The \beta-transformation with a hole

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March 2016, 36(3): 1249-1269. doi: 10.3934/dcds.2016.36.1249

The $\beta$-transformation with a hole

Lyndsey Clark ^1,

1.	School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom

Received January 2015 Revised May 2015 Published August 2015

Abstract
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This paper extends those of Glendinning and Sidorov [3] and of Hare and Sidorov [6] from the case of the doubling map to the more general $\beta$-transformation. Let $\beta \in (1,2)$ and consider the $\beta$-transformation $T_\beta(x)=\beta x$ mod 1. Let $\mathcal{J}_\beta(a,b) := \{ x \in (0,1) : T_\beta^n(x) \notin (a,b) \text{ for all } n \geq 0 \}$. An integer $n$ is bad for $(a,b)$ if every periodic point of period $n$ for $T_\beta$ intersects $(a,b)$. Denote the set of all bad $n$ for $(a,b)$ by $B_\beta(a,b)$. In this paper we completely describe the following sets: \begin{align*} D_0(\beta) &= \{ (a,b) \in [0,1)^2 : \mathcal{J}(a,b) \neq \emptyset \}, \\ D_1(\beta) &= \{ (a,b) \in [0,1)^2 : \mathcal{J}(a,b) \text{ is uncountable} \}, \\ D_2(\beta) &= \{ (a,b) \in [0,1)^2 : B_\beta(a,b) \text{ is finite} \}. \end{align*}

Keywords: extremal pairs., symbolic dynamics, beta expansions, Open dynamical systems, combinatorics on words.

Mathematics Subject Classification: Primary: 28D05; Secondary: 37B10, 68R1.

Citation: Lyndsey Clark. The $\beta$-transformation with a hole. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1249-1269. doi: 10.3934/dcds.2016.36.1249

References:

[1]	P. Boyland, A. de Carvalho and T. Hall, On digit frequencies in $\beta$-expansions,, Transactions of the AMS, ().
[2]	S. Bullett and P. Sentenac, Ordered orbits of the shift, square roots, and the devil's staircase, Math. Proc. Camb. Phil. Soc., 115 (1994), 451-481. doi: 10.1017/S0305004100072236.
[3]	P. Glendinning and N. Sidorov, The doubling map with asymmetrical holes, Ergodic Theory and Dynamical Systems, 35 (2015), 1208-1228. doi: 10.1017/etds.2013.98.
[4]	P. Glendinning and C. T. Sparrow, Prime and renormalisable kneading invariants and the dynamics of expanding Lorenz maps, Physica D, 62 (1993), 22-50. doi: 10.1016/0167-2789(93)90270-B.
[5]	L. Goldberg and C. Tresser, Rotation and the Farey tree, Ergodic Theory and Dynamical Systems, 16 (1996), 1011-1029. doi: 10.1017/S0143385700010154.
[6]	K. G. Hare and N. Sidorov, On cycles for the doubling map which are disjoint from an interval, Monatsh. Math., 175 (2014), 347-365. doi: 10.1007/s00605-014-0646-y.
[7]	J. H. Hubbard and C. T. Sparrow, The classification of topologically expansive Lorenz maps, Comm. Pure Appl. Math., 43 (1990), 431-443. doi: 10.1002/cpa.3160430402.
[8]	M. Lothaire, Algebraic Combinatorics on Words, Encyclopedia of Mathematics and its Applications, 90, Cambridge University Press, 2002. doi: 10.1017/CBO9781107326019.
[9]	W. Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hung., 11 (1960), 401-416. doi: 10.1007/BF02020954.
[10]	A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung., 8 (1957), 477-493. doi: 10.1007/BF02020331.
[11]	N. Sidorov, Supercritical holes for the doubling map, Acta Mathematica Hungarica, 143 (2014), 298-312. doi: 10.1007/s10474-014-0403-7.
[12]	L. Vuillon, Balanced words, Bull. Belg. Math. Soc. Simon Stevin, 10 (2003), 787-805.

show all references

References:

[1]	P. Boyland, A. de Carvalho and T. Hall, On digit frequencies in $\beta$-expansions,, Transactions of the AMS, ().
[2]	S. Bullett and P. Sentenac, Ordered orbits of the shift, square roots, and the devil's staircase, Math. Proc. Camb. Phil. Soc., 115 (1994), 451-481. doi: 10.1017/S0305004100072236.
[3]	P. Glendinning and N. Sidorov, The doubling map with asymmetrical holes, Ergodic Theory and Dynamical Systems, 35 (2015), 1208-1228. doi: 10.1017/etds.2013.98.
[4]	P. Glendinning and C. T. Sparrow, Prime and renormalisable kneading invariants and the dynamics of expanding Lorenz maps, Physica D, 62 (1993), 22-50. doi: 10.1016/0167-2789(93)90270-B.
[5]	L. Goldberg and C. Tresser, Rotation and the Farey tree, Ergodic Theory and Dynamical Systems, 16 (1996), 1011-1029. doi: 10.1017/S0143385700010154.
[6]	K. G. Hare and N. Sidorov, On cycles for the doubling map which are disjoint from an interval, Monatsh. Math., 175 (2014), 347-365. doi: 10.1007/s00605-014-0646-y.
[7]	J. H. Hubbard and C. T. Sparrow, The classification of topologically expansive Lorenz maps, Comm. Pure Appl. Math., 43 (1990), 431-443. doi: 10.1002/cpa.3160430402.
[8]	M. Lothaire, Algebraic Combinatorics on Words, Encyclopedia of Mathematics and its Applications, 90, Cambridge University Press, 2002. doi: 10.1017/CBO9781107326019.
[9]	W. Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hung., 11 (1960), 401-416. doi: 10.1007/BF02020954.
[10]	A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung., 8 (1957), 477-493. doi: 10.1007/BF02020331.
[11]	N. Sidorov, Supercritical holes for the doubling map, Acta Mathematica Hungarica, 143 (2014), 298-312. doi: 10.1007/s10474-014-0403-7.
[12]	L. Vuillon, Balanced words, Bull. Belg. Math. Soc. Simon Stevin, 10 (2003), 787-805.

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