Pure discrete spectrum for one-dimensional substitution systems of Pisot type

Canadian Mathematical Bulletin

Volumn 45, Issue 4, 2002, Pages 697-710

  Sirvent, V F ^a   Solomyak, B ^b  

^a UNIVERSIDAD SIMÓN BOLÍVAR   (Venezuela)

^b UNIVERSITY OF WASHINGTON   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036898301     PISSN: 00084395     EISSN: None     Source Type: Journal    
DOI: 10.4153/CMB-2002-062-3     Document Type: Article

References (29)

1. P. Arnoux and S. Ito, Pisot substitutions and Rauzy fractats. Bull. Belg. Math. Soc. Simon Stevin 8(2001), 181-207.
2. M. Barge and B. Diamond, Coincidence for substitutions of Pisot type. 2001, preprint.
3. V. Canterini and A. Siegel, Geometric representation of substitutions of Pisot type. Trans. Amer. Math. Soc. 353(2001), 5121-5144.
4. F.M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length. Z. Wahrsch. Verw. Gebiete 41 (1978), 221-239.
5. S. Dworkin, Spectral theory and X-ray diffraction. J. Math. Phys. 34(1993), 2965-2967.
6. S. Ferenczi, C. Mauduit and A. Nogueira, Substitution dynamical systems: algebraic characterization of eigenvalues. Ann. Sci. École Norm. Sup. 29(1996), 519-533.
7. C. Frougny and B. Solomyak, Finite beta-expansions. Ergodic Theory Dynamical Systems 12(1992), 713-723.
8. A. Hof, Diffraction by aperiodic structures. In: The Mathematics of Long-Range Aperiodic Order, (ed. R.V. Moody), Kluwer, 1997, 239-268.
9. M. Hollander, Linear numeration systems, finite β-expansions, and discrete spectrum for substitution dynamical systems. Ph.D. Thesis, University of Washington, 1996.
10. M. Hollander and B. Solomyak, Two-symbol Pisot substitutions have pure discrete spectrum. 2001, preprint.
11. B. Host, Valeurs propres de systèmes dynamiques définis par de substitutions de longueur variable. Ergodic Theory Dynamical Systems 6(1986), 529-540.
12. _, Représentation géométrique des substitutions sur 2 lettres. Unpublished manuscript, 1992.
13. J.Y. Lee, R.V. Moody and B. Solomyak, Diffraction and Multi-dimensional Substitution Systems. 2001, preprint.
14. D. Lind and B. Marcus, Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.
15. A.N. Livshits, On the spectra of adic transformations of Markov compacta. Russian Math. Surveys 42(1987), 222 -223.
16. _, Some examples of adic transformations and substitutions. Selecta Math. Soviet. 11 (1992), 83-104.
17. P. Michel, Coincidence values and spectra of substitutions. Z. Wahrsch. Verw. Gebiete 42(1978), 205-227.
18. M. Queffélec, Substitution Dynamical Systems-Spectral Analysis. Lecture Notes in Math. 1294, Springer-Verlag, Berlin, 1987.
19. C. Rodin, Space tilings and substitutions. Geom. Dedicata 55(1995), 257-264.
20. C. Rodin and M. Wolff, Space tilings and local isomorphism. Geom. Dedicata 42 (1992), 355-360.
21. G. Rauzy, Nombres algébriques et substitutions. Bull. Soc. Math. Prance 110(1982), 147-178.
22. A. Siegel, Chapter 7: Spectral theory and geometric representation of substitutions. In: Substitutions in Dynamics, Arithmetics and Combinatorics, 199-252, (eds. V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel), Lecture Notes in Math., Springer, to appear.
23. _, Répresentation des systèmes dynamiques substitutifs non unimodulaires. Ergodic Theory Dynamical Systems, to appear.
24. V.F. Sirvent, Modelos geométricos asociados a substituciones. Trabajo de ascenso, Universidad Simón Bolívar, 1998.
25. V.F. Sirvent and Y. Wang, Self-affine tiling via substitution dynamical systems and Rauzy fractals. Pacific J. Math., to appear.
26. B. Solomyak, On the spectral theory of adic transformations. Adv. Soviet Math. 9(1992), 217-230.
27. _, Dynamics of self-similar tilings. Ergodic Theory Dynamical Systems 17(1997), 695-738. Corrections, ibid. 19(1999), 1685.
28. _, Dynamics of self-similar tilings. Ergodic Theory Dynamical Systems 17(1997), 695-738. Corrections, ibid. 19(1999), 1685.
29. P. Walters, An Introduction to Ergodic Theory. Springer Graduates Texts in Math., Springer-Verlag, New York, 1982.