Intersecting Two-Dimensional Fractals with Lines

The Twin Dragon and Rauzy fractals are intersected with the real axis. In the Twin Dragon case, unexpectedly from its fractal nature, the intersection is an interval characterized by a finite automaton. For the case of the Rauzy fractal, it is proved that the intersection has infinitely many components.

[1]  Andreas Stein,et al.  High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams , 2004 .

[2]  Shigeki Akiyama,et al.  On canonical number systems , 2002, Theor. Comput. Sci..

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

[4]  Jean Berstel,et al.  Transductions and context-free languages , 1979, Teubner Studienbücher : Informatik.

[5]  H. Prodinger,et al.  The Sum‐of‐Digits Function for Complex Bases , 1998 .

[6]  Yoichiro Takahashi,et al.  Markov subshifts and realization of β-expansions , 1974 .

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

[8]  Shigeki Akiyama.  A self-similar tiling generated by the minimal Pisot number , 1998 .

[9]  J. M. Marstrand.  Some Fundamental Geometrical Properties of Plane Sets of Fractional Dimensions , 1954 .

[10]  Jörg M. Thuswaldner,et al.  Canonical number systems, counting automata and fractals , 2002, Mathematical Proceedings of the Cambridge Philosophical Society.

[11]  Robert F. Tichy,et al.  Fractal properties of number systems , 2001, Period. Math. Hung..

[12]  E. T..  An Introduction to the Theory of Numbers , 1946, Nature.

[13]  S. Akiyama.  On a generalization of the radix representation-a survey , 2004 .

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

[15]  J. Karhumäki,et al.  ALGEBRAIC COMBINATORICS ON WORDS (Encyclopedia of Mathematics and its Applications 90) By M. LOTHAIRE: 504 pp., 60.00, ISBN 0 521 81220 8 (Cambridge University Press, 2002) , 2003 .

[16]  Jun Luo.  A NOTE ON A SELF-SIMILAR TILING GENERATED BY THE MINIMAL PISOT NUMBER , 2002 .

[17]  Berndt Farwer,et al.  ω-automata , 2002 .

[18]  J. Thuswaldner,et al.  On the characterization of canonical number systems , 2004 .

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

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

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

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

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

[24]  J. Keesling,et al.  The Hausdorff Dimension of the Boundary of a Self‐Similar Tile , 2000 .

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

[26]  A. Brauer,et al.  On algebraic equations with all but one root in the interior of the unit circle. To my teacher and former colleague Erhard Schmidt on his 75th birthday , 1950 .

[27]  Jörg M. Thuswaldner,et al.  Neighbours of Self-affine Tiles in Lattice Tilings , 2003 .

[28]  I. Kátai,et al.  Canonical number systems in imaginary quadratic fields , 1981 .

[29]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

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

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

[32]  Samuel Eilenberg,et al.  Automata, languages, and machines. A , 1974, Pure and applied mathematics.