Skip to main content
SpringerLink
Log in

Introduction to Hierarchical Tiling Dynamical Systems

  • Chapter
  • First Online:
Substitution and Tiling Dynamics: Introduction to Self-inducing Structures

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2273))

Abstract

This chapter is about the tiling dynamical systems approach to the study of aperiodic order. We compare and contrast four related types of systems: ordinary (one-dimensional) symbolic systems, one-dimensional tiling systems, multidimensional \(\mathbb {Z}^d\)-systems, and multidimensional tiling systems. Aperiodically ordered structures are often hierarchical in nature, and there are a number of different yet related ways to define them. We will focus on what we are calling “supertile construction methods”: symbolic substitution in one and many dimensions, S-adic sequences, self-similar and pseudo-self-similar tilings, and fusion rules. The techniques of dynamical analysis of these systems are discussed and a number of results are surveyed. We conclude with a discussion of the spectral theory of supertile systems from both the dynamical and diffraction perspectives.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 29.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 39.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Ordinarily in the literature the shift map is given with notation like σ(x), so that x − j = σ j(x). We use the notation “x − j” instead to be consistent with the more general case.

  2. 2.

    FLC is a common restriction, but if you want to learn about the infinite local complexity case see [47] and references within.

  3. 3.

    Also known in the literature as \(\mathcal {T}\) being uniformly recurrent, almost periodic, and having the local isomorphism property.

  4. 4.

    For a more indepth discussion of the meaning of the word ‘frequency’ and the appropriate ergodic theorem for this setting, see [50, Section 3.3].

  5. 5.

    In the literature (notably [100, 102]) it is taken as given that ϕ is orientation preserving, which can be assumed by squaring any substitution that is not.

  6. 6.

    These are also known as inflate-and-subdivide rules and tiling substitution rules.

  7. 7.

    In general, non-primitive substitutions can have more complicated structure.

  8. 8.

    Actually, it is possible to see the process of fusion as a cutting and stacking process.

  9. 9.

    There is a separate lexicon in which what are known to some as “substitutions” are known to others as “non-erasing morphisms” and the set \({\mathcal A}^*\) is called the “free monoid” instead of the set of all finite words from \({\mathcal A}\) [6, 76].

  10. 10.

    See [64, 89] for symbolic substitutions, [87] for self-affine tilings, and [50] for fusions.

  11. 11.

    How to adapt the analysis appears in [50, section 3.7].

  12. 12.

    The general results on spectral theory of dynamical systems in this section can be found in many places, for instance [56, 71]; specialization to the tiling case first appears in [100].

  13. 13.

    One can show that this term is independent of the choice of vertex.

  14. 14.

    Recall that this is the largest topological factor of the dynamical system that is a rotation of a compact group.

  15. 15.

    This is also true for substitution sequences [62], but not for general fusions.

  16. 16.

    These sequences were actually defined using number-theoretic constructions, but have simple substitution rules also.

  17. 17.

    This is indicative of the ‘odometer’-like structure of constant-length substitutions: the shift map augments until a fixed number and then resets to 0, augmenting elsewhere. In general the supertile structure of any constant-length substitution looks like an odometer, see for example [45].

  18. 18.

    This volume contains an overview of the history and development of tilings and Delone sets in [101].

  19. 19.

    This is also known as the “natural” autocorrelation measure because the averaging sets used are balls centered at the origin as opposed to an arbitrary van Hove sequence.

  20. 20.

    Not a particularly restrictive subclass according to Section 5.1 of [70].

References

  1. Tilings encyclopedia (2018). http://tilings.math.uni-bielefeld.de/. Accessed Dec 2018

  2. S. Akiyama, M. Barge, V. Berthé, J.-Y. Lee, A. Siegel, On the Pisot Substitution Conjecture. Mathematics of Aperiodic Order, Progress in Mathematics, vol. 309 (Birkhäuser/Springer, Basel, 2015), pp. 33–72, MR 3381478

    Google Scholar 

  3. S. Akiyama, Strong coincidence and overlap coincidence. Discrete Contin. Dyn. Syst. 36(10), 5223–5230 (2016). MR 3543543

    Google Scholar 

  4. S. Akiyama, J.-Y. Lee, Algorithm for determining pure pointedness of self-affine tilings. Adv. Math. 226(4), 2855–2883 (2011). MR 2764877

    Google Scholar 

  5. S. Akiyama, J.-Y. Lee, Overlap coincidence to strong coincidence in substitution tiling dynamics. Eur. J. Combin. 39, 233–243 (2014). MR 3168528

    Google Scholar 

  6. J.-P. Allouche, J. Shallit, Automatic Sequences. Theory, Applications, Generalizations (Cambridge University Press, Cambridge, 2003), MR 1997038

    Google Scholar 

  7. P. Arnoux, M. Furukado, E. Harriss, S. Ito, Algebraic numbers, free group automorphisms and substitutions on the plane. Trans. Amer. Math. Soc. 363(9), 4651–4699 (2011). MR 2806687

    Google Scholar 

  8. P. Arnoux, S. Ito, Pisot substitutions and Rauzy fractals. Bull. Belg. Math. Soc. Simon Stevin 8(2), 181–207 (2001). Journées Montoises d’Informatique Théorique (Marne-la-Vallée, 2000). MR 1838930

    Google Scholar 

  9. M. Baake, N.P. Frank, U. Grimm, E.A. Robinson Jr., Geometric properties of a binary non-pisot inflation and absence of absolutely continuous diffraction. Studia Math. 247, 109–154 (2019)

    Article  MathSciNet  Google Scholar 

  10. M. Baake, F. Gähler, Pair correlations of aperiodic inflation rules via renormalisation: some interesting examples. Topology Appl. 205, 4–27 (2016). MR 3493304

    Google Scholar 

  11. M. Baake, U. Grimm, Aperiodic Order. Vol. 1. Encyclopedia of Mathematics and its Applications, vol. 149 (Cambridge University Press, Cambridge, 2013), A Mathematical Invitation, With a foreword by Roger Penrose. MR 3136260

    Google Scholar 

  12. M. Baake, U. Grimm, Squirals and beyond: substitution tilings with singular continuous spectrum. Ergodic Theory Dynam. Syst. 34(4), 1077–1102 (2014). MR 3227148

    Google Scholar 

  13. M. Baake, U. Grimm, Diffraction of a binary non-pisot inflation tiling. J. Phys. Conf. Ser. 809(1), 012026 (2017)

    Google Scholar 

  14. M. Baake, U. Grimm, N. Mañibo, Spectral analysis of a family of binary inflation rules. Lett. Math. Phys. 108(8), 1783–1805 (2018). MR 3814725

    Google Scholar 

  15. M. Baake, D. Lenz, Spectral notions of aperiodic order. Discrete Contin. Dyn. Syst. Ser. S 10(2), 161–190 (2017). MR 3600642

    Google Scholar 

  16. M. Baake, D. Lenz, A. van Enter, Dynamical versus diffraction spectrum for structures with finite local complexity. Ergodic Theory Dyn. Syst. 35(7), 2017–2043 (2015). MR 3394105

    Google Scholar 

  17. M. Barge, Pure discrete spectrum for a class of one-dimensional substitution tiling systems. Discrete Contin. Dyn. Syst. 36(3), 1159–1173 (2016). MR 3431249

    Google Scholar 

  18. M. Barge, B. Diamond, Coincidence for substitutions of Pisot type. Bull. Soc. Math. France 130(4), 619–626 (2002). MR 1947456

    Google Scholar 

  19. A. Bartlett, Spectral theory of \(\mathbb {Z}^d\) substitutions. Ergodic Theory Dyn. Syst. 38(4), 1289–1341 (2018). MR 3789166

    Google Scholar 

  20. N. Bédaride, A. Hilion, Geometric realizations of two-dimensional substitutive tilings. Q. J. Math. 64(4), 955–979 (2013). MR 3151599

    Google Scholar 

  21. J. Bellissard, A. Julien, J. Savinien, Tiling groupoids and Bratteli diagrams. Ann. Henri Poincaré 11(1–2), 69–99 (2010). MR 2658985

    Google Scholar 

  22. R. Berger, The undecidability of the domino problem. Mem. Amer. Math. Soc. No. 66, 72 (1966). MR 0216954

    Google Scholar 

  23. A. Berlinkov, B. Solomyak, Singular substitutions of constant length (2017). ArXiv:1705.00899v2

    Google Scholar 

  24. V. Berthé, V. Delecroix, Beyond Substitutive Dynamical Systems:S-adic Expansions (Numeration and Substitution 2012). RIMS Kôkyûroku Bessatsu, vol. B46, (Research Institute for Mathematical Sciences (RIMS), Kyoto, 2014), pp. 81–123. MR 3330561

    Google Scholar 

  25. S. Bezuglyi, J. Kwiatkowski, K. Medynets, Aperiodic substitution systems and their Bratteli diagrams. Ergod. Theory Dyn. Syst. 29(1), 37–72 (2009). MR 2470626

    Google Scholar 

  26. S. Bezuglyi, J. Kwiatkowski, K. Medynets, B. Solomyak, Invariant measures on stationary Bratteli diagrams. Ergod. Theory Dyn. Syst. 30(4), 973–1007 (2010). MR 2669408

    Google Scholar 

  27. S. Bezuglyi, J. Kwiatkowski, K. Medynets, B. Solomyak, Finite rank Bratteli diagrams: structure of invariant measures. Trans. Amer. Math. Soc. 365(5), 2637–2679 (2013). MR 3020111

    Google Scholar 

  28. E. Bombieri, J.E. Taylor, Which distributions of matter diffract? An initial investigation. J. Phys. 47(7) , Suppl. Colloq. C3, C3–19–C3–28 (1986). International workshop on aperiodic crystals (Les Houches, 1986). MR 866320

    Google Scholar 

  29. R.V. Chacon, A geometric construction of measure preserving transformations, in Proceedings of the Fifth Berkeley Symposium Mathematical. Statistics and Probability (Berkeley, California, 1965/66), Vol. II: Contributions to Probability Theory, Part 2 (University of California Press, Berkeley, 1967), pp. 335–360. MR 0212158

    Google Scholar 

  30. L. Chan, U. Grimm, Spectrum of a Rudin-Shapiro-like sequence. Adv. Appl. Math. 87, 16–23 (2017). MR 3629260

    Google Scholar 

  31. A. Clark, L. Sadun, When size matters: subshifts and their related tiling spaces. Ergod. Theory Dyn. Syst. 23(4), 1043–1057 (2003). MR 1997967

    Google Scholar 

  32. E.M. Coven, M.S. Keane, The structure of substitution minimal sets. Trans. Amer. Math. Soc. 162, 89–102 (1971). MR 0284995

    Google Scholar 

  33. F.M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 41(3), 221–239 (1977/1978). MR 0461470

    Google Scholar 

  34. F.M. Dekking, M. Keane, Mixing properties of substitutions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 42(1), 23–33 (1978). MR 0466485

    Google Scholar 

  35. F. Durand, Linearly recurrent subshifts have a finite number of non-periodic subshift factors. Ergod. Theory Dyn. Syst. 20(4), 1061–1078 (2000). MR 1779393

    Google Scholar 

  36. F. Durand, Corrigendum and addendum to: “Linearly recurrent subshifts have a finite number of non-periodic subshift factors” [Ergodic Theory Dynam. Systems 20(4), 1061–1078 (2000); MR1779393 (2001m:37022)]. Ergod. Theory Dyn. Syst. 23(2), 663–669 (2003). MR 1972245

    Google Scholar 

  37. S. Dworkin, Spectral theory and x-ray diffraction. J. Math. Phys. 34(7), 2965–2967 (1993). MR 1224190

    Google Scholar 

  38. S. Ferenczi, Rank and symbolic complexity. Ergod. Theory Dyn. Syst. 16(4), 663–682 (1996). MR 1406427

    Google Scholar 

  39. T. Fernique, Local rule substitutions and stepped surfaces. Theoret. Comput. Sci. 380(3), 317–329 (2007). MR 2331001

    Google Scholar 

  40. T. Fernique, N. Ollinger, Combinatorial substitutions and sofic tilings. Journées Automates Cellulaires (2010) (Turku). arXiv:1009.5167

    Google Scholar 

  41. A.M. Fisher, Nonstationary mixing and the unique ergodicity of adic transformations. Stoch. Dyn. 9(3), 335–391 (2009). MR 2566907

    Google Scholar 

  42. N.P. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, ed. by V. Berthé, S. Ferenczi, C. Mauduit, A. Siegel. Lecture Notes in Mathematics, vol. 1794 (Springer, Berlin, 2002). MR 1970385

    Google Scholar 

  43. N.P. Frank, Detecting combinatorial hierarchy in tilings using derived Voronoï tesselations. Discrete Comput. Geom. 29(3), 459–476 (2003). MR 1961011

    Google Scholar 

  44. N.P. Frank, Substitution sequences in \(\mathbb Z^d\) with a non-simple Lebesgue component in the spectrum. Ergod. Theory Dyn. Syst. 23(2), 519–532 (2003). MR 1972236

    Google Scholar 

  45. N.P. Frank, Multidimensional constant-length substitution sequences. Topol. Appl. 152(1–2), 44–69 (2005). MR 2160805

    Google Scholar 

  46. N.P. Frank, A primer of substitution tilings of the Euclidean plane. Expo. Math. 26(4), 295–326 (2008). MR 2462439

    Google Scholar 

  47. N.P. Frank, Tilings with Infinite Local Complexity. Mathematics of Aperiodic Order, Progress in Mathematics, vol. 309 (Birkhäuser/Springer, Basel, 2015), pp. 223–257. MR 3381483

    Google Scholar 

  48. N.P. Frank, E.A. Robinson, Jr., Generalized β-expansions, substitution tilings, and local finiteness. Trans. Amer. Math. Soc. 360(3), 1163–1177 (2008). MR 2357692

    Google Scholar 

  49. N.P. Frank, L. Sadun, Topology of some tiling spaces without finite local complexity. Discrete Contin. Dyn. Syst. 23(3), 847–865 (2009). MR 2461829

    Google Scholar 

  50. N.P. Frank, L. Sadun, Fusion: a general framework for hierarchical tilings of \(\mathbb {R}^d\). Geom. Dedicata 171, 149–186 (2014). MR 3226791

    Google Scholar 

  51. D. Frettlöh, More Inflation Tilings, in Aperiodic Order, Vol. 2. Encyclopedia of Mathematics and its Applications, vol. 166 (Cambridge University Press, Cambridge, 2017), pp. 1–37. MR 3791847

    Google Scholar 

  52. F. Gähler, R. Klitzing, The diffraction pattern of self-similar tilings, in The Mathematics of Long-Range Aperiodic Order (Waterloo, ON, 1995). NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 489 (Kluwer Academic Publishers, Dordrecht, 1997), pp. 141–174. MR 1460023

    Google Scholar 

  53. F. Gähler, E.E. Kwan, G.R. Maloney, A computer search for planar substitution tilings with n-fold rotational symmetry. Discrete Comput. Geom. 53(2), 445–465 (2015). MR 3316232

    Google Scholar 

  54. M. Gardner, Mathematical games. Sci. Amer. 236(1), 110–121 (1977)

    Article  Google Scholar 

  55. C. Godrèche, F. Lançon, A simple example of a non-Pisot tiling with five-fold symmetry. J. Physique I 2(2), 207–220 (1992). MR 1185612

    Google Scholar 

  56. G.R. Goodson, A survey of recent results in the spectral theory of ergodic dynamical systems. J. Dyn. Control Syst. 5(2), 173–226 (1999). MR 1693318

    Google Scholar 

  57. W.H. Gottschalk, Substitution minimal sets. Trans. Amer. Math. Soc. 109, 467–491 (1963). MR 0190915

    Google Scholar 

  58. B. Grünbaum, G.C. Shephard, Tilings and Patterns (W. H. Freeman and Company, New York, 1987). MR 857454

    Google Scholar 

  59. J.L. Herning, Spectrum and factors of substitution dynamical systems, ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)–The George Washington University, 2013. MR 3167382

    Google Scholar 

  60. A. Hof, On diffraction by aperiodic structures. Comm. Math. Phys. 169(1), 25–43 (1995). MR 1328260

    Google Scholar 

  61. M. Hollander, B. Solomyak, Two-symbol Pisot substitutions have pure discrete spectrum. Ergod. Theory Dyn. Syst. 23(2), 533–540 (2003). MR 1972237

    Google Scholar 

  62. B. Host, Valeurs propres des systèmes dynamiques définis par des substitutions de longueur variable. Ergod. Theory Dyn. Syst. 6(4), 529–540 (1986). MR 873430

    Google Scholar 

  63. E. Jeandel, P. Vanier, The Undecidability of the Domino Problem. This volume, Lecture Notes in Mathematics (Springer, Berlin, 2019)

    Google Scholar 

  64. T. Kamae, Spectrum of a substitution minimal set. J. Math. Soc. Japan 22, 567–578 (1970). MR 0286092

    Google Scholar 

  65. J. Kellendonk, L. Sadun, Meyer sets, topological eigenvalues, and Cantor fiber bundles. J. Lond. Math. Soc. (2) 89(1), 114–130 (2014). MR 3174736

    Google Scholar 

  66. R. Kenyon, B. Solomyak, On the characterization of expansion maps for self-affine tilings. Discrete Comput. Geom. 43(3), 577–593 (2010). MR 2587839

    Google Scholar 

  67. B.P. Kitchens, Symbolic Dynamics: One-sided, Two-sided and Countable State Markov Shifts. Universitext (Springer, Berlin, 1998). MR 1484730

    Google Scholar 

  68. J. Kwapisz, Inflations of self-affine tilings are integral algebraic Perron. Invent. Math. 205(1), 173–220 (2016). MR 3514961

    Google Scholar 

  69. J.-Y. Lee, R.V. Moody, B. Solomyak, Pure point dynamical and diffraction spectra. Ann. Henri Poincaré 3(5), 1003–1018 (2002). MR 1937612

    Google Scholar 

  70. J.-Y. Lee, R.V. Moody, B. Solomyak, Consequences of pure point diffraction spectra for multiset substitution systems. Discrete Comput. Geom. 29(4), 525–560 (2003). MR 1976605

    Google Scholar 

  71. M. Lemańczyk, Spectral Theory of Dynamical Systems. Mathematics of Complexity and Dynamical Systems, Vol. 1–3 (Springer, New York, 2012), pp. 1618–1638. MR 3220776

    Google Scholar 

  72. D. Lenz, Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks. Comm. Math. Phys. 287(1), 225–258 (2009). MR 2480747

    Google Scholar 

  73. S. Lidin, The Discovery of Quasicrystals: Scientific Background on the Nobel Prize in Chemistry 2011 (The Royal Swedish Academy of Sciences, Stockholm, 2011)

    Google Scholar 

  74. D. Lind, B. Marcus, An Introduction to Symbolic Dynamics and Coding (Cambridge University Press, Cambridge, 1995). MR 1369092

    Google Scholar 

  75. A.N. Livshits, On the spectra of adic transformations of Markov compact sets. Uspekhi Mat. Nauk 42(3(255)), 189–190 (1987). MR 896889

    Google Scholar 

  76. M. Lothaire, Algebraic Combinatorics on Words. Encyclopedia of Mathematics and its Applications, vol. 90 (Cambridge University Press, Cambridge, 2002), A collective work by J. Berstel, D. Perrin, P. Seebold, J. Cassaigne, A. De Luca, S. Varricchio, A. Lascoux, B. Leclerc, J.-Y. Thibon, V. Bruyere, C. Frougny, F. Mignosi, A. Restivo, C. Reutenauer, D. Foata, G.-N. Han, J. Desarmenien, V. Diekert, T. Harju, J. Karhumaki, W. Plandowski, With a preface by Berstel and Perrin. MR 1905123

    Google Scholar 

  77. A.L. Mackay, Crystallography and the Penrose pattern. Phys. A 114(1–3), 609–613 (1982). MR 678468

    Google Scholar 

  78. G.R. Maloney, D. Rust, Beyond primitivity for one-dimensional substitution subshifts and tiling spaces. Ergod. Theory Dyn. Syst. 38(3), 1086–1117 (2018). MR 3784255

    Google Scholar 

  79. B.F. Martensen, Generalized balanced pair algorithm. Topology Proc. 28(1), 163–178 (2004). Spring Topology and Dynamical Systems Conference. MR 2105455

    Google Scholar 

  80. J.C. Martin, Substitution minimal flows. Amer. J. Math. 93, 503–526 (1971). MR 0300261

    Google Scholar 

  81. B. Mossé, Puissances de mots et reconnaissabilité des points fixes d’une substitution. Theoret. Comput. Sci. 99(2), 327–334 (1992). MR 1168468

    Google Scholar 

  82. R. Penrose, The rôle of aesthetics in pure and applied mathematical research. Bull. Inst. Math. Appl. 10, 266–271 (1974)

    Google Scholar 

  83. R. Penrose, Pentaplexity: a class of nonperiodic tilings of the plane. Math. Intell. 2(1), 32–37 (1979/1980). MR 558670

    Google Scholar 

  84. K. Petersen, Ergodic Theory. Cambridge Studies in Advanced Mathematics, vol. 2 (Cambridge University Press, Cambridge, 1989, Corrected Reprint of the 1983 Original). MR 1073173

    Google Scholar 

  85. K. Petersen, Factor maps between tiling dynamical systems. Forum Math. 11(4), 503–512 (1999). MR 1699171

    Google Scholar 

  86. J. Peyrière, Frequency of patterns in certain graphs and in Penrose tilings. J. Physique 47(7), Suppl. Colloq. C3, C3–41–C3–62 (1986). International workshop on aperiodic crystals (Les Houches, 1986). MR 866322

    Google Scholar 

  87. B. Praggastis, Numeration systems and Markov partitions from self-similar tilings. Trans. Amer. Math. Soc. 351(8), 3315–3349 (1999). MR 1615950

    Google Scholar 

  88. N. Priebe, B. Solomyak, Characterization of planar pseudo-self-similar tilings. Discrete Comput. Geom. 26(3), 289–306 (2001). MR 1854103

    Google Scholar 

  89. M. Queffélec, Substitution Dynamical Systems—Spectral Analysis. Lecture Notes in Mathematics, vol. 1294 (Springer, Berlin, 1987). MR 924156

    Google Scholar 

  90. C. Radin, Miles of Tiles. Student Mathematical Library, vol. 1 (American Mathematical Society, Providence, 1999). MR 1707270

    Google Scholar 

  91. C. Radin, L. Sadun, Isomorphism of hierarchical structures. Ergod. Theory Dyn. Syst. 21(4), 1239–1248 (2001). MR 1849608

    Google Scholar 

  92. C. Radin, M. Wolff, Space tilings and local isomorphism. Geom. Dedicata 42(3), 355–360 (1992). MR 1164542

    Google Scholar 

  93. E.A. Robinson, Jr., Symbolic Dynamics and Tilings of \(\mathbb R^d\)s. Symbolic Dynamics and Its Applications, Proceedings of Symposia in Applied Mathematics, vol. 60 (American Mathematical Society, Providence, 2004), pp. 81–119. MR 2078847

    Google Scholar 

  94. R.M. Robinson, Undecidability and nonperiodicity for tilings of the plane. Invent. Math. 12, 177–209 (1971). MR 0297572

    Google Scholar 

  95. L. Sadun, Topology of Tiling Spaces. University Lecture Series, vol. 46 (American Mathematical Society, Providence, 2008). MR 2446623

    Google Scholar 

  96. M. Senechal, Quasicrystals and Geometry (Cambridge University Press, Cambridge, 1995). MR 1340198

    Google Scholar 

  97. D. Shechtman, I. Blech, D. Gratias, J.W. Cahn, Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951–1953 (1984)

    Article  Google Scholar 

  98. B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings. Discrete Comput. Geom. 20(2), 265–279 (1998). MR 1637896

    Google Scholar 

  99. B. Solomyak, Pseudo-self-affine tilings in \(\mathbb R^d\). Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. 326 (2005), Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 13, 198–213, 282–283 (2005); Translation in J. Math. Sci. 140(3), 452–460 (2007). MR 2183221

    Google Scholar 

  100. B. Solomyak, Dynamics of self-similar tilings. Ergodic Theory Dynam. Syst. 17(3), 695–738 (1997). MR 1452190

    Google Scholar 

  101. B. Solomyak, Lecture on Delone Sets and Tilings. This Volume, Lecture Notes in Mathematics (Springer, Berlin, 2019)

    Google Scholar 

  102. W. Thurston, Groups, Tilings, and Finite State Automata. AMS Colloquium Lecture Notes (American Mathematical Society, Providence, 1989)

    Google Scholar 

  103. J. Thuswaldner, S-adic Sequences. A Bridge Between Dynamics, Arithmetic, and Geometry. This Volume, Lecture Notes in Mathematics (Springer, Berlin, 2019)

    Google Scholar 

  104. P. Walters, An Introduction to Ergodic Theory. Graduate Texts in Mathematics, vol. 79 (Springer, New York, 1982). MR 648108

    Google Scholar 

  105. H. Wang, Proving theorems by pattern recognition, II. Bell Syst. Tech. J. 40, 1–41 (1961).

    Article  Google Scholar 

Download references

Acknowledgements

The author would like to thank Michael Baake, Franz Gähler, E. A. Robinson, Jr., Dan Rust, Lorenzo Sadun, and Boris Solomyak for their comments on drafts of this work.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Vassar College, Poughkeepsie, NY, USA

    Natalie Priebe Frank

Authors
  1. Natalie Priebe Frank
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Natalie Priebe Frank .

Editor information

Editors and Affiliations

  1. Institute of Mathematics, University of Tsukuba, Tsukuba, Japan

    Shigeki Akiyama

  2. Institut de Mathématiques de Marseille (I2M), Aix-Marseille University, Marseille Cedex 09, France

    Pierre Arnoux

Rights and permissions

Reprints and permissions

Copyright information

© 2020 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Frank, N.P. (2020). Introduction to Hierarchical Tiling Dynamical Systems. In: Akiyama, S., Arnoux, P. (eds) Substitution and Tiling Dynamics: Introduction to Self-inducing Structures. Lecture Notes in Mathematics, vol 2273. Springer, Cham. https://doi.org/10.1007/978-3-030-57666-0_2

Download citation

Publish with us

Policies and ethics

Search

Navigation