Pure discrete spectrum for a class of one-dimensional substitution tiling systems

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March 2016, 36(3): 1159-1173. doi: 10.3934/dcds.2016.36.1159

Pure discrete spectrum for a class of one-dimensional substitution tiling systems

Marcy Barge ^1,

1.	Department of Mathematics, Montana State University, Bozeman, MT 59717

Received November 2014 Revised June 2015 Published August 2015

Abstract
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We prove that if a primitive and non-periodic substitution is injective on initial letters, constant on final letters, and has Pisot inflation, then the $\mathbb{R}$-action on the corresponding tiling space has pure discrete spectrum. As a consequence, all $\beta$-substitutions for $\beta$ a Pisot simple Parry number have tiling dynamical systems with pure discrete spectrum, as do the Pisot systems arising, for example, from substitutions associated with the Jacobi-Perron and Brun continued fraction algorithms.

Keywords: Substitution, tiling space, maximal equicontinuous factor., discrete spectrum.

Mathematics Subject Classification: Primary: 37B05, 37B50; Secondary: 11A55, 11A6.

Citation: Marcy Barge. Pure discrete spectrum for a class of one-dimensional substitution tiling systems. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1159-1173. doi: 10.3934/dcds.2016.36.1159

References:

[1]	S. Akiyama, M. Barge, V. Berthé, J.-Y. Lee and A. Siegel, On the Pisot Conjecture, in Mathematics of Aperiodic Order (eds. J. Kellendonk, D. Lenz and J. Savinien), Progress in Mathematics, 309, 2015, 33-72.
[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]	S. Akiyama and J.-Y. Lee, Computation of pure discrete spectrum of self-affine tilings,, preprint., ().
[4]	J. E. Anderson and I. F. Putnam, Topological invariants for substitution tilings and their associated $C^$-algebras, Ergodic Theory & Dynamical Systems*, 18 (1998), 509-537. doi: 10.1017/S0143385798100457.
[5]	P. Arnoux and S. Ito, Pisot Substitutions and Rauzy fractals, Bull. Belg. Math Soc., 8 (2001), 181-207.
[6]	A. Avila and V. Delecroix, Some monoids of Pisot matrices, preprint,, , ().
[7]	V. Baker, M. Barge and J. Kwapisz, Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to $\beta$-shifts, Ann. Inst. Fourier (Grenoble), 56 (2006), 2213-2248. doi: 10.5802/aif.2238.
[8]	M. Barge, Factors of Pisot tiling spaces and the coincidence rank conjecture,, , ().
[9]	M. Barge and B. Diamond, A complete invariant for the topology of one- dimensional substitution tiling spaces, Ergod. Th. & Dyn. Sys., 21 (2001), 1333-1358. doi: 10.1017/S0143385701001638.
[10]	M. Barge and B. Diamond, Coincidence for substitutions of Pisot type, Bull. Soc. Math. France, 130 (2002), 619-626.
[11]	M. Barge and J. Kellendonk, Proximality and pure point spectrum for tiling dynamical systems, Michigan Math. J., 62 (2013), 793-822. doi: 10.1307/mmj/1387226166.
[12]	M. Barge and J. Kwapisz, Geometric theory of unimodular Pisot substitutions, Amer J. Math., {\bf128} (2006), 1219-1282. doi: 10.1353/ajm.2006.0037.
[13]	M. Barge, S. Štimac and R. F. Williams, Pure discrete spectrum in substitution tiling spaces, Disc. and Cont. Dynam. Sys. - A, 2 (2013), 579-597. doi: 10.3934/dcds.2013.33.579.
[14]	V. Berthé, J. Bourdon, T. Jolivet and A. Siegel, A combinatorial approach to products of Pisot substitutions,, , ().
[15]	V. Berthé, S. Ferenczi and L. Q. Zamboni, Interactions between dynamics, arihtmetics and combinatorics: The good, the bad, and the ugly, in Algebraic and Topological Dynamics, Contemp. Math., 385, Amer. Math. Soc., Providence, RI, 2005, 333-364. doi: 10.1090/conm/385/07205.
[16]	V. Berthé, T. Jolivet and A. Siegel, Substitutive Arnoux-Rauzy substitutions have pure discrete spectrum, Unif. Distrib. Theory, 7 (2012), 173-197.
[17]	V. Berthé, Multidimensional Euclidean algorithms, numeration and substitutions, Integers, 11B (2011), Paper No. A2, 34pp.
[18]	A. Bertrand, Développements en base de Pisot et répartition modulo 1, C. R. Acad. Sci. Paris, 280 (1979), A1-A4.
[19]	J. Cassaigne and N. Chekhova, Fonctions de récurrence des suites d'Arnoux-Rauzy et réponse à une questionde Morse et Hedlund, Ann. Inst. Fourier (Grenoble), Numération, Pavages, Substitutions, 56 (2006), 2249-2270. doi: 10.5802/aif.2239.
[20]	A. Clark and L. Sadun, When size matters: Subshifts and their related tiling spaces, Ergodic Theory Dynam. Systems, 23 (2003), 1043-1057. doi: 10.1017/S0143385702001633.
[21]	E. Dubois, A. Farhane and R. Paysant-Le Roux, The Jacobi-Perron algorithm and Pisot numbers, Acta Arith., 111 (2004), 269-275. doi: 10.4064/aa111-3-4.
[22]	H. Ei and S. Ito, Tilings from some non-irreducible, Pisot substitutions, Discrete Math. and Theo. Comp. Science, 7 (2005), 81-121.
[23]	M. Hollander and B. Solomyak, Two-symbol Pisot substitutions have pure discrete spectrum, Ergodic Theory & Dynamical Systems, 23 (2003), 533-540. doi: 10.1017/S0143385702001384.
[24]	B. Mossé, Puissances de mots et reconnaissabilité des points fixes d'une substitution, Theoretical Computer Science, 99 (1992), 327-334. doi: 10.1016/0304-3975(92)90357-L.
[25]	F. Schweiger, Multidimensional Continued Fractions, Oxford Science Publications, Oxford University Press, Oxford, 2000.
[26]	K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc., 12 (1980), 269-278. doi: 10.1112/blms/12.4.269.
[27]	N. Sidorov, Arithmetic dynamics, in Topics in Dynamics and Ergodic Theory, London Math. Soc. Lecture Note Ser., 310, Cambridge Univ. Press, Cambridge, 2003, 145-189. doi: 10.1017/CBO9780511546716.010.
[28]	B. Solomyak, Dynamics of self-similar tilings, Ergod. Th. & Dynam. Sys., 17 (1997), 695-738. doi: 10.1017/S0143385797084988.
[29]	B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete Comput. Geometry, 20 (1998), 265-279. doi: 10.1007/PL00009386.
[30]	P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York, 1982.

show all references

References:

[1]	S. Akiyama, M. Barge, V. Berthé, J.-Y. Lee and A. Siegel, On the Pisot Conjecture, in Mathematics of Aperiodic Order (eds. J. Kellendonk, D. Lenz and J. Savinien), Progress in Mathematics, 309, 2015, 33-72.
[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]	S. Akiyama and J.-Y. Lee, Computation of pure discrete spectrum of self-affine tilings,, preprint., ().
[4]	J. E. Anderson and I. F. Putnam, Topological invariants for substitution tilings and their associated $C^$-algebras, Ergodic Theory & Dynamical Systems*, 18 (1998), 509-537. doi: 10.1017/S0143385798100457.
[5]	P. Arnoux and S. Ito, Pisot Substitutions and Rauzy fractals, Bull. Belg. Math Soc., 8 (2001), 181-207.
[6]	A. Avila and V. Delecroix, Some monoids of Pisot matrices, preprint,, , ().
[7]	V. Baker, M. Barge and J. Kwapisz, Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to $\beta$-shifts, Ann. Inst. Fourier (Grenoble), 56 (2006), 2213-2248. doi: 10.5802/aif.2238.
[8]	M. Barge, Factors of Pisot tiling spaces and the coincidence rank conjecture,, , ().
[9]	M. Barge and B. Diamond, A complete invariant for the topology of one- dimensional substitution tiling spaces, Ergod. Th. & Dyn. Sys., 21 (2001), 1333-1358. doi: 10.1017/S0143385701001638.
[10]	M. Barge and B. Diamond, Coincidence for substitutions of Pisot type, Bull. Soc. Math. France, 130 (2002), 619-626.
[11]	M. Barge and J. Kellendonk, Proximality and pure point spectrum for tiling dynamical systems, Michigan Math. J., 62 (2013), 793-822. doi: 10.1307/mmj/1387226166.
[12]	M. Barge and J. Kwapisz, Geometric theory of unimodular Pisot substitutions, Amer J. Math., {\bf128} (2006), 1219-1282. doi: 10.1353/ajm.2006.0037.
[13]	M. Barge, S. Štimac and R. F. Williams, Pure discrete spectrum in substitution tiling spaces, Disc. and Cont. Dynam. Sys. - A, 2 (2013), 579-597. doi: 10.3934/dcds.2013.33.579.
[14]	V. Berthé, J. Bourdon, T. Jolivet and A. Siegel, A combinatorial approach to products of Pisot substitutions,, , ().
[15]	V. Berthé, S. Ferenczi and L. Q. Zamboni, Interactions between dynamics, arihtmetics and combinatorics: The good, the bad, and the ugly, in Algebraic and Topological Dynamics, Contemp. Math., 385, Amer. Math. Soc., Providence, RI, 2005, 333-364. doi: 10.1090/conm/385/07205.
[16]	V. Berthé, T. Jolivet and A. Siegel, Substitutive Arnoux-Rauzy substitutions have pure discrete spectrum, Unif. Distrib. Theory, 7 (2012), 173-197.
[17]	V. Berthé, Multidimensional Euclidean algorithms, numeration and substitutions, Integers, 11B (2011), Paper No. A2, 34pp.
[18]	A. Bertrand, Développements en base de Pisot et répartition modulo 1, C. R. Acad. Sci. Paris, 280 (1979), A1-A4.
[19]	J. Cassaigne and N. Chekhova, Fonctions de récurrence des suites d'Arnoux-Rauzy et réponse à une questionde Morse et Hedlund, Ann. Inst. Fourier (Grenoble), Numération, Pavages, Substitutions, 56 (2006), 2249-2270. doi: 10.5802/aif.2239.
[20]	A. Clark and L. Sadun, When size matters: Subshifts and their related tiling spaces, Ergodic Theory Dynam. Systems, 23 (2003), 1043-1057. doi: 10.1017/S0143385702001633.
[21]	E. Dubois, A. Farhane and R. Paysant-Le Roux, The Jacobi-Perron algorithm and Pisot numbers, Acta Arith., 111 (2004), 269-275. doi: 10.4064/aa111-3-4.
[22]	H. Ei and S. Ito, Tilings from some non-irreducible, Pisot substitutions, Discrete Math. and Theo. Comp. Science, 7 (2005), 81-121.
[23]	M. Hollander and B. Solomyak, Two-symbol Pisot substitutions have pure discrete spectrum, Ergodic Theory & Dynamical Systems, 23 (2003), 533-540. doi: 10.1017/S0143385702001384.
[24]	B. Mossé, Puissances de mots et reconnaissabilité des points fixes d'une substitution, Theoretical Computer Science, 99 (1992), 327-334. doi: 10.1016/0304-3975(92)90357-L.
[25]	F. Schweiger, Multidimensional Continued Fractions, Oxford Science Publications, Oxford University Press, Oxford, 2000.
[26]	K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc., 12 (1980), 269-278. doi: 10.1112/blms/12.4.269.
[27]	N. Sidorov, Arithmetic dynamics, in Topics in Dynamics and Ergodic Theory, London Math. Soc. Lecture Note Ser., 310, Cambridge Univ. Press, Cambridge, 2003, 145-189. doi: 10.1017/CBO9780511546716.010.
[28]	B. Solomyak, Dynamics of self-similar tilings, Ergod. Th. & Dynam. Sys., 17 (1997), 695-738. doi: 10.1017/S0143385797084988.
[29]	B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete Comput. Geometry, 20 (1998), 265-279. doi: 10.1007/PL00009386.
[30]	P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York, 1982.

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