Denseness of intermediate 𝛽-shifts of finite-type

Li, Bing; Sahlsten, Tuomas; Samuel, Tony; Steiner, Wolfgang

doi:10.1090/proc/14279

Denseness of intermediate $\beta$-shifts of finite-type
HTML articles powered by AMS MathViewer

by Bing Li, Tuomas Sahlsten, Tony Samuel and Wolfgang Steiner

Proc. Amer. Math. Soc. 147 (2019), 2045-2055

DOI: https://doi.org/10.1090/proc/14279

Published electronically: February 6, 2019

PDF | Request permission

Abstract:

We determine the structure of the set of intermediate $\beta$-shifts of finite-type. Specifically, we show that this set is dense in the parameter space \begin{align*} \Delta \coloneq \{ (\beta , \alpha ) \in \mathbb {R}^{2} \colon \beta \in (1, 2) \; \text {and} \; 0 \leq \alpha \leq 2 - \beta \}. \end{align*} This generalises the classical result of Parry from 1960 for greedy $\beta$-shifts.

References

Ll. Alsedà and F. Mañosas, Kneading theory for a family of circle maps with one discontinuity, Acta Math. Univ. Comenian. (N.S.) 65 (1996), no. 1, 11–22. MR 1422291
Jan Awrejcewicz and Claude-Henri Lamarque, Bifurcation and chaos in nonsmooth mechanical systems, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, vol. 45, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. MR 2015431, DOI 10.1142/9789812564801
Simon Baker, On small bases which admit countably many expansions, J. Number Theory 147 (2015), 515–532. MR 3276338, DOI 10.1016/j.jnt.2014.08.003
Yuru Zou, Lijin Wang, Jian Lu, and Simon Baker, On small bases for which 1 has countably many expansions, Mathematika 62 (2016), no. 2, 362–377. MR 3451258, DOI 10.1112/S002557931500025X
S. Banerjee and G. C. Verfghese, Nonlinear Phenomena in Power Electronics: Attractors, Bifurcations, Chaos and Nonlinear Control, Wiley-IEEE Press, 2001.
Michael Barnsley, Brendan Harding, and Andrew Vince, The entropy of a special overlapping dynamical system, Ergodic Theory Dynam. Systems 34 (2014), no. 2, 483–500. MR 3233701, DOI 10.1017/etds.2012.140
M. F. Barnsley and N. Mihalache, Symmetric itinerary sets, preprint, arXiv:1110.2817v1.
Michael Barnsley, Wolfgang Steiner, and Andrew Vince, Critical itineraries of maps with constant slope and one discontinuity, Math. Proc. Cambridge Philos. Soc. 157 (2014), no. 3, 547–565. MR 3286523, DOI 10.1017/S0305004114000486
F. Blanchard, $\beta$-expansions and symbolic dynamics, Theoret. Comput. Sci. 65 (1989), no. 2, 131–141. MR 1020481, DOI 10.1016/0304-3975(89)90038-8
Karma Dajani and Charlene Kalle, Random $\beta$-expansions with deleted digits, Discrete Contin. Dyn. Syst. 18 (2007), no. 1, 199–217. MR 2276494, DOI 10.3934/dcds.2007.18.199
Karma Dajani and Cor Kraaikamp, Random $\beta$-expansions, Ergodic Theory Dynam. Systems 23 (2003), no. 2, 461–479. MR 1972232, DOI 10.1017/S0143385702001141
Karma Dajani and Martijn de Vries, Measures of maximal entropy for random $\beta$-expansions, J. Eur. Math. Soc. (JEMS) 7 (2005), no. 1, 51–68. MR 2120990, DOI 10.4171/JEMS/21
I. Daubechies, R. DeVore, C.S.Gunturk and V.A. Vaishampayan, Beta expansions: a new approach to digitally corrected A/D conversion, IEEE International Symposium on Circuits and Systems. Proceedings (Cat. No. 02CH37353) 2 (2002) II-784–II-787.
Bruno Eckhardt and Gerolf Ott, Periodic orbit analysis of the Lorenz attractor, Z. Phys. B 93 (1994), no. 2, 259–266. MR 1259284, DOI 10.1007/BF01316970
Pál Erdös, István Joó, and Vilmos Komornik, Characterization of the unique expansions $1=\sum ^\infty _{i=1}q^{-n_i}$ and related problems, Bull. Soc. Math. France 118 (1990), no. 3, 377–390 (English, with French summary). MR 1078082
Kenneth Falconer, Fractal geometry, 2nd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2003. Mathematical foundations and applications. MR 2118797, DOI 10.1002/0470013850
Kenneth Falconer, Techniques in fractal geometry, John Wiley & Sons, Ltd., Chichester, 1997. MR 1449135
Paul Glendinning, Topological conjugation of Lorenz maps by $\beta$-transformations, Math. Proc. Cambridge Philos. Soc. 107 (1990), no. 2, 401–413. MR 1027793, DOI 10.1017/S0305004100068675
Paul Glendinning and Toby Hall, Zeros of the kneading invariant and topological entropy for Lorenz maps, Nonlinearity 9 (1996), no. 4, 999–1014. MR 1399483, DOI 10.1088/0951-7715/9/4/010
John H. Hubbard and Colin T. Sparrow, The classification of topologically expansive Lorenz maps, Comm. Pure Appl. Math. 43 (1990), no. 4, 431–443. MR 1047331, DOI 10.1002/cpa.3160430402
Charlene Kalle and Wolfgang Steiner, Beta-expansions, natural extensions and multiple tilings associated with Pisot units, Trans. Amer. Math. Soc. 364 (2012), no. 5, 2281–2318. MR 2888207, DOI 10.1090/S0002-9947-2012-05362-1
K. Keller, T. Mangold, I. Stolz, and J. Werner, Permutation Entropy: New Ideas and Challenges, Entropy 19 (2017), 134.
Vilmos Komornik, Expansions in noninteger bases, Integers 11B (2011), Paper No. A9, 30. MR 3054428
Vilmos Komornik and Paola Loreti, Unique developments in non-integer bases, Amer. Math. Monthly 105 (1998), no. 7, 636–639. MR 1633077, DOI 10.2307/2589246
JinJun Li and Bing Li, Hausdorff dimensions of some irregular sets associated with $\beta$-expansions, Sci. China Math. 59 (2016), no. 3, 445–458. MR 3457047, DOI 10.1007/s11425-015-5046-9
Bing Li, Tuomas Sahlsten, and Tony Samuel, Intermediate $\beta$-shifts of finite type, Discrete Contin. Dyn. Syst. 36 (2016), no. 1, 323–344. MR 3369224, DOI 10.3934/dcds.2016.36.323
D. A. Lind, The entropies of topological Markov shifts and a related class of algebraic integers, Ergodic Theory Dynam. Systems 4 (1984), no. 2, 283–300. MR 766106, DOI 10.1017/S0143385700002443
Douglas Lind and Brian Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995. MR 1369092, DOI 10.1017/CBO9780511626302
E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci. 20 (1963) 130–141.
M. R. Palmer, On the classification of measure preserving transformations of Lebesgue spaces, Ph.D. thesis, University of Warwick, 1979.
W. Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416 (English, with Russian summary). MR 142719, DOI 10.1007/BF02020954
W. Parry, Representations for real numbers, Acta Math. Acad. Sci. Hungar. 15 (1964), 95–105. MR 166332, DOI 10.1007/BF01897025
William Parry, The Lorenz attractor and a related population model, Ergodic theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978) Lecture Notes in Math., vol. 729, Springer, Berlin, 1979, pp. 169–187. MR 550420
B. Quackenbush, Fiber density of intermediate $\beta$-shifts of finite type, M.Sc. thesis, California Polytechnic State University, 2018.
A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493. MR 97374, DOI 10.1007/BF02020331
Nikita Sidorov, Arithmetic dynamics, Topics in dynamics and ergodic theory, London Math. Soc. Lecture Note Ser., vol. 310, Cambridge Univ. Press, Cambridge, 2003, pp. 145–189. MR 2052279, DOI 10.1017/CBO9780511546716.010
Nikita Sidorov, Almost every number has a continuum of $\beta$-expansions, Amer. Math. Monthly 110 (2003), no. 9, 838–842. MR 2024754, DOI 10.2307/3647804
Nikita Sidorov and Anatoly Vershik, Ergodic properties of the Erdős measure, the entropy of the golden shift, and related problems, Monatsh. Math. 126 (1998), no. 3, 215–261. MR 1651776, DOI 10.1007/BF01367764
Divakar Viswanath, Symbolic dynamics and periodic orbits of the Lorenz attractor, Nonlinearity 16 (2003), no. 3, 1035–1056. MR 1975795, DOI 10.1088/0951-7715/16/3/314
R. F. Williams, The structure of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 73–99. MR 556583
Zhanybai T. Zhusubaliyev and Erik Mosekilde, Bifurcations and chaos in piecewise-smooth dynamical systems, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, vol. 44, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. MR 2001701, DOI 10.1142/5313