophical Society.

[12]  A. Rényi.  Representations for real numbers and their ergodic properties , 1957 .

[13]  Brenda Praggastis,et al.  Numeration systems and Markov partitions from self similar tilings , 1999 .

[14]  Jeffrey C. Lagarias,et al.  Self-affine tiles in ℝn , 1996 .

[15]  Hui Rao,et al.  A CERTAIN FINITENESS PROPERTY OF PISOT NUMBER SYSTEMS , 2004 .

[16]  Nicolau C. Saldanha,et al.  Remarks on Self-Affine Tilings , 1994, Exp. Math..

[17]  K. Schmidt,et al.  On Periodic Expansions of Pisot Numbers and Salem Numbers , 1980 .

[18]  G. Rauzy.  Nombres algébriques et substitutions , 1982 .

[19]  Boris Solomyak,et al.  Finite beta-expansions , 1992, Ergodic Theory and Dynamical Systems.

[20]  Kenneth Falconer,et al.  Fractal Geometry: Mathematical Foundations and Applications , 1990 .

[21]  J. Lagarias.  Self-Affine Tiles in , 1994 .

[22]  Masayoshi Hata,et al.  On the structure of self-similar sets , 1985 .

[23]  Nikita Sidorov,et al.  Bijective and General Arithmetic Codings for Pisot Toral Automorphisms , 2001 .

[24]  Shigeki Akiyama,et al.  On the boundary of self affine tilings generated by Pisot numbers , 2002 .

[25]  N. Sidorov.  Ergodic-theoretic properties of certain Bernoulli convolutions , 2002, math/0203056.

[26]  V Canterini.  Connectedness of geometric representation of substitutions of Pisot type , 2003 .

[27]  Christoph Bandt.  Self-Similar Tilings and Patterns Described by Mappings , 1997 .

[28]  Andrew Haas,et al.  Self-Similar Lattice Tilings , 1994 .

[29]  R. Daniel Mauldin,et al.  Hausdorff dimension in graph directed constructions , 1988 .

[30]  K. Lau,et al.  On the Connectedness of Self‐Affine Tiles , 2000 .