
-
Previous Article
Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent
- DCDS Home
- This Issue
-
Next Article
On the Gevrey regularity of solutions to the 3D ideal MHD equations
An algebraic approach to entropy plateaus in non-integer base expansions
Mathematics Department, University of North Texas, 1155 Union Cir #311430, Denton, TX 76203-5017, USA |
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.
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. |
[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. |
[3] |
P. Allaart, S. Baker and D. Kong, Bifurcation sets arising from non-integer base expansions, in J. Fractal Geom., (2018), arXiv: 1706.05190. |
[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. |
[5] |
P. Allaart and D. Kong, Relative bifurcation sets and the local dimension of univoque bases, preprint, 2018, arXiv: 1809.00323. |
[6] |
S. Baker, Generalized golden ratios over integer alphabets, Integers, 14 (2014), 28 pp. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511626302.![]() ![]() ![]() |
[19] |
W. Parry,
On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11 (1960), 401-416.
doi: 10.1007/BF02020954. |
[20] |
A. Rényi,
Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., 8 (1957), 477-493.
doi: 10.1007/BF02020331. |
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. |
[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. |
[3] |
P. Allaart, S. Baker and D. Kong, Bifurcation sets arising from non-integer base expansions, in J. Fractal Geom., (2018), arXiv: 1706.05190. |
[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. |
[5] |
P. Allaart and D. Kong, Relative bifurcation sets and the local dimension of univoque bases, preprint, 2018, arXiv: 1809.00323. |
[6] |
S. Baker, Generalized golden ratios over integer alphabets, Integers, 14 (2014), 28 pp. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511626302.![]() ![]() ![]() |
[19] |
W. Parry,
On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11 (1960), 401-416.
doi: 10.1007/BF02020954. |
[20] |
A. Rényi,
Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., 8 (1957), 477-493.
doi: 10.1007/BF02020331. |

[1] |
Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461 |
[2] |
Silvère Gangloff, Benjamin Hellouin de Menibus. Effect of quantified irreducibility on the computability of subshift entropy. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1975-2000. doi: 10.3934/dcds.2019083 |
[3] |
Dominik Kwietniak. Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2451-2467. doi: 10.3934/dcds.2013.33.2451 |
[4] |
Katrin Gelfert. Lower bounds for the topological entropy. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555 |
[5] |
Jaume Llibre. Brief survey on the topological entropy. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3363-3374. doi: 10.3934/dcdsb.2015.20.3363 |
[6] |
Michał Misiurewicz, Peter Raith. Strict inequalities for the entropy of transitive piecewise monotone maps. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 451-468. doi: 10.3934/dcds.2005.13.451 |
[7] |
Boris Hasselblatt, Zbigniew Nitecki, James Propp. Topological entropy for nonuniformly continuous maps. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 201-213. doi: 10.3934/dcds.2008.22.201 |
[8] |
Michał Misiurewicz. On Bowen's definition of topological entropy. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 827-833. doi: 10.3934/dcds.2004.10.827 |
[9] |
Lluís Alsedà, David Juher, Francesc Mañosas. Forward triplets and topological entropy on trees. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 623-641. doi: 10.3934/dcds.2021131 |
[10] |
Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 545-557 . doi: 10.3934/dcds.2011.31.545 |
[11] |
Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547 |
[12] |
Jan Philipp Schröder. Ergodicity and topological entropy of geodesic flows on surfaces. Journal of Modern Dynamics, 2015, 9: 147-167. doi: 10.3934/jmd.2015.9.147 |
[13] |
Xiaomin Zhou. Relative entropy dimension of topological dynamical systems. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6631-6642. doi: 10.3934/dcds.2019288 |
[14] |
Yun Zhao, Wen-Chiao Cheng, Chih-Chang Ho. Q-entropy for general topological dynamical systems. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2059-2075. doi: 10.3934/dcds.2019086 |
[15] |
Eva Glasmachers, Gerhard Knieper, Carlos Ogouyandjou, Jan Philipp Schröder. Topological entropy of minimal geodesics and volume growth on surfaces. Journal of Modern Dynamics, 2014, 8 (1) : 75-91. doi: 10.3934/jmd.2014.8.75 |
[16] |
Yujun Ju, Dongkui Ma, Yupan Wang. Topological entropy of free semigroup actions for noncompact sets. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 995-1017. doi: 10.3934/dcds.2019041 |
[17] |
César J. Niche. Topological entropy of a magnetic flow and the growth of the number of trajectories. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 577-580. doi: 10.3934/dcds.2004.11.577 |
[18] |
Tao Wang, Yu Huang. Weighted topological and measure-theoretic entropy. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3941-3967. doi: 10.3934/dcds.2019159 |
[19] |
Kendry J. Vivas, Víctor F. Sirvent. Metric entropy for set-valued maps. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022010 |
[20] |
José S. Cánovas. Topological sequence entropy of $\omega$–limit sets of interval maps. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 781-786. doi: 10.3934/dcds.2001.7.781 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]