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

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

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

Jeong-Yup Lee ^1, and Boris Solomyak ^2,

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

Abstract
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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.

Keywords: no weakly mixing, Meyer sets., Pisot family, discrete spectrum, self-affine tilings.

Mathematics Subject Classification: Primary: 37B50; Secondary: 52C2.

Citation: Jeong-Yup Lee, Boris Solomyak. Pisot family self-affine tilings, discrete spectrum, and the Meyer property. Discrete and Continuous Dynamical Systems, 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, Vol. 19, Academic Press, New York-London, 1966.
[2]	S. Akiyama and J.-Y. Lee, Algorithm for determining pure pointedness of self-affine tilings, Adv. Math., 226 (2011), 2855-2883. doi: 10.1016/j.aim.2010.07.019.
[3]	J. Andersen and I. Putnam, Topological invariants for substitution tilings and their associated $C^\mathbf{star}$-algebras, Ergodic Theory Dynam. Systems, 18 (1998), 509-537. doi: 10.1017/S0143385798100457.
[4]	M. Baake and D. Lenz, Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra, Ergodic Theory Dynam. Systems, 24 (2004), 1867-1893. doi: 10.1017/S0143385704000318.
[5]	M. Baake, D. Lenz and R. V. Moody, Characterization of model sets by dynamical systems, Ergodic Theory Dynam. Systems, 27 (2007), 341-382. doi: 10.1017/S0143385706000800.
[6]	R. Benedetti and J.-M. Gambaudo, On the dynamics of $\mathbb G$-solenoids. Applications to Delone sets, Ergodic Theory Dynam. Systems, 23 (2003), 673-691. doi: 10.1017/S0143385702001578.
[7]	A. Clark and L. Sadun, When shape matters: Deformations of tiling spaces, Ergodic Theory Dynam. Systems, 26 (2006), 69-86. doi: 10.1017/S0143385705000623.
[8]	L. Danzer, Inflation species of planar tilings which are not of locally finite complexity, Proc. Steklov Inst. Math., 239 (2002), 108-116.
[9]	S. Dworkin, Spectral theory and $x$-ray diffraction, J. Math. Phys., 34 (1993), 2965-2967. doi: 10.1063/1.530108.
[10]	N. P. Frank, A primer of substitution tilings of the Euclidean plane, Expo. Math., 26 (2008), 295-326. doi: 10.1016/j.exmath.2008.02.001.
[11]	N. P. Frank and E. A. Robinson, Jr., Generalized $\beta$-expansions, substitution tilings, and local finiteness, Trans. Amer. Math. Soc., 360 (2008), 1163-1177. doi: 10.1090/S0002-9947-07-04527-8.
[12]	J.-M. Gambaudo, A note on tilings and translation surfaces, Ergodic Theory Dynam. Systems, 26 (2006), 179-188. doi: 10.1017/S0143385705000404.
[13]	J.-B. Gouéré, Quasicrystals and almost periodicity, Comm. Math. Phys., 255 (2005), 655-681. doi: 10.1007/s00220-004-1271-8.
[14]	M. Hirsch and S. Smale, "Differential Equations, Dynamical Systems, and Linear Algebra,'' Pure and Applied Mathematics, Vol. 60, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974.
[15]	C. Holton, C. Radin and L. Sadun, Conjugacies for tiling dynamical systems, Comm. Math. Phys., 254 (2005), 343-359. doi: 10.1007/s00220-004-1195-3.
[16]	J. Kellendonk, Pattern equivariant functions, deformations and equivalence of tiling spaces, Ergodic Theory Dynam. Systems, 28 (2008), 1153-1176. doi: 10.1017/S014338570700065X.
[17]	R. Kenyon, Self-replicating tilings, in "Symbolic Dynamics and its Application" (New Haven, CT, 1991), 239-263,
[18]	R. Kenyon, Inflationary tilings with a similarity structure, Comment. Math. Helv., 69 (1994), 169-198. doi: 10.1007/BF02564481.
[19]	R. Kenyon, The construction of self-similar tilings, Geom. Funct. Anal., 6 (1996), 471-488. doi: 10.1007/BF02249260.
[20]	R. Kenyon, "Self-Similar Tilings,'' Ph.D Thesis, Princeton University, NJ, 1990.
[21]	R. Kenyon and B. Solomyak, On the characterization of expansion maps for self-affine tilings, Discrete Comput. Geom., 43 (2010), 577-593.
[22]	J. C. Lagarias, Mathematical quasicrystals and the problem of diffraction, in "Directions in Mathematical Quasicrystals'' (ed. M. Baake and R. V. Moody), CRM Monograph Series, Vol. 13, AMS, Providence, RI, (2000), 61-93.
[23]	J. C. Lagarias and Y. Wang, Substitution Delone sets, Discrete Comput. Geom., 29 (2003), 175-209. doi: 10.1007/s00454-002-2820-6.
[24]	J.-Y. Lee, Substitution Delone sets with pure point spectrum are inter-model sets, J. Geom. Phys., 57 (2007), 2263-2285. doi: 10.1016/j.geomphys.2007.07.003.
[25]	J.-Y. Lee, R. V. Moody and B. Solomyak, Pure point dynamical and diffraction spectra, Ann. Henri Poincaré, 3 (2002), 1003-1018. doi: 10.1007/s00023-002-8646-1.
[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), 525-560. doi: 10.1007/s00454-003-0781-z.
[27]	J.-Y Lee and B. Solomyak, Pure point diffractive substitution Delone sets have the Meyer property, Discrete Comput. Geom., 39 (2008), 319-338. doi: 10.1007/s00454-008-9054-1.
[28]	D. Lind, The entropies of topological Markov shifts and a related class of algebraic integers, Ergodic Theory Dynam. Systems, 4 (1984), 283-300.
[29]	C. Mauduit, Caractérisation des ensembles normaux substitutifs, Invent. Math., 95 (1989), 133-147. doi: 10.1007/BF01394146.
[30]	R. V. Moody, Meyer sets and their duals, in "The Mathematics of Long-Range Aperiodic Order" (Waterloo, ON, 1995) (ed. R. V. Moody), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 489, Kluwer Acad. Publ., Dordrecht, (1997), 403-441.
[31]	S. Mozes, Tilings, substitution systems and dynamical systems generated by them, J. Anal. Math., 53 (1989), 139-186. doi: 10.1007/BF02793412.
[32]	K. Petersen, Factor maps between tiling dynamical systems, Forum Math., 11 (1999), 503-512. doi: 10.1515/form.1999.011.
[33]	B. Praggastis, Numeration systems and Markov partitions from self-similar tilings, Trans. Amer. Math. Soc., 351 (1999), 3315-3349. doi: 10.1090/S0002-9947-99-02360-0.
[34]	C. Radin, The pinwheel tilings of the plane, Annals of Math., 139 (1994), 661-702. doi: 10.2307/2118575.
[35]	E. A. Robinson, Symbolic dynamics and tilings of $\mathbbR^d$, in "Symbolic Dynamics and its Applications," 81-119, Proc. Sympos. Appl. Math., 60, Amer. Math. Soc., Providence, RI, 2004.
[36]	L. Sadun, Some generalizations of the Pinwheel tiling, Discrete Comput. Geom., 20 (1998), 79-110. doi: 10.1007/PL00009379.
[37]	L. Sadun, "Topology of Tiling Spaces,'' University Lecture Series, 46, Amer. Math. Soc., Providence, RI, 2008.
[38]	B. Solomyak, Corrections to: "Dynamics of self-similar tilings", [Ergodic Theory Dynam. Systems 17 (1997), 695-738], Ergodic Theory Dynam. Systems, 19 (1999), 1685. doi: 10.1017/S014338579917161X.
[39]	B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete Comput. Geom., 20 (1998), 265-279. doi: 10.1007/PL00009386.
[40]	B. Solomyak, Eigenfunctions for substitution tiling systems, in "Probability and Number Theory-Kanazawa 2005,'' 433-454, Adv. Stud. Pure Math., 49, Math. Soc., Japan, Tokyo, 2007.
[41]	B. Solomyak, Tilings and dynamics, Lecture Notes, EMS Summer School on Combinatorics, Automata and Number Theory, 8-19 May 2006, unpublished manuscript. Available from: http://www.math.washington.edu/ solomyak/PREPRINTS/notes6.pdf.
[42]	E. M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,'' With the assistance of Timothy S. Murphy, Princeton Mathematical Series, 43, Monographs in Harmonic Analysis, III, Princeton University Press, Princeton, NJ, 1993.
[43]	W. Thurston, "Groups, Tilings, and Finite State Automata,'' AMS lecture notes, 1989.
[44]	T. Vijayaraghavan, On the fractional parts of the powers of a number. II, Proc. Cambridge Philos. Soc., 37 (1941), 349-357. doi: 10.1017/S0305004100017989.

show all references

References:

[1]	J. Aczél, "Lectures on Functional Equations and Their Applications,'' Mathematics in Science and Engineering, Vol. 19, Academic Press, New York-London, 1966.
[2]	S. Akiyama and J.-Y. Lee, Algorithm for determining pure pointedness of self-affine tilings, Adv. Math., 226 (2011), 2855-2883. doi: 10.1016/j.aim.2010.07.019.
[3]	J. Andersen and I. Putnam, Topological invariants for substitution tilings and their associated $C^\mathbf{star}$-algebras, Ergodic Theory Dynam. Systems, 18 (1998), 509-537. doi: 10.1017/S0143385798100457.
[4]	M. Baake and D. Lenz, Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra, Ergodic Theory Dynam. Systems, 24 (2004), 1867-1893. doi: 10.1017/S0143385704000318.
[5]	M. Baake, D. Lenz and R. V. Moody, Characterization of model sets by dynamical systems, Ergodic Theory Dynam. Systems, 27 (2007), 341-382. doi: 10.1017/S0143385706000800.
[6]	R. Benedetti and J.-M. Gambaudo, On the dynamics of $\mathbb G$-solenoids. Applications to Delone sets, Ergodic Theory Dynam. Systems, 23 (2003), 673-691. doi: 10.1017/S0143385702001578.
[7]	A. Clark and L. Sadun, When shape matters: Deformations of tiling spaces, Ergodic Theory Dynam. Systems, 26 (2006), 69-86. doi: 10.1017/S0143385705000623.
[8]	L. Danzer, Inflation species of planar tilings which are not of locally finite complexity, Proc. Steklov Inst. Math., 239 (2002), 108-116.
[9]	S. Dworkin, Spectral theory and $x$-ray diffraction, J. Math. Phys., 34 (1993), 2965-2967. doi: 10.1063/1.530108.
[10]	N. P. Frank, A primer of substitution tilings of the Euclidean plane, Expo. Math., 26 (2008), 295-326. doi: 10.1016/j.exmath.2008.02.001.
[11]	N. P. Frank and E. A. Robinson, Jr., Generalized $\beta$-expansions, substitution tilings, and local finiteness, Trans. Amer. Math. Soc., 360 (2008), 1163-1177. doi: 10.1090/S0002-9947-07-04527-8.
[12]	J.-M. Gambaudo, A note on tilings and translation surfaces, Ergodic Theory Dynam. Systems, 26 (2006), 179-188. doi: 10.1017/S0143385705000404.
[13]	J.-B. Gouéré, Quasicrystals and almost periodicity, Comm. Math. Phys., 255 (2005), 655-681. doi: 10.1007/s00220-004-1271-8.
[14]	M. Hirsch and S. Smale, "Differential Equations, Dynamical Systems, and Linear Algebra,'' Pure and Applied Mathematics, Vol. 60, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974.
[15]	C. Holton, C. Radin and L. Sadun, Conjugacies for tiling dynamical systems, Comm. Math. Phys., 254 (2005), 343-359. doi: 10.1007/s00220-004-1195-3.
[16]	J. Kellendonk, Pattern equivariant functions, deformations and equivalence of tiling spaces, Ergodic Theory Dynam. Systems, 28 (2008), 1153-1176. doi: 10.1017/S014338570700065X.
[17]	R. Kenyon, Self-replicating tilings, in "Symbolic Dynamics and its Application" (New Haven, CT, 1991), 239-263,
[18]	R. Kenyon, Inflationary tilings with a similarity structure, Comment. Math. Helv., 69 (1994), 169-198. doi: 10.1007/BF02564481.
[19]	R. Kenyon, The construction of self-similar tilings, Geom. Funct. Anal., 6 (1996), 471-488. doi: 10.1007/BF02249260.
[20]	R. Kenyon, "Self-Similar Tilings,'' Ph.D Thesis, Princeton University, NJ, 1990.
[21]	R. Kenyon and B. Solomyak, On the characterization of expansion maps for self-affine tilings, Discrete Comput. Geom., 43 (2010), 577-593.
[22]	J. C. Lagarias, Mathematical quasicrystals and the problem of diffraction, in "Directions in Mathematical Quasicrystals'' (ed. M. Baake and R. V. Moody), CRM Monograph Series, Vol. 13, AMS, Providence, RI, (2000), 61-93.
[23]	J. C. Lagarias and Y. Wang, Substitution Delone sets, Discrete Comput. Geom., 29 (2003), 175-209. doi: 10.1007/s00454-002-2820-6.
[24]	J.-Y. Lee, Substitution Delone sets with pure point spectrum are inter-model sets, J. Geom. Phys., 57 (2007), 2263-2285. doi: 10.1016/j.geomphys.2007.07.003.
[25]	J.-Y. Lee, R. V. Moody and B. Solomyak, Pure point dynamical and diffraction spectra, Ann. Henri Poincaré, 3 (2002), 1003-1018. doi: 10.1007/s00023-002-8646-1.
[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), 525-560. doi: 10.1007/s00454-003-0781-z.
[27]	J.-Y Lee and B. Solomyak, Pure point diffractive substitution Delone sets have the Meyer property, Discrete Comput. Geom., 39 (2008), 319-338. doi: 10.1007/s00454-008-9054-1.
[28]	D. Lind, The entropies of topological Markov shifts and a related class of algebraic integers, Ergodic Theory Dynam. Systems, 4 (1984), 283-300.
[29]	C. Mauduit, Caractérisation des ensembles normaux substitutifs, Invent. Math., 95 (1989), 133-147. doi: 10.1007/BF01394146.
[30]	R. V. Moody, Meyer sets and their duals, in "The Mathematics of Long-Range Aperiodic Order" (Waterloo, ON, 1995) (ed. R. V. Moody), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 489, Kluwer Acad. Publ., Dordrecht, (1997), 403-441.
[31]	S. Mozes, Tilings, substitution systems and dynamical systems generated by them, J. Anal. Math., 53 (1989), 139-186. doi: 10.1007/BF02793412.
[32]	K. Petersen, Factor maps between tiling dynamical systems, Forum Math., 11 (1999), 503-512. doi: 10.1515/form.1999.011.
[33]	B. Praggastis, Numeration systems and Markov partitions from self-similar tilings, Trans. Amer. Math. Soc., 351 (1999), 3315-3349. doi: 10.1090/S0002-9947-99-02360-0.
[34]	C. Radin, The pinwheel tilings of the plane, Annals of Math., 139 (1994), 661-702. doi: 10.2307/2118575.
[35]	E. A. Robinson, Symbolic dynamics and tilings of $\mathbbR^d$, in "Symbolic Dynamics and its Applications," 81-119, Proc. Sympos. Appl. Math., 60, Amer. Math. Soc., Providence, RI, 2004.
[36]	L. Sadun, Some generalizations of the Pinwheel tiling, Discrete Comput. Geom., 20 (1998), 79-110. doi: 10.1007/PL00009379.
[37]	L. Sadun, "Topology of Tiling Spaces,'' University Lecture Series, 46, Amer. Math. Soc., Providence, RI, 2008.
[38]	B. Solomyak, Corrections to: "Dynamics of self-similar tilings", [Ergodic Theory Dynam. Systems 17 (1997), 695-738], Ergodic Theory Dynam. Systems, 19 (1999), 1685. doi: 10.1017/S014338579917161X.
[39]	B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete Comput. Geom., 20 (1998), 265-279. doi: 10.1007/PL00009386.
[40]	B. Solomyak, Eigenfunctions for substitution tiling systems, in "Probability and Number Theory-Kanazawa 2005,'' 433-454, Adv. Stud. Pure Math., 49, Math. Soc., Japan, Tokyo, 2007.
[41]	B. Solomyak, Tilings and dynamics, Lecture Notes, EMS Summer School on Combinatorics, Automata and Number Theory, 8-19 May 2006, unpublished manuscript. Available from: http://www.math.washington.edu/ solomyak/PREPRINTS/notes6.pdf.
[42]	E. M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,'' With the assistance of Timothy S. Murphy, Princeton Mathematical Series, 43, Monographs in Harmonic Analysis, III, Princeton University Press, Princeton, NJ, 1993.
[43]	W. Thurston, "Groups, Tilings, and Finite State Automata,'' AMS lecture notes, 1989.
[44]	T. Vijayaraghavan, On the fractional parts of the powers of a number. II, Proc. Cambridge Philos. Soc., 37 (1941), 349-357. doi: 10.1017/S0305004100017989.

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