Abstract
We show that every intermediate \(\beta\)-transformation is topologically conjugate to a greedy \(\beta\)-transformation with a hole at zero, and provide a counterexample illustrating that the correspondence is not one-to-one. This characterisation is employed to (1) build a Krieger embedding theorem for intermediate \(\beta\)-transformation, complementing the result of Li, Sahlsten, Samuel and Steiner [22], and (2) obtain new metric and topological results on survivor sets of intermediate \(\beta\)-transformations with a hole at zero, extending the work of Kalle, Kong, Langeveld and Li [18]. Further, we derive a method to calculate the Hausdorff dimension of such survivor sets as well as results on certain bifurcation sets. Moreover, by taking unions of survivor sets of intermediate \(\beta\)-transformations one obtains an important class of sets arising in metric number theory, namely sets of badly approximable numbers in non-integer bases. We prove, under the assumption that the underlying symbolic space is of finite type, that these sets of badly approximable numbers are winning in the sense of Schmidt games, and hence have the countable intersection property, extending the results of Hu and Yu [15], Tseng [35] and Färm, Persson and Schmeling [11].
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References
P. Allaart and D. Kong, Critical values for the β-transformation with a hole at 0, arXiv: 2109.10012 (2021).
L. Alsedá and F. Manosas, Kneading theory for a family of circle maps with one discontinuity, Acta Math. Univ. Comenian. (N.S.), 65 (1996), 11–22.
W. Bahsoun, C. Bose and G. Froyland, Ergodic Theory, Open Dynamics and Coherent Structures, Springer-Verlag (New York, 2014).
M. Barnsley, B. Harding and A. Vince, The entropy of a special overlapping dynamical system, Ergodic Theory Dynam. Systems, 34 (2014), 483–500.
M. Barnsley, W. Steiner and A. Vince, Critical itineraries of maps with constant slope and one discontinuity, Math. Proc. Cambridge Philos. Soc., 157 (2014), 547–565.
M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press (2002).
S. Bundfuss, T. Krüger and S. Troubetzkoy, Topological and symbolic dynamics for hyperbolic systems with holes, Ergodic Theory Dynam. Systems, 31 (2011), 1305–1323.
Z. Cooperband, E. P. J. Pearse, B. Quackenbush, J. Rowley, T. Samuel and M. A. West, Continuity of entropy for Lorenz maps, Indag. Math. (N.S.), 31 (2020), 96–105.
I. Daubechies, R. DeVore, C. Gunturk and V. Vaishampayan, Beta expansions: a new approach to digitally corrected a/d conversion, in: 2002 IEEE International Symposium on Circuits and Systems. Proceedings (Cat. No. 02CH37353), vol. 2 (2002).
K. J. Falconer, Fractal geometry: Mathematical Foundations and Applications, 2nd ed., John Wiley and Sons (2003).
D. Färm, T. Persson and J. Schmeling, Dimension of countable intersections of some sets arising in expansions in non-integer bases, Fund. Math., 209 (2010), 157–176.
P. Glendinning, Topological conjugation of Lorenz maps by β-transformations, Math. Proc. Cambridge Philos. Soc., 107 (1990), 401–413.
P. Glendinning and T. Hall, Zeros of the kneading invariant and topological entropy for Lorenz maps, Nonlinearity, 9 (1996), 999–1014.
F. Hofbauer, Maximal measures for simple piecewise monotonic transformations, Z. Wahrsch. Verw. Gebiete, 52 (1989), 289–300.
H. Hu and Y. Yu, On Schmidt’s game and the set of points with non-dense orbits under a class of expanding maps, J. Math. Anal. Appl., 418 (2014), 906–920.
J. H. Hubbard and C. T. Sparrow, The classification of topologically expansive Lorenz maps, Comm. Pure Appl. Math., 43 (1990), 431–443.
Y. Jitsumatsu and k. Matsumura, A β-ary to binary conversion for random number generation using a β encoder, Nonlinear Theory Appl. IEICE, 7 (2016), 38–55.
C. Kalle, D. Kong, N. Langeveld and W. Li, The β-transformation with a hole at 0, Ergodic Theory Dynam. Systems, 40 (2020), 2482–2514.
C. Kalle and W. Steiner, Beta-expansions, natural extensions and multiple tilings associated with pisot units, Trans. Amer. Math. Soc., 364 (2012), 2281–2318.
V. Komornik, Expansions in noninteger bases, Integers, 11B (2011), Paper No. A9, 30 pp.
B. Li, T. Sahlsten and T. Samuel, Intermediate β-shifts of finite type, Discrete Contin. Dyn. Syst., 36 (2016), 323–344.
B. Li, T. Sahlsten, T. Samuel and W. Steiner, Denseness of intermediate β-shifts of finite-type, Proc. Amer. Math. Soc., 147 (2019), 2045–2055.
J. Li and B. Li, Hausdorff dimensions of some irregular sets associated with β-expansions, Sci. China Math., 59 (2016), 445–458.
D. Lind and B. Marcus, An introduction to Symbolic Dynamics and Coding, Cambridge University Press (1995).
A. Mosbach, Finite and infinite rotation sequences and beyond, Ph.D. thesis, Universität Bremen (2019).
J. Nilsson, On numbers badly approximable by dyadic rationals, Israel J. Math., 171 (2009), 93–110.
R. Palmer, On the classification of measure preserving transformations of Lebesgue spaces, Ph.D. thesis, University of Warwick (1979).
W. Parry, On the β-expansions of real numbers, Acta Math. Hungar., 11 (1960), 401– 416.
W. Parry, Representations for real numbers, Acta Math. Hungar., 15 (1964), 95–105.
P. Raith, Continuity of the Hausdorff dimension for invariant subsets of interval maps, Acta Math. Univ. Comenian. (N.S.), 63 (1994), 39–53.
J. Schmeling, Symbolic dynamics for β-shifts and self-normal numbers, Ergodic Theory Dynam. Systems, 17 (1997), 675–694.
W. M. Schmidt, On badly approximable numbers and certain games, Trans. Amer. Math. Soc., 123 (1966), 178–199.
N. Sidorov, Topics in Dynamics and Ergodic Theory, London Math. Soc. Lecture Note Series, Cambridge University Press (2003).
C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, AppliedMathematical Sciences, vol. 41, Springer-Verlag (New York–Berlin, 1982).
J. Tseng, Schmidt games and Markov partitions, Nonlinearity, 22 (2009), 525–543.
M. Urbański, On Hausdorff dimension of invariant sets for expanding maps of a circle, Ergodic Theory Dynam. Systems, 6 (1986), 295–309.
P. Walters, An Introduction to Ergodic Theory, Springer (1982).
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Langeveld, N., Samuel, T. Intermediate β-shifts as greedy β-shifts with a hole. Acta Math. Hungar. 170, 269–301 (2023). https://doi.org/10.1007/s10474-023-01337-3
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DOI: https://doi.org/10.1007/s10474-023-01337-3