On the specification property and synchronization of unique \(q\)-expansions. (English) Zbl 1477.37019
Summary: Given a positive integer \(M\) and \(q\in (1,M+1]\) we consider expansions in base \(q\) for real numbers \(x\in [0,M/{q-1}]\) over the alphabet \(\{0,\dots,M\}\). In particular, we study some dynamical properties of the natural occurring subshift \((\boldsymbol{V}_q,\sigma)\) related to unique expansions in such base \(q\). We characterize the set of \(q\in\mathcal{V}\subset (1,M+1]\) such that \((\boldsymbol{V}_q,\sigma)\) has the specification property and the set of \(q\in\mathcal{V}\) such that \((\boldsymbol{V}_q,\sigma)\) is a synchronized subshift. Such properties are studied by analysing the combinatorial and dynamical properties of the quasi-greedy expansion of \(q\). We also calculate the size of such classes as subsets of \(\mathcal{V}\) giving similar results to those shown by F. Blanchard in [Theor. Comput. Sci. 65, No. 2, 131–141 (1989; Zbl 0682.68081)]
and J. Schmeling in [Ergodic Theory Dyn. Syst. 17, No. 3, 675–694 (1997; Zbl 0908.58017)]
in the context of \(\beta\)-transformations.
MSC:
| 37B10 | Symbolic dynamics |
| 37B51 | Multidimensional shifts of finite type |
| 37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |
| 11A63 | Radix representation; digital problems |
| 37B40 | Topological entropy |
| 37C45 | Dimension theory of smooth dynamical systems |
| 68R15 | Combinatorics on words |
Keywords:
expansions; non-integer bases; specification property; synchronized systems; Hausdorff dimensionReferences:
| [1] | Alcaraz Barrera, R.. Topological and ergodic properties of symmetric sub-shifts. Discrete Contin. Dyn. Syst.34(11) (2014), 4459-4486. · Zbl 1351.37051 |
| [2] | Alcaraz Barrera, R., Baker, S. and Kong, D.. Entropy, topological transitivity, and dimensional properties of unique \(q\) -expansions. Trans. Amer. Math. Soc.371(5) (2019), 3209-3258. · Zbl 1435.11016 |
| [3] | Allaart, P. and Kong, D.. On the continuity of the Hausdorff dimension of the univoque set. Adv. Math.354 (2019), 106729. · Zbl 1473.11015 |
| [4] | Allaart, P. C.. On univoque and strongly univoque sets. Adv. Math.308 (2017), 575-598. · Zbl 1362.11074 |
| [5] | Allouche, J., Clarke, M. and Sidorov, N.. Periodic unique beta-expansions: the Sharkovskǐ ordering. Ergod. Th. & Dynam. Sys.29(4) (2009), 1055-1074. · Zbl 1279.11009 |
| [6] | Baiocchi, C. and Komornik, V.. Greedy and quasi-greedy expansions in non-integer bases. Preprint, 2007, https://arxiv.org/pdf/0710.3001.pdf. |
| [7] | Baker, S.. Generalized golden ratios over integer alphabets. Integers14 (2014), paper a15, 28. · Zbl 1285.11049 |
| [8] | Baker, S. and Ghenciu, A. E.. Dynamical properties of \(S\) -gap shifts and other shift spaces. J. Math. Anal. Appl.430(2) (2015), 633-647. · Zbl 1371.37022 |
| [9] | Bertrand-Mathis, A.. Développement en base \(\theta \) ; répartition modulo un de la suite \({(x{\theta}^n)}_{n\ge 0}\) ; langages codés et \(\theta \) -shift. Bull. Soc. Math. France114(3) (1986), 271-323. · Zbl 0628.58024 |
| [10] | Blanchard, F.. \( \beta \) -expansions and symbolic dynamics. Theoret. Comput. Sci.65(2) (1989), 131-141. · Zbl 0682.68081 |
| [11] | Boyle, M.. Algebraic aspects of symbolic dynamics. Topics in Symbolic Dynamics and Applications (Temuco, 1997)(London Mathematical Society Lecture Note Series, 279). Cambridge University Press, Cambridge, 2000, pp. 57-88. · Zbl 0967.37008 |
| [12] | Allaart, P. C.. An algebraic approach to entropy plateaus in non-integer base expansions. Discrete Contin. Dyn. Syst.39(11) (2019), 6507-6522. · Zbl 1473.11014 |
| [13] | Daróczy, Z. and Kátai, I.. On the structure of univoque numbers. Publ. Math. Debrecen46(3-4) (1995), 385-408. · Zbl 0874.11013 |
| [14] | De Vries, M. and Komornik, V.. Unique expansions of real numbers. Adv. Math.221(2) (2009), 390-427. · Zbl 1166.11007 |
| [15] | De Vries, M., Komornik, V. and Loreti, P.. Topology of the set of univoque bases. Topology Appl.205 (2016), 117-137. The Pisot Substitution Conjecture. · Zbl 1362.11010 |
| [16] | Erdős, P. and Joó, I.. On the number of expansions \(1=\sum{q}^{-{n}_i} \). Ann. Univ. Sci. Budapest. Eötvös Sect. Math.35 (1992), 129-132. · Zbl 0805.11011 |
| [17] | Erdős, P., Joó, I. and Komornik, V.. Characterization of the unique expansions \(1={\sum}_{i=1}^{\infty }{q}^{-{n}_i}\) and related problems. Bull. Soc. Math. France118(3) (1990), 377-390. · Zbl 0721.11005 |
| [18] | Falconer, K.. Fractal Geometry: Mathematical Foundations and Applications. John Wiley, Chichester, 1990. · Zbl 0689.28003 |
| [19] | Faller, B. and Pfister, C.-E.. A point is normal for almost all maps \(\beta\text{x}+\alpha mod1\) or generalized \(\beta \) -transformations. Ergod. Th. & Dynam. Sys.29(5) (2009), 1529-1547. · Zbl 1184.37034 |
| [20] | Fiebig, D. and Fiebig, U.. Invariants for subshifts via nested sequences of shifts of finite type. Ergod. Th. & Dynam. Sys.21(6) (2001), 1731-1758. · Zbl 1021.37006 |
| [21] | García-Ramos, F. and Pavlov, R.. Extender sets and measures of maximal entropy for subshifts. J. Lond. Math. Soc. (2)100(3) (2019), 1013-1033. · Zbl 1450.37011 |
| [22] | Ge, Y. and Tan, B.. Periodicity of the univoque \(\beta \) -expansions. Acta Math. Sci. Ser. B (Engl. Ed.)37(1) (2017), 33-46. · Zbl 1389.11017 |
| [23] | Kalle, C., Kong, D., Li, W. and Lü, F.. On the bifurcation set of unique expansions. Acta Arith.188(4) (2019), 367-399. · Zbl 1441.11018 |
| [24] | Kalle, C. and Steiner, W.. Beta-expansions, natural extensions and multiple tilings associated with Pisot units. Trans. Amer. Math. Soc.364(5) (2012), 2281-2318. · Zbl 1295.11010 |
| [25] | Komornik, V.. Expansions in noninteger bases. Integers11B (2011), paper a9, 30. · Zbl 1301.11008 |
| [26] | Komornik, V., Kong, D. and Li, W.. Hausdorff dimension of univoque sets and devil’s staircase. Adv. Math.305 (2017), 165-196. · Zbl 1362.11075 |
| [27] | Komornik, V. and Loreti, P.. Subexpansions, superexpansions and uniqueness properties in non-integer bases. Period. Math. Hungar.44(2) (2002), 197-218. · Zbl 1017.11008 |
| [28] | Komornik, V. and Loreti, P.. On the topological structure of univoque sets. J. Number Theory122(1) (2007), 157-183. · Zbl 1111.11005 |
| [29] | Kong, D. and Li, W.. Hausdorff dimension of unique beta expansions. Nonlinearity28(1) (2015), 187-209. · Zbl 1346.37011 |
| [30] | Krieger, W.. On Subshifts and Topological Markov Chains. Springer US, Boston, MA, 2000, pp. 453-472. · Zbl 0957.60073 |
| [31] | Kwietniak, D., Ła̧Cka, M. and Oprocha, P.. A panorama of specification-like properties and their consequences. Dynamics and Numbers(Contemporary Mathematics, 669). American Mathematical Society, Providence, RI, 2016, pp. 155-186. · Zbl 1376.37024 |
| [32] | Li, B., Sahlsten, T. and Samuel, T.. Intermediate \(\beta \) -shifts of finite type. Discrete Contin. Dyn. Syst.36(1) (2016), 323-344. · Zbl 1371.37023 |
| [33] | Lind, D. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995. · Zbl 1106.37301 |
| [34] | Parry, W.. On the \(\beta \) -expansions of real numbers. Acta Math. Acad. Sci. Hungar.11 (1960), 401-416. · Zbl 0099.28103 |
| [35] | Rényi, A.. Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar.8 (1957), 477-493. · Zbl 0079.08901 |
| [36] | Schmeling, J.. Symbolic dynamics for \(\beta \) -shifts and self-normal numbers. Ergod. Th. & Dynam. Sys.176 (1997), 675-694. · Zbl 0908.58017 |
| [37] | Sidorov, N.. Almost every number has a continuum of \(\beta \) -expansions. Amer. Math. Monthly110(9) (2003), 838-842. · Zbl 1049.11085 |
| [38] | Sidorov, N.. Arithmetic dynamics. Topics in Dynamics and Ergodic Theory(London Mathematical Society Lecture Note Series, 310). Cambridge University Press, Cambridge, 2003, pp. 145-189. · Zbl 1051.37007 |
| [39] | Sidorov, N.. Expansions in non-integer bases: lower, middle and top orders. J. Number Theory129(4) (2009), 741-754. · Zbl 1230.11090 |
| [40] | Urbański, M.. Invariant subsets of expanding mappings of the circleErgod. Th. & Dynam. Sys.7(4) (1987), 627-645. · Zbl 0653.58031 |
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