# American Institute of Mathematical Sciences

March  2012, 32(3): 935-959. doi: 10.3934/dcds.2012.32.935

## Pisot family self-affine tilings, discrete spectrum, and the Meyer property

 1 Dept. of Math. Edu., Kwandong University, 522 Naegok-dong, Gangneung, Gangwon 210-701, South Korea 2 Box 354350, Department of Mathematics, University of Washington, Seattle WA 98195, United States

Received  September 2010 Revised  March 2011 Published  October 2011

We consider self-affine tilings in the Euclidean space and the associated tiling dynamical systems, namely, the translation action on the orbit closure of the given tiling. We investigate the spectral properties of the system. It turns out that the presence of the discrete component depends on the algebraic properties of the eigenvalues of the expansion matrix $\phi$ for the tiling. Assuming that $\phi$ is diagonalizable over $\mathbb{C}$ and all its eigenvalues are algebraic conjugates of the same multiplicity, we show that the dynamical system has a relatively dense discrete spectrum if and only if it is not weakly mixing, and if and only if the spectrum of $\phi$ is a "Pisot family." Moreover, this is equivalent to the Meyer property of the associated discrete set of "control points" for the tiling.
Citation: Jeong-Yup Lee, Boris Solomyak. Pisot family self-affine tilings, discrete spectrum, and the Meyer property. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 935-959. doi: 10.3934/dcds.2012.32.935
##### References:
 [1] J. Aczél, "Lectures on Functional Equations and Their Applications,'', Mathematics in Science and Engineering, (1966). Google Scholar [2] S. Akiyama and J.-Y. Lee, Algorithm for determining pure pointedness of self-affine tilings,, Adv. Math., 226 (2011), 2855. doi: 10.1016/j.aim.2010.07.019. Google Scholar [3] J. Andersen and I. Putnam, Topological invariants for substitution tilings and their associated $C^\mathbf{star}$-algebras,, Ergodic Theory Dynam. Systems, 18 (1998), 509. doi: 10.1017/S0143385798100457. Google Scholar [4] M. Baake and D. Lenz, Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra,, Ergodic Theory Dynam. Systems, 24 (2004), 1867. doi: 10.1017/S0143385704000318. Google Scholar [5] M. Baake, D. Lenz and R. V. Moody, Characterization of model sets by dynamical systems,, Ergodic Theory Dynam. Systems, 27 (2007), 341. doi: 10.1017/S0143385706000800. Google Scholar [6] R. Benedetti and J.-M. Gambaudo, On the dynamics of $\mathbb G$-solenoids. Applications to Delone sets,, Ergodic Theory Dynam. Systems, 23 (2003), 673. doi: 10.1017/S0143385702001578. Google Scholar [7] A. Clark and L. Sadun, When shape matters: Deformations of tiling spaces,, Ergodic Theory Dynam. Systems, 26 (2006), 69. doi: 10.1017/S0143385705000623. Google Scholar [8] L. Danzer, Inflation species of planar tilings which are not of locally finite complexity,, Proc. Steklov Inst. Math., 239 (2002), 108. Google Scholar [9] S. Dworkin, Spectral theory and $x$-ray diffraction,, J. Math. Phys., 34 (1993), 2965. doi: 10.1063/1.530108. Google Scholar [10] N. P. Frank, A primer of substitution tilings of the Euclidean plane,, Expo. Math., 26 (2008), 295. doi: 10.1016/j.exmath.2008.02.001. Google Scholar [11] N. P. Frank and E. A. Robinson, Jr., Generalized $\beta$-expansions, substitution tilings, and local finiteness,, Trans. Amer. Math. Soc., 360 (2008), 1163. doi: 10.1090/S0002-9947-07-04527-8. Google Scholar [12] J.-M. Gambaudo, A note on tilings and translation surfaces,, Ergodic Theory Dynam. Systems, 26 (2006), 179. doi: 10.1017/S0143385705000404. Google Scholar [13] J.-B. Gouéré, Quasicrystals and almost periodicity,, Comm. Math. Phys., 255 (2005), 655. doi: 10.1007/s00220-004-1271-8. Google Scholar [14] M. Hirsch and S. Smale, "Differential Equations, Dynamical Systems, and Linear Algebra,'', Pure and Applied Mathematics, (1974). Google Scholar [15] C. Holton, C. Radin and L. Sadun, Conjugacies for tiling dynamical systems,, Comm. Math. Phys., 254 (2005), 343. doi: 10.1007/s00220-004-1195-3. Google Scholar [16] J. Kellendonk, Pattern equivariant functions, deformations and equivalence of tiling spaces,, Ergodic Theory Dynam. Systems, 28 (2008), 1153. doi: 10.1017/S014338570700065X. Google Scholar [17] R. Kenyon, Self-replicating tilings,, in, (1991), 239. Google Scholar [18] R. Kenyon, Inflationary tilings with a similarity structure,, Comment. Math. Helv., 69 (1994), 169. doi: 10.1007/BF02564481. Google Scholar [19] R. Kenyon, The construction of self-similar tilings,, Geom. Funct. Anal., 6 (1996), 471. doi: 10.1007/BF02249260. Google Scholar [20] R. Kenyon, "Self-Similar Tilings,'', Ph.D Thesis, (1990). Google Scholar [21] R. Kenyon and B. Solomyak, On the characterization of expansion maps for self-affine tilings,, Discrete Comput. Geom., 43 (2010), 577. Google Scholar [22] J. C. Lagarias, Mathematical quasicrystals and the problem of diffraction,, in, (2000), 61. Google Scholar [23] J. C. Lagarias and Y. Wang, Substitution Delone sets,, Discrete Comput. Geom., 29 (2003), 175. doi: 10.1007/s00454-002-2820-6. Google Scholar [24] J.-Y. Lee, Substitution Delone sets with pure point spectrum are inter-model sets,, J. Geom. Phys., 57 (2007), 2263. doi: 10.1016/j.geomphys.2007.07.003. Google Scholar [25] J.-Y. Lee, R. V. Moody and B. Solomyak, Pure point dynamical and diffraction spectra,, Ann. Henri Poincaré, 3 (2002), 1003. doi: 10.1007/s00023-002-8646-1. Google Scholar [26] J.-Y. Lee, R. V. Moody and B. Solomyak, Consequences of pure point diffraction spectra for multiset substitution systems,, Discrete Comp. Geom., 29 (2003), 525. doi: 10.1007/s00454-003-0781-z. Google Scholar [27] J.-Y Lee and B. Solomyak, Pure point diffractive substitution Delone sets have the Meyer property,, Discrete Comput. Geom., 39 (2008), 319. doi: 10.1007/s00454-008-9054-1. Google Scholar [28] D. Lind, The entropies of topological Markov shifts and a related class of algebraic integers,, Ergodic Theory Dynam. Systems, 4 (1984), 283. Google Scholar [29] C. Mauduit, Caractérisation des ensembles normaux substitutifs,, Invent. Math., 95 (1989), 133. doi: 10.1007/BF01394146. Google Scholar [30] R. V. Moody, Meyer sets and their duals,, in, (1997), 403. Google Scholar [31] S. Mozes, Tilings, substitution systems and dynamical systems generated by them,, J. Anal. Math., 53 (1989), 139. doi: 10.1007/BF02793412. Google Scholar [32] K. Petersen, Factor maps between tiling dynamical systems,, Forum Math., 11 (1999), 503. doi: 10.1515/form.1999.011. Google Scholar [33] B. Praggastis, Numeration systems and Markov partitions from self-similar tilings,, Trans. Amer. Math. Soc., 351 (1999), 3315. doi: 10.1090/S0002-9947-99-02360-0. Google Scholar [34] C. Radin, The pinwheel tilings of the plane,, Annals of Math., 139 (1994), 661. doi: 10.2307/2118575. Google Scholar [35] E. A. Robinson, Symbolic dynamics and tilings of $\mathbbR^d$, in "Symbolic Dynamics and its Applications,", 81-119, 60 (2004), 81. Google Scholar [36] L. Sadun, Some generalizations of the Pinwheel tiling,, Discrete Comput. Geom., 20 (1998), 79. doi: 10.1007/PL00009379. Google Scholar [37] L. Sadun, "Topology of Tiling Spaces,'', University Lecture Series, 46 (2008). Google Scholar [38] B. Solomyak, Corrections to: "Dynamics of self-similar tilings",, [Ergodic Theory Dynam. Systems 17 (1997), 17 (1997), 695. doi: 10.1017/S014338579917161X. Google Scholar [39] B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings,, Discrete Comput. Geom., 20 (1998), 265. doi: 10.1007/PL00009386. Google Scholar [40] B. Solomyak, Eigenfunctions for substitution tiling systems,, in, 49 (2005), 433. Google Scholar [41] B. Solomyak, Tilings and dynamics,, Lecture Notes, (2006), 8. Google Scholar [42] E. M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,'', With the assistance of Timothy S. Murphy, 43 (1993). Google Scholar [43] W. Thurston, "Groups, Tilings, and Finite State Automata,'', AMS lecture notes, (1989). Google Scholar [44] T. Vijayaraghavan, On the fractional parts of the powers of a number. II,, Proc. Cambridge Philos. Soc., 37 (1941), 349. doi: 10.1017/S0305004100017989. Google Scholar

show all references

##### References:
 [1] J. Aczél, "Lectures on Functional Equations and Their Applications,'', Mathematics in Science and Engineering, (1966). Google Scholar [2] S. Akiyama and J.-Y. Lee, Algorithm for determining pure pointedness of self-affine tilings,, Adv. Math., 226 (2011), 2855. doi: 10.1016/j.aim.2010.07.019. Google Scholar [3] J. Andersen and I. Putnam, Topological invariants for substitution tilings and their associated $C^\mathbf{star}$-algebras,, Ergodic Theory Dynam. Systems, 18 (1998), 509. doi: 10.1017/S0143385798100457. Google Scholar [4] M. Baake and D. Lenz, Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra,, Ergodic Theory Dynam. Systems, 24 (2004), 1867. doi: 10.1017/S0143385704000318. Google Scholar [5] M. Baake, D. Lenz and R. V. Moody, Characterization of model sets by dynamical systems,, Ergodic Theory Dynam. Systems, 27 (2007), 341. doi: 10.1017/S0143385706000800. Google Scholar [6] R. Benedetti and J.-M. Gambaudo, On the dynamics of $\mathbb G$-solenoids. Applications to Delone sets,, Ergodic Theory Dynam. Systems, 23 (2003), 673. doi: 10.1017/S0143385702001578. Google Scholar [7] A. Clark and L. Sadun, When shape matters: Deformations of tiling spaces,, Ergodic Theory Dynam. Systems, 26 (2006), 69. doi: 10.1017/S0143385705000623. Google Scholar [8] L. Danzer, Inflation species of planar tilings which are not of locally finite complexity,, Proc. Steklov Inst. Math., 239 (2002), 108. Google Scholar [9] S. Dworkin, Spectral theory and $x$-ray diffraction,, J. Math. Phys., 34 (1993), 2965. doi: 10.1063/1.530108. Google Scholar [10] N. P. Frank, A primer of substitution tilings of the Euclidean plane,, Expo. Math., 26 (2008), 295. doi: 10.1016/j.exmath.2008.02.001. Google Scholar [11] N. P. Frank and E. A. Robinson, Jr., Generalized $\beta$-expansions, substitution tilings, and local finiteness,, Trans. Amer. Math. Soc., 360 (2008), 1163. doi: 10.1090/S0002-9947-07-04527-8. Google Scholar [12] J.-M. Gambaudo, A note on tilings and translation surfaces,, Ergodic Theory Dynam. Systems, 26 (2006), 179. doi: 10.1017/S0143385705000404. Google Scholar [13] J.-B. Gouéré, Quasicrystals and almost periodicity,, Comm. Math. Phys., 255 (2005), 655. doi: 10.1007/s00220-004-1271-8. Google Scholar [14] M. Hirsch and S. Smale, "Differential Equations, Dynamical Systems, and Linear Algebra,'', Pure and Applied Mathematics, (1974). Google Scholar [15] C. Holton, C. Radin and L. Sadun, Conjugacies for tiling dynamical systems,, Comm. Math. Phys., 254 (2005), 343. doi: 10.1007/s00220-004-1195-3. Google Scholar [16] J. Kellendonk, Pattern equivariant functions, deformations and equivalence of tiling spaces,, Ergodic Theory Dynam. Systems, 28 (2008), 1153. doi: 10.1017/S014338570700065X. Google Scholar [17] R. Kenyon, Self-replicating tilings,, in, (1991), 239. Google Scholar [18] R. Kenyon, Inflationary tilings with a similarity structure,, Comment. Math. Helv., 69 (1994), 169. doi: 10.1007/BF02564481. Google Scholar [19] R. Kenyon, The construction of self-similar tilings,, Geom. Funct. Anal., 6 (1996), 471. doi: 10.1007/BF02249260. Google Scholar [20] R. Kenyon, "Self-Similar Tilings,'', Ph.D Thesis, (1990). Google Scholar [21] R. Kenyon and B. Solomyak, On the characterization of expansion maps for self-affine tilings,, Discrete Comput. Geom., 43 (2010), 577. Google Scholar [22] J. C. Lagarias, Mathematical quasicrystals and the problem of diffraction,, in, (2000), 61. Google Scholar [23] J. C. Lagarias and Y. Wang, Substitution Delone sets,, Discrete Comput. Geom., 29 (2003), 175. doi: 10.1007/s00454-002-2820-6. Google Scholar [24] J.-Y. Lee, Substitution Delone sets with pure point spectrum are inter-model sets,, J. Geom. Phys., 57 (2007), 2263. doi: 10.1016/j.geomphys.2007.07.003. Google Scholar [25] J.-Y. Lee, R. V. Moody and B. Solomyak, Pure point dynamical and diffraction spectra,, Ann. Henri Poincaré, 3 (2002), 1003. doi: 10.1007/s00023-002-8646-1. Google Scholar [26] J.-Y. Lee, R. V. Moody and B. Solomyak, Consequences of pure point diffraction spectra for multiset substitution systems,, Discrete Comp. Geom., 29 (2003), 525. doi: 10.1007/s00454-003-0781-z. Google Scholar [27] J.-Y Lee and B. Solomyak, Pure point diffractive substitution Delone sets have the Meyer property,, Discrete Comput. Geom., 39 (2008), 319. doi: 10.1007/s00454-008-9054-1. Google Scholar [28] D. Lind, The entropies of topological Markov shifts and a related class of algebraic integers,, Ergodic Theory Dynam. Systems, 4 (1984), 283. Google Scholar [29] C. Mauduit, Caractérisation des ensembles normaux substitutifs,, Invent. Math., 95 (1989), 133. doi: 10.1007/BF01394146. Google Scholar [30] R. V. Moody, Meyer sets and their duals,, in, (1997), 403. Google Scholar [31] S. Mozes, Tilings, substitution systems and dynamical systems generated by them,, J. Anal. Math., 53 (1989), 139. doi: 10.1007/BF02793412. Google Scholar [32] K. Petersen, Factor maps between tiling dynamical systems,, Forum Math., 11 (1999), 503. doi: 10.1515/form.1999.011. Google Scholar [33] B. Praggastis, Numeration systems and Markov partitions from self-similar tilings,, Trans. Amer. Math. Soc., 351 (1999), 3315. doi: 10.1090/S0002-9947-99-02360-0. Google Scholar [34] C. Radin, The pinwheel tilings of the plane,, Annals of Math., 139 (1994), 661. doi: 10.2307/2118575. Google Scholar [35] E. A. Robinson, Symbolic dynamics and tilings of $\mathbbR^d$, in "Symbolic Dynamics and its Applications,", 81-119, 60 (2004), 81. Google Scholar [36] L. Sadun, Some generalizations of the Pinwheel tiling,, Discrete Comput. Geom., 20 (1998), 79. doi: 10.1007/PL00009379. Google Scholar [37] L. Sadun, "Topology of Tiling Spaces,'', University Lecture Series, 46 (2008). Google Scholar [38] B. Solomyak, Corrections to: "Dynamics of self-similar tilings",, [Ergodic Theory Dynam. Systems 17 (1997), 17 (1997), 695. doi: 10.1017/S014338579917161X. Google Scholar [39] B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings,, Discrete Comput. Geom., 20 (1998), 265. doi: 10.1007/PL00009386. Google Scholar [40] B. Solomyak, Eigenfunctions for substitution tiling systems,, in, 49 (2005), 433. Google Scholar [41] B. Solomyak, Tilings and dynamics,, Lecture Notes, (2006), 8. Google Scholar [42] E. M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,'', With the assistance of Timothy S. Murphy, 43 (1993). Google Scholar [43] W. Thurston, "Groups, Tilings, and Finite State Automata,'', AMS lecture notes, (1989). Google Scholar [44] T. Vijayaraghavan, On the fractional parts of the powers of a number. II,, Proc. Cambridge Philos. Soc., 37 (1941), 349. doi: 10.1017/S0305004100017989. Google Scholar
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