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

1. 

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

Received  November 2014 Revised  June 2015 Published  August 2015

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.
Citation: Marcy Barge. Pure discrete spectrum for a class of one-dimensional substitution tiling systems. Discrete & Continuous Dynamical Systems - A, 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, (2015), 33. 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]

S. Akiyama and J.-Y. Lee, Computation of pure discrete spectrum of self-affine tilings,, preprint., (). Google Scholar

[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. doi: 10.1017/S0143385798100457. Google Scholar

[5]

P. Arnoux and S. Ito, Pisot Substitutions and Rauzy fractals,, Bull. Belg. Math Soc., 8 (2001), 181. Google Scholar

[6]

A. Avila and V. Delecroix, Some monoids of Pisot matrices, preprint,, , (). Google Scholar

[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. doi: 10.5802/aif.2238. Google Scholar

[8]

M. Barge, Factors of Pisot tiling spaces and the coincidence rank conjecture,, , (). Google Scholar

[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. doi: 10.1017/S0143385701001638. Google Scholar

[10]

M. Barge and B. Diamond, Coincidence for substitutions of Pisot type,, Bull. Soc. Math. France, 130 (2002), 619. Google Scholar

[11]

M. Barge and J. Kellendonk, Proximality and pure point spectrum for tiling dynamical systems,, Michigan Math. J., 62 (2013), 793. doi: 10.1307/mmj/1387226166. Google Scholar

[12]

M. Barge and J. Kwapisz, Geometric theory of unimodular Pisot substitutions,, Amer J. Math., (2006), 1219. doi: 10.1353/ajm.2006.0037. Google Scholar

[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. doi: 10.3934/dcds.2013.33.579. Google Scholar

[14]

V. Berthé, J. Bourdon, T. Jolivet and A. Siegel, A combinatorial approach to products of Pisot substitutions,, , (). Google Scholar

[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, (2005), 333. doi: 10.1090/conm/385/07205. Google Scholar

[16]

V. Berthé, T. Jolivet and A. Siegel, Substitutive Arnoux-Rauzy substitutions have pure discrete spectrum,, Unif. Distrib. Theory, 7 (2012), 173. Google Scholar

[17]

V. Berthé, Multidimensional Euclidean algorithms, numeration and substitutions,, Integers, 11B (2011). Google Scholar

[18]

A. Bertrand, Développements en base de Pisot et répartition modulo 1,, C. R. Acad. Sci. Paris, 280 (1979). Google Scholar

[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), 56 (2006), 2249. doi: 10.5802/aif.2239. Google Scholar

[20]

A. Clark and L. Sadun, When size matters: Subshifts and their related tiling spaces,, Ergodic Theory Dynam. Systems, 23 (2003), 1043. doi: 10.1017/S0143385702001633. Google Scholar

[21]

E. Dubois, A. Farhane and R. Paysant-Le Roux, The Jacobi-Perron algorithm and Pisot numbers,, Acta Arith., 111 (2004), 269. doi: 10.4064/aa111-3-4. Google Scholar

[22]

H. Ei and S. Ito, Tilings from some non-irreducible, Pisot substitutions,, Discrete Math. and Theo. Comp. Science, 7 (2005), 81. Google Scholar

[23]

M. Hollander and B. Solomyak, Two-symbol Pisot substitutions have pure discrete spectrum,, Ergodic Theory & Dynamical Systems, 23 (2003), 533. doi: 10.1017/S0143385702001384. Google Scholar

[24]

B. Mossé, Puissances de mots et reconnaissabilité des points fixes d'une substitution,, Theoretical Computer Science, 99 (1992), 327. doi: 10.1016/0304-3975(92)90357-L. Google Scholar

[25]

F. Schweiger, Multidimensional Continued Fractions,, Oxford Science Publications, (2000). Google Scholar

[26]

K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers,, Bull. London Math. Soc., 12 (1980), 269. doi: 10.1112/blms/12.4.269. Google Scholar

[27]

N. Sidorov, Arithmetic dynamics,, in Topics in Dynamics and Ergodic Theory, (2003), 145. doi: 10.1017/CBO9780511546716.010. Google Scholar

[28]

B. Solomyak, Dynamics of self-similar tilings,, Ergod. Th. & Dynam. Sys., 17 (1997), 695. doi: 10.1017/S0143385797084988. Google Scholar

[29]

B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings,, Discrete Comput. Geometry, 20 (1998), 265. doi: 10.1007/PL00009386. Google Scholar

[30]

P. Walters, An Introduction to Ergodic Theory,, Springer-Verlag, (1982). Google Scholar

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, (2015), 33. 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]

S. Akiyama and J.-Y. Lee, Computation of pure discrete spectrum of self-affine tilings,, preprint., (). Google Scholar

[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. doi: 10.1017/S0143385798100457. Google Scholar

[5]

P. Arnoux and S. Ito, Pisot Substitutions and Rauzy fractals,, Bull. Belg. Math Soc., 8 (2001), 181. Google Scholar

[6]

A. Avila and V. Delecroix, Some monoids of Pisot matrices, preprint,, , (). Google Scholar

[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. doi: 10.5802/aif.2238. Google Scholar

[8]

M. Barge, Factors of Pisot tiling spaces and the coincidence rank conjecture,, , (). Google Scholar

[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. doi: 10.1017/S0143385701001638. Google Scholar

[10]

M. Barge and B. Diamond, Coincidence for substitutions of Pisot type,, Bull. Soc. Math. France, 130 (2002), 619. Google Scholar

[11]

M. Barge and J. Kellendonk, Proximality and pure point spectrum for tiling dynamical systems,, Michigan Math. J., 62 (2013), 793. doi: 10.1307/mmj/1387226166. Google Scholar

[12]

M. Barge and J. Kwapisz, Geometric theory of unimodular Pisot substitutions,, Amer J. Math., (2006), 1219. doi: 10.1353/ajm.2006.0037. Google Scholar

[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. doi: 10.3934/dcds.2013.33.579. Google Scholar

[14]

V. Berthé, J. Bourdon, T. Jolivet and A. Siegel, A combinatorial approach to products of Pisot substitutions,, , (). Google Scholar

[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, (2005), 333. doi: 10.1090/conm/385/07205. Google Scholar

[16]

V. Berthé, T. Jolivet and A. Siegel, Substitutive Arnoux-Rauzy substitutions have pure discrete spectrum,, Unif. Distrib. Theory, 7 (2012), 173. Google Scholar

[17]

V. Berthé, Multidimensional Euclidean algorithms, numeration and substitutions,, Integers, 11B (2011). Google Scholar

[18]

A. Bertrand, Développements en base de Pisot et répartition modulo 1,, C. R. Acad. Sci. Paris, 280 (1979). Google Scholar

[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), 56 (2006), 2249. doi: 10.5802/aif.2239. Google Scholar

[20]

A. Clark and L. Sadun, When size matters: Subshifts and their related tiling spaces,, Ergodic Theory Dynam. Systems, 23 (2003), 1043. doi: 10.1017/S0143385702001633. Google Scholar

[21]

E. Dubois, A. Farhane and R. Paysant-Le Roux, The Jacobi-Perron algorithm and Pisot numbers,, Acta Arith., 111 (2004), 269. doi: 10.4064/aa111-3-4. Google Scholar

[22]

H. Ei and S. Ito, Tilings from some non-irreducible, Pisot substitutions,, Discrete Math. and Theo. Comp. Science, 7 (2005), 81. Google Scholar

[23]

M. Hollander and B. Solomyak, Two-symbol Pisot substitutions have pure discrete spectrum,, Ergodic Theory & Dynamical Systems, 23 (2003), 533. doi: 10.1017/S0143385702001384. Google Scholar

[24]

B. Mossé, Puissances de mots et reconnaissabilité des points fixes d'une substitution,, Theoretical Computer Science, 99 (1992), 327. doi: 10.1016/0304-3975(92)90357-L. Google Scholar

[25]

F. Schweiger, Multidimensional Continued Fractions,, Oxford Science Publications, (2000). Google Scholar

[26]

K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers,, Bull. London Math. Soc., 12 (1980), 269. doi: 10.1112/blms/12.4.269. Google Scholar

[27]

N. Sidorov, Arithmetic dynamics,, in Topics in Dynamics and Ergodic Theory, (2003), 145. doi: 10.1017/CBO9780511546716.010. Google Scholar

[28]

B. Solomyak, Dynamics of self-similar tilings,, Ergod. Th. & Dynam. Sys., 17 (1997), 695. doi: 10.1017/S0143385797084988. Google Scholar

[29]

B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings,, Discrete Comput. Geometry, 20 (1998), 265. doi: 10.1007/PL00009386. Google Scholar

[30]

P. Walters, An Introduction to Ergodic Theory,, Springer-Verlag, (1982). Google Scholar

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