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Combinatorics of one-dimensional simple Toeplitz subshifts

Part of: Global differential geometry Discrete mathematics in relation to computer science Topological dynamics

Published online by Cambridge University Press:  13 November 2018

DANIEL SELL
DANIEL SELL*
Affiliation:
Friedrich-Schiller-Universität Jena, Institut für Mathematik, 07743 Jena, Germany email [email protected]
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Abstract

This paper provides a systematic study of fundamental combinatorial properties of one-dimensional, two-sided infinite simple Toeplitz subshifts. Explicit formulas for the complexity function, the palindrome complexity function and the repetitivity function are proved. Moreover, a complete description of the de Bruijn graphs of the subshifts is given. Finally, the Boshernitzan condition is characterized in terms of combinatorial quantities, based on a recent result of Liu and Qu [Uniform convergence of Schrödinger cocycles over simple Toeplitz subshift. Ann. Henri Poincaré12(1) (2011), 153–172]. Particular simple characterizations are provided for simple Toeplitz subshifts that correspond to the orbital Schreier graphs of the family of Grigorchuk’s groups, a class of subshifts that serves as the main example throughout the paper.

Keywords

symbolic dynamicssimple Toeplitz wordscombinatorics on words

MSC classification

Primary: 68R15: Combinatorics on words
Secondary: 37B10: Symbolic dynamics 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 6 , June 2020 , pp. 1673 - 1714
Copyright
© Cambridge University Press, 2018

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References

Baake, M. and Grimm, U.. Aperiodic Order. Vol. 1 (Encyclopedia of Mathematics and its Applications, 149). Cambridge University Press, Cambridge, 2013, a mathematical invitation, with a foreword by Roger Penrose.Google Scholar
Baake, M., Jäger, T. and Lenz, D.. Toeplitz flows and model sets. Bull. Lond. Math. Soc. 48(4) (2016), 691–698.Google Scholar
Bartholdi, L., Grigorchuk, R. and Nekrashevych, V.. From fractal groups to fractal sets. Fractals in Graz 2001 (Trends in Mathematics). Birkhäuser, Basel, 2003, pp. 25–118.Google Scholar
Bartholdi, L., Grigorchuk, R. and Šuniḱ, Z.. Branch groups. Handbook of Algebra. Vol. 3 (Handbook of Algebra, 3). Elsevier/North-Holland, Amsterdam, 2003, pp. 989–1112.Google Scholar
Beckus, S. and Pogorzelski, F.. Spectrum of Lebesgue measure zero for Jacobi matrices of quasicrystals. Math. Phys. Anal. Geom. 16(3) (2013), 289–308.Google Scholar
Bellissard, J., Iochum, B., Scoppola, E. and Testard, D.. Spectral properties of one-dimensional quasi-crystals. Comm. Math. Phys. 125(3) (1989), 527–543.Google Scholar
Bon, N. M.. Topological full groups of minimal subshifts with subgroups of intermediate growth. J. Mod. Dyn. 9 (2015), 67–80.Google Scholar
Boshernitzan, M.. A unique ergodicity of minimal symbolic flows with linear block growth. J. Anal. Math. 44 (1984/85), 77–96.Google Scholar
Casdagli, M.. Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation. Comm. Math. Phys. 107(2) (1986), 295–318.Google Scholar
Cassaigne, J. and Karhumäki, J.. Toeplitz words, generalized periodicity and periodically iterated morphisms. Computing and Combinatorics (Xi’an, 1995) (Lecture Notes in Computer Science, 959). Springer, Berlin, 1995, pp. 244–253.Google Scholar
Damanik, D., Killip, R. and Lenz, D.. Uniform spectral properties of one-dimensional quasicrystals. III. 𝛼-continuity. Comm. Math. Phys. 212(1) (2000), 191–204.Google Scholar
Damanik, D. and Lenz, D.. Uniform spectral properties of one-dimensional quasicrystals. I. Absence of eigenvalues. Comm. Math. Phys. 207(3) (1999), 687–696.Google Scholar
Damanik, D. and Lenz, D.. Uniform spectral properties of one-dimensional quasicrystals. II. The Lyapunov exponent. Lett. Math. Phys. 50(4) (1999), 245–257.Google Scholar
Damanik, D. and Lenz, D.. A condition of Boshernitzan and uniform convergence in the multiplicative ergodic theorem. Duke Math. J. 133(1) (2006), 95–123.Google Scholar
Damanik, D., Liu, Q. and Qu, Y.. Spectral properties of Schrödinger operators with pattern Sturmian potentials. Preprint, 2015, arXiv:1511.03834.Google Scholar
Downarowicz, T.. Survey of odometers and Toeplitz flows. Algebraic and Topological Dynamics (Contemporary Mathematics, 385). American Mathematical Society, Providence, RI, 2005, pp. 7–37.Google Scholar
Dreher, F., Kesseböhmer, M., Mosbach, A., Samuel, T. and Steffens, M.. Regularity of aperiodic minimal subshifts. Bull. Math. Sci. https://doi.org/10.1007/s13373-017-0102-0. Published online 29 March 2017.Google Scholar
Francoeur, D., Nagnibeda, T. and Pérez, A.. (2017), in preparation/private communication.Google Scholar
Furman, A.. On the multiplicative ergodic theorem for uniquely ergodic systems. Ann. Inst. H. Poincaré Probab. Stat. 33(6) (1997), 797–815.Google Scholar
Gjini, N., Kamae, T., Bo, T. and Yu-Mei, X.. Maximal pattern complexity for Toeplitz words. Ergod. Th. & Dynam. Sys. 26(4) (2006), 1073–1086.Google Scholar
Grigorchuk, R.. On Burnside’s problem on periodic groups. Funktsional. Anal. i Prilozhen. 14(1) (1980), 53–54, English translation: Funct. Anal. Appl. 14(1) (1980), 41–43.Google Scholar
Grigorchuk, R.. Degrees of growth of finitely generated groups and the theory of invariant means. Izv. Akad. Nauk SSSR Ser. Mat. 48(5) (1984), 939–985, English translation: Mathematics of the USSR-Izvestiya 25(2) (1985), 259–300.Google Scholar
R. Grigorchuk, D. Lenz and T. Nagnibeda, in preparation.Google Scholar
Grigorchuk, R., Lenz, D. and Nagnibeda, T.. Combinatorics of the subshift associated with Grigorchuk’s group. Tr. Mat. Inst. Steklova 297(Poryadok i Khaos v Dinamicheskikh Sistemakh) (2017), 158–164, English translation: Proc. Steklov Inst. Math. 297(1) (2017), 138–144.Google Scholar
Grigorchuk, R., Lenz, D. and Nagnibeda, T.. Schreier graphs of Grigorchuk’s group and a subshift associated to a nonprimitive substitution. Groups, Graphs and Random Walks (London Mathematical Society Lecture Note Series, 436). Cambridge University Press, Cambridge, 2017, pp. 250–299.Google Scholar
Grigorchuk, R., Lenz, D. and Nagnibeda, T.. Spectra of Schreier graphs of Grigorchuk’s group and Schrödinger operators with aperiodic order. Math. Ann. 370(3) (2018), 1607–1637.Google Scholar
Gröger, M. and Jäger, T.. Some remarks on modified power entropy. Dynamics and Numbers (Contemporary Mathematics, 669). American Mathematical Society, Providence, RI, 2016, pp. 105–122.Google Scholar
Gröger, M., Kesseböhmer, M., Mosbach, A., Samuel, T. and Steffens, M.. A classification of aperiodic order via spectral metrics & Jarník sets. Ergod. Th. & Dynam. Sys. https://doi.org/10.1017/etds.2018.7. Published online 13 March 2018.Google Scholar
Jacobs, K. and Keane, M.. 0–1-sequences of Toeplitz type. Z. Wahrscheinlichkeitstheor. Verw. Geb. 13 (1969), 123–131.Google Scholar
Kamae, T. and Zamboni, L.. Maximal pattern complexity for discrete systems. Ergod. Th. & Dynam. Sys. 22(4) (2002), 1201–1214.Google Scholar
Kamae, T. and Zamboni, L.. Sequence entropy and the maximal pattern complexity of infinite words. Ergod. Th. & Dynam. Sys. 22(4) (2002), 1191–1199.Google Scholar
Kellendonk, J., Lenz, D. and Savinien, J. (Eds). Mathematics of Aperiodic Order (Progress in Mathematics, 309). Birkhäuser/Springer, Basel, 2015.Google Scholar
Kohmoto, M., Kadanoff, L. P. and Tang, C.. Localization problem in one dimension: mapping and escape. Phys. Rev. Lett. 50(23) (1983), 1870–1872.Google Scholar
Koskas, M.. Complexités de suites de Toeplitz. Discrete Math. 183(1–3) (1998), 161–183.Google Scholar
Liu, Q. and Qu, Y.. Uniform convergence of Schrödinger cocycles over simple Toeplitz subshift. Ann. Henri Poincaré 12(1) (2011), 153–172.Google Scholar
Liu, Q. and Qu, Y.. Uniform convergence of Schrödinger cocycles over bounded Toeplitz subshift. Ann. Henri Poincaré 13(6) (2012), 1483–1500.Google Scholar
Morse, M. and Hedlund, G. A.. Symbolic dynamics. Amer. J. Math. 60(4) (1938), 815–866.Google Scholar
Nekrashevych, V.. Self-similar Groups (Mathematical Surveys and Monographs, 117). American Mathematical Society, Providence, RI, 2005.Google Scholar
Ostlund, S., Pandit, R., Rand, D., Schellnhuber, H. J. and Siggia, E. D.. One-dimensional Schrödinger equation with an almost periodic potential. Phys. Rev. Lett. 50(23) (1983), 1873–1876.Google Scholar
Qu, Y., Rao, H., Wen, Z. and Xue, Y.. Maximal pattern complexity of higher dimensional words. J. Combin. Theory Ser. A 117(5) (2010), 489–506.Google Scholar
Sütő, A.. The spectrum of a quasiperiodic Schrödinger operator. Comm. Math. Phys. 111(3) (1987), 409–415.Google Scholar
Sütő, A.. Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian. J. Statist. Phys. 56(3–4) (1989), 525–531.Google Scholar
Vorobets, Y.. On a substitution subshift related to the Grigorchuk group. Tr. Mat. Inst. Steklova 271(Differentsialnye Uravneniya i Topologiya. II) (2010), 319–334; Engl. transl. Proc. Steklov Inst. Math. 271(1) (2010), 306–321.Google Scholar
Williams, S.. Toeplitz minimal flows which are not uniquely ergodic. Z. Wahrscheinlichkeitstheor. Verw. Geb. 67(1) (1984), 95–107.Google Scholar