Dynamics of self-similar tilings

Ergodic Theory and Dynamical Systems

Volumn 17, Issue 3, 1997, Pages 695-738

  Solomyak, Boris ^a  

^a UNIVERSITY OF WASHINGTON   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031506167     PISSN: 01433857     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0143385797084988     Document Type: Article

References (84)

1. C. Bandt. Self-similar tilings and patterns decribed by mappings. The Mathematics of Aperiodic Order - Proceedings of a NATO-ASI in Waterloo, ON, August 1995. To appear.
2. C. Bandt and G. Gelbrich. Classification of self-affine lattice tilings. J. London Math. Soc. 50 (1994), 581-593.
3. T. Bedford. Crinkly curves, Markov partitions, and dimension. PhD thesis, Warwick University, 1984.
4. T. Bedford. Generating special Markov partitions for hyperbolic toral automorphisms using fractals. Ergod. Th. & Dynam. Sys. 6 (1986), 325-333.
5. D. Berend and C. Radin. Are there chaotic tilings? Commun. Math. Phys. 152 (1993), 215-219.
6. F. Blanchard. β-expansions and symbolic dynamics. Theoretical Comp. Sci. 65 (1989), 131-141.
7. E. Bombieri and J. E. Taylor. Quasicrystals, tilings and algebraic number theory: some preliminary connections. Contemp. Math. 64 (1987), 241-264.
8. A. Brauer. On algebraic equations with all but one root in the interior of the unit circle. Math. Nachri. 4 (1951), 250-257.
9. N. G. de Bruijn. Algebraic theory of Penrose's nonperiodic tilings of the plane I, II. Indagationes Math. 43 (1981), 39-66.
10. N. G. de Bruijn. Quasicrystals and their Fourier transform. Indagationes Math. 48 (1986), 123-152.
11. N. G. de Bruijn. Updown generations of Penrose patterns. Indagationes Math., N.S. 1 (1990), 201-219.
12. F. M. Dekking. The spectrum of dynamical systems arising from substitutions of constant length. Zeit. Wahr. 41 (1978), 221-239.
13. F. M. Dekking. Recurrent sets. Adv. in Math. 44 (1982), 78-104.
14. F. M. Dekking. Replicating superfigures and endomorphisms of free groups. J. Combinat. Theory A 32 (1982), 315-320.
15. F. M. Dekking and M. Keane. Mixing properties of substitutions. Zeit. Wahr. 42 (1978), 23-33.
16. S. Dworkin. Spectral theory and X-ray diffraction. J. Math. Phys. 34 (1993), 2965-2967.
17. S. Dworkin. The local structure and translational dynamics of tilings. Preprint, University of Texas, 1995.
18. S. Ferenczi, C. Mauduit and A. Nogueira. Substitution dynamical systems: algebraic characterization of eigenvalues. Ann. Sci. L'École Norm. Sup. 29 (1996), 519-533.
19. C. Frougny and B. Solomyak. Finite beta-expansions. Ergod. Th. & Dynam. Sys. 12 (1992), 713-723.
20. H. Furstenberg. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton, 1978.
21. F. Gähler and R. Klitzing. The diffraction pattern of selfsimilar tilings. The Mathematics of Aperiodic Order - Proceedings of a NATO-ASI in Waterloo, ON, August 1995. To appear.
22. A. Garsia. Arithmetic properties of Bernoulli convolutions. Trans. Amer. Math. Soc. 102 (1962), 409-432.
23. G. Gelbrich. Crystallographic reptiles. Geometriae Dedicata 51 (1994), 235-256.
24. G. Gelbrich. Galois duality of self-similar tilings. Preprint, University of Greifswald, 1996.
25. C. P. M. Geerse and A. Hof. Lattice gas models on self-similar aperiodic tilings. Rev. Math. Phys. 3 (1991), 163-221.
26. W. J. Gilbert. The fractal dimension of sets derived from complex basis. Canad. Math. Bull. 29 (1986), 495-500.
27. J. Giles, Jr. Construction of replicating superfigures. J. Combinat. Theory A 26 (1979), 328-334.
28. C. Godrèche. The sphinx: a limit-periodic tiling of the plane. J. Phys. A: Math. Gen. 22 (1989), L1163-L1166.
29. C. Godrèche. Non Pisot tilings and singular scattering. Phase Transitions 32 (1991), 45-79.
30. C. Godrèche and F. Lançon. A simple example of a non-Pisot tiling with five-fold symmetry. J. Phys. I France 2 (1992), 207-220.
31. C. Godrèche and J. M. Luck. Multifractal analysis in reciprocal space and the nature of the Fourier spectrum. J. Phys. A: Math. Gen. 23 (1990), 3769-3797.
32. B. Grünbaum and G. S. Shephard. Tilings and Patterns. Freeman, New York, 1986.
33. A. Hof. On diffraction by aperiodic structures. Comm. Math. Phys. 169 (1995), 25-43.
34. A. Hof. Diffraction by aperiodic structures. The Mathematics of Aperiodic Order - Proceedings of a NATO-ASI in Waterloo, ON, August 1995. To appear.
35. B. Host. Valeurs propres de systèmes dynamiques définis par de substitutions de longueur variable. Ergod. Th. & Dynam. Sys. 6 (1986), 529-540.
36. S. Ito and M. Kimura. On Rauzy fractal. Japan J. Indust. Appl. Math. 8 (1991), 461-486.
37. S. Ito and M. Ohtsuki. On the fractal curves induced from endomorphisms on a free group of rank 2. Tokyo J. Math. 14 (1991), 277-304.
38. S. Ito and M. Ohtsuki. Modified Jacobi-Perron algorithm and generating Markov partitions for special hyperbolic toral automorphisms. Tokyo J. Math. 16 (1993), 441-473.
39. S. Ito and M. Ohtsuki. Parallelogram tilings and Jacobi-Perron algorithm. Tokyo J. Math. 17 (1994), 33-59.
40. R. Kenyon. Self-similar tilings. Ph.D. Thesis, Princeton University, 1990.
41. R. Kenyon. Self-replicating tilings. Contemp. Math. 135 (1992), 239-263.
42. R. Kenyon. Inflationary tilings with similarity structure. Comment. Math. Helv. 69 (1994), 169-198.
43. R. Kenyon. The construction of self-similar tilings. Geometric and Funct. Anal. 6 (1996), 471-488.
44. n. Advances in Math. 121 (1996), 21-49.
45. n. I. Standard and nonstandard digit sets. J. London Math. Soc. 53 (1996).
46. J. C. Lagarias and Y. Wang. Tiling the line with translates of one tile. Preprint, 1995.
47. D. Lind. The entropies of topological Markov shifts and a related class of algebraic integers. Ergod. Th. & Dynam. Sys. 4 (1984), 283-300.
48. A. N. Livshits. On the spectra of adic transformations of Markov compacta. Russian Math. Surveys 42 (1987), 222-223.
49. A. N. Livshits. A sufficient condition for weak mixing of substitutions and stationary adic transformations. Math. Notes 44 (1988), 920-925.
50. A. N. Livshits. Some examples of adic transformations and substitutions. Selecta Math. Sovietica 11 (1992), 83-104.
51. J. M. Luck, C. Godrèche, A. Janner and T. Janssen. The nature of atomic surfaces of quasiperiodic self-similar structures. J. Phys. A: Math. Gen. 26 (1993), 1951-1999.
52. W. F. Lunnon and P. A. B. Pleasants. Quasicrystallographic tilings. J. Math. Pures et Appl. 66 (1987), 217-263.
53. C. Mauduit. Caractérisation des ensembles normaux substitutifs. Inventiones Math. 95 (1989), 133-147.
54. P. Michel. Stricte ergodicite d'ensembles minimaux de substitutions. C. R. Acad. Sci. Paris 278 (1974), 811-813.
55. P. Michel. Coincidence values and spectra of substitutions. Zeit. Wahr. 42 (1978), 205-227.
56. B. Mossé. Puissances de mots et reconnaisabilité des point fixes d'une substitution. Theor. Comp. Sci. 99 (1992), 327-334.
57. S. Mozes. Tilings, substitution systems and dynamical systems generated by them. J. d'Analyse Math. 53 (1989), 139-186.
58. P. Narbel. Limits and boundaries of word and tiling substitutions. Thèse, Université Paris 7, 1993.
59. W. Parry. On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401-416.
60. R. Penrose. The role of aesthetics in pure and applied mathematical research. Bull. Inst. Math. Appl. 10 (1974), 266-271.
61. C. Petronio. A class of self-similar tilings of the plane. Preprint, Scuola Normale Superiore, Pisa, 1992.
62. J. Peyrière. Frequency of patterns in certain graphs and in Penrose tilings. J. Phys. Colloque 47 (1986), 217-263.
63. B. Praggastis. Markov partitions for hyperbolic toral automorphisms. Ph.D. Thesis, University of Washington, 1992.
64. M. Queffelec. Substitution Dynamical Systems - Spectral Analysis (Lecture Notes in Mathematics 1294) Springer, 1987.
65. C. Radin. Symmetry of tilings of the plane. Bull. AMS 29 (1993), 213-217.
66. C. Radin. Space tilings and substitutions. Geometriae Dedicata 55 (1995), 257-264.
67. C. Radin. The pinwheel tiling of the plane. Annals of Math. 139 (1994), 661-702.
68. C. Radin and M. Wolff. Space tilings and local isomorphism. Geometriae Dedicata 42 (1992), 355-360.
69. G. Rauzy. Nombres algébriques et substitutions. Bull. Soc. Math. France 110 (1982), 147-178.
70. A. Rényi. Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957), 477-493.
71. d-Actions. Eds M. Pollicott and K. Schmidt. Cambridge University Press, pp. 451-173.
72. E. A. Robinson, Jr. The dynamical properties of Penrose tilings. Trans. Amer. Math. Soc. 348 (1996), 4447-4464.
73. E. A. Robinson, Jr. The point spectra of quasilattices. In preparation.
74. n actions. Contemp. Math. 94 (1989), 271-290.
75. D. Ruelle. Thermodynamic Formalism. Addison-Wesley, 1978.
76. R. Salem. Algebraic Numbers and Fourier Analysis. D. C. Heath and Co., Boston, 1963.
77. M. Senechal. Quasicrystals and Geometry. Cambridge University Press, 1995.
78. E. Seneta. Non-negative Matrices and Markov Chains. Springer, New York, 1981.
79. B. Solomyak. On the spectral theory of adic transformations. Advances of Soviet Math. 9 (1992), 217-230.
80. B. Solomyak. Substitutions, adic transformations and beta-expansions. Contemp. Math. 135 (1992), 361-372.
81. B. Solomyak. Non-periodic self-similar tilings have the unique composition property. Preprint, University of Washington, 1996. To appear in J. Disc. Comp. Geom.
82. B. Solomyak. Spectrum of dynamical systems arising from Delone sets. Preprint, University of Washington, 1996.
83. W. Thurston. Groups, tilings, and finite state automata. AMS Colloquium Lecture Notes, Boulder, 1989.
84. A. Vince. Replicating tesselations. SIAM J. Disc. Math. 6 (1993), 501-521.