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November  2019, 39(11): 6507-6522. doi: 10.3934/dcds.2019282

An algebraic approach to entropy plateaus in non-integer base expansions

Pieter C. Allaart

Mathematics Department, University of North Texas, 1155 Union Cir #311430, Denton, TX 76203-5017, USA

Received  December 2018 Revised  June 2019 Published  August 2019

For a positive integer $ M $ and a real base $ q\in(1, M+1] $, let $ {\mathcal{U}}_q $ denote the set of numbers having a unique expansion in base $ q $ over the alphabet $ \{0, 1, \dots, M\} $, and let $ \mathbf{U}_q $ denote the corresponding set of sequences in $ \{0, 1, \dots, M\}^ {\mathbb{N}} $. Komornik et al. [ Adv. Math. 305 (2017), 165–196] showed recently that the Hausdorff dimension of $ {\mathcal{U}}_q $ is given by $ h(\mathbf{U}_q)/\log q $, where $ h(\mathbf{U}_q) $ denotes the topological entropy of $ \mathbf{U}_q $. They furthermore showed that the function $ H: q\mapsto h(\mathbf{U}_q) $ is continuous, nondecreasing and locally constant almost everywhere. The plateaus of $ H $ were characterized by Alcaraz Barrera et al. [ Trans. Amer. Math. Soc., 371 (2019), 3209–3258]. In this article we reinterpret the results of Alcaraz Barrera et al. by introducing a notion of composition of fundamental words, and use this to obtain new information about the structure of the function $ H $. This method furthermore leads to a more streamlined proof of their main theorem.

Keywords: Beta-expansion, univoque set, topological entropy, entropy plateau, transitive subshift, composition of fundamental words.
Mathematics Subject Classification: Primary: 11A63; Secondary: 37B10, 37B40, 68R15.
Citation: Pieter C. Allaart. An algebraic approach to entropy plateaus in non-integer base expansions. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6507-6522. doi: 10.3934/dcds.2019282
References:
[1]

R. Alcaraz Barrera, Topological and ergodic properties of symmetric sub-shifts, Discrete Contin. Dyn. Syst., 34 (2014), 4459-4486.  doi: 10.3934/dcds.2014.34.4459.  Google Scholar

[2]

R. Alcaraz Barrera, S. Baker and D. Kong, Entropy, topological transitivity, and dimensional properties of unique $q$-expansions, Trans. Amer. Math. Soc., 371 (2019), 3209-3258.  doi: 10.1090/tran/7370.  Google Scholar

[3]

P. Allaart, S. Baker and D. Kong, Bifurcation sets arising from non-integer base expansions, in J. Fractal Geom., (2018), arXiv: 1706.05190. Google Scholar

[4]

P. Allaart and D. Kong, On the continuity of the Hausdorff dimension of the univoque set, Advances in Mathematics, 354 (2019), 106729, arXiv: 1804.02879. doi: 10.1016/j.aim.2019.106729.  Google Scholar

[5]

P. Allaart and D. Kong, Relative bifurcation sets and the local dimension of univoque bases, preprint, 2018, arXiv: 1809.00323. Google Scholar

[6]

S. Baker, Generalized golden ratios over integer alphabets, Integers, 14 (2014), 28 pp.  Google Scholar

[7]

M. de Vries and V. Komornik, Unique expansions of real numbers, Adv. Math., 221 (2009), 390-427.  doi: 10.1016/j.aim.2008.12.008.  Google Scholar

[8]

P. Erdős, M. Horváth and I. Joó, On the uniqueness of the expansions $1=\sum q^{-n_i}$, Acta Math. Hungar., 58 (1991), 333-342.  doi: 10.1007/BF01903963.  Google Scholar

[9]

P. Erdős and I. Joó, On the number of expansions $1=\sum q^{-n_i}$, Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 35 (1992), 129-132.   Google Scholar

[10]

P. Erdős, I. Joó and V. Komornik, Characterization of the unique expansions $1=\sum_{i=1}^\infty q^{-n_i}$ and related problems, Bull. Soc. Math. France, 118 (1990), 377-390.  doi: 10.24033/bsmf.2151.  Google Scholar

[11]

P. Glendinning and T. Hall, Zeros of the kneading invariant and topological entropy for Lorenz maps, Nonlinearity, 9 (1996), 999-1014.  doi: 10.1088/0951-7715/9/4/010.  Google Scholar

[12]

P. Glendinning and N. Sidorov, Unique representations of real numbers in non-integer bases, Math. Res. Lett., 8 (2001), 535-543.  doi: 10.4310/MRL.2001.v8.n4.a12.  Google Scholar

[13]

V. Komornik, D. Kong and W. X. Li, Hausdorff dimension of univoque sets and devil's staircase, Adv. Math., 305 (2017), 165-196.  doi: 10.1016/j.aim.2016.03.047.  Google Scholar

[14]

V. Komornik and P. Loreti, Unique developments in non-integer bases, Amer. Math. Monthly, 105 (1998), 636-639.  doi: 10.1080/00029890.1998.12004937.  Google Scholar

[15]

V. Komornik and P. Loreti, Subexpansions, superexpansions and uniqueness properties in non-integer bases, Period. Math. Hungar., 44 (2002), 197-218.  doi: 10.1023/A:1019696514372.  Google Scholar

[16]

D. Kong and W. X. Li, Hausdorff dimension of unique beta expansions, Nonlinearity, 28 (2015), 187-209.  doi: 10.1088/0951-7715/28/1/187.  Google Scholar

[17]

D. Kong, W. X. Li and F. M. Dekking, Intersections of homogeneous Cantor sets and beta-expansions, Nonlinearity, 23 (2010), 2815-2834.  doi: 10.1088/0951-7715/23/11/005.  Google Scholar

[18] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511626302.  Google Scholar
[19]

W. Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11 (1960), 401-416.  doi: 10.1007/BF02020954.  Google Scholar

[20]

A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., 8 (1957), 477-493.  doi: 10.1007/BF02020331.  Google Scholar

show all references

References:
[1]

R. Alcaraz Barrera, Topological and ergodic properties of symmetric sub-shifts, Discrete Contin. Dyn. Syst., 34 (2014), 4459-4486.  doi: 10.3934/dcds.2014.34.4459.  Google Scholar

[2]

R. Alcaraz Barrera, S. Baker and D. Kong, Entropy, topological transitivity, and dimensional properties of unique $q$-expansions, Trans. Amer. Math. Soc., 371 (2019), 3209-3258.  doi: 10.1090/tran/7370.  Google Scholar

[3]

P. Allaart, S. Baker and D. Kong, Bifurcation sets arising from non-integer base expansions, in J. Fractal Geom., (2018), arXiv: 1706.05190. Google Scholar

[4]

P. Allaart and D. Kong, On the continuity of the Hausdorff dimension of the univoque set, Advances in Mathematics, 354 (2019), 106729, arXiv: 1804.02879. doi: 10.1016/j.aim.2019.106729.  Google Scholar

[5]

P. Allaart and D. Kong, Relative bifurcation sets and the local dimension of univoque bases, preprint, 2018, arXiv: 1809.00323. Google Scholar

[6]

S. Baker, Generalized golden ratios over integer alphabets, Integers, 14 (2014), 28 pp.  Google Scholar

[7]

M. de Vries and V. Komornik, Unique expansions of real numbers, Adv. Math., 221 (2009), 390-427.  doi: 10.1016/j.aim.2008.12.008.  Google Scholar

[8]

P. Erdős, M. Horváth and I. Joó, On the uniqueness of the expansions $1=\sum q^{-n_i}$, Acta Math. Hungar., 58 (1991), 333-342.  doi: 10.1007/BF01903963.  Google Scholar

[9]

P. Erdős and I. Joó, On the number of expansions $1=\sum q^{-n_i}$, Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 35 (1992), 129-132.   Google Scholar

[10]

P. Erdős, I. Joó and V. Komornik, Characterization of the unique expansions $1=\sum_{i=1}^\infty q^{-n_i}$ and related problems, Bull. Soc. Math. France, 118 (1990), 377-390.  doi: 10.24033/bsmf.2151.  Google Scholar

[11]

P. Glendinning and T. Hall, Zeros of the kneading invariant and topological entropy for Lorenz maps, Nonlinearity, 9 (1996), 999-1014.  doi: 10.1088/0951-7715/9/4/010.  Google Scholar

[12]

P. Glendinning and N. Sidorov, Unique representations of real numbers in non-integer bases, Math. Res. Lett., 8 (2001), 535-543.  doi: 10.4310/MRL.2001.v8.n4.a12.  Google Scholar

[13]

V. Komornik, D. Kong and W. X. Li, Hausdorff dimension of univoque sets and devil's staircase, Adv. Math., 305 (2017), 165-196.  doi: 10.1016/j.aim.2016.03.047.  Google Scholar

[14]

V. Komornik and P. Loreti, Unique developments in non-integer bases, Amer. Math. Monthly, 105 (1998), 636-639.  doi: 10.1080/00029890.1998.12004937.  Google Scholar

[15]

V. Komornik and P. Loreti, Subexpansions, superexpansions and uniqueness properties in non-integer bases, Period. Math. Hungar., 44 (2002), 197-218.  doi: 10.1023/A:1019696514372.  Google Scholar

[16]

D. Kong and W. X. Li, Hausdorff dimension of unique beta expansions, Nonlinearity, 28 (2015), 187-209.  doi: 10.1088/0951-7715/28/1/187.  Google Scholar

[17]

D. Kong, W. X. Li and F. M. Dekking, Intersections of homogeneous Cantor sets and beta-expansions, Nonlinearity, 23 (2010), 2815-2834.  doi: 10.1088/0951-7715/23/11/005.  Google Scholar

[18] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511626302.  Google Scholar
[19]

W. Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11 (1960), 401-416.  doi: 10.1007/BF02020954.  Google Scholar

[20]

A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., 8 (1957), 477-493.  doi: 10.1007/BF02020331.  Google Scholar

Figure 1.  The labeled graph $ \mathcal G_ \mathbf{a} = (G, \mathcal L_ \mathbf{a}) $ with labeling $ \mathcal L_ \mathbf{a}: E\to L_ \mathbf{a}: = \big\{ \mathbf{a}, \mathbf{a}^+, \overline{ \mathbf{a}}, \overline{ \mathbf{a}^+}\big\} $, and the labeled graph $ \mathcal G^* = (G, \mathcal L^*) $ with labeling $ \mathcal L^*: E\to\left\{{0, 1}\right\} $
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