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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

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 & 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, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

L. Goldberg and C. Tresser, Rotation and the Farey tree, Ergodic Theory and Dynamical Systems, 16 (1996), 1011-1029. doi: 10.1017/S0143385700010154.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

M. Lothaire, Algebraic Combinatorics on Words, Encyclopedia of Mathematics and its Applications, 90, Cambridge University Press, 2002. doi: 10.1017/CBO9781107326019.  Google Scholar

[9]

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

[10]

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

[11]

N. Sidorov, Supercritical holes for the doubling map, Acta Mathematica Hungarica, 143 (2014), 298-312. doi: 10.1007/s10474-014-0403-7.  Google Scholar

[12]

L. Vuillon, Balanced words, Bull. Belg. Math. Soc. Simon Stevin, 10 (2003), 787-805.  Google Scholar

show all references

References:
[1]

P. Boyland, A. de Carvalho and T. Hall, On digit frequencies in $\beta$-expansions,, Transactions of the AMS, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

L. Goldberg and C. Tresser, Rotation and the Farey tree, Ergodic Theory and Dynamical Systems, 16 (1996), 1011-1029. doi: 10.1017/S0143385700010154.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

M. Lothaire, Algebraic Combinatorics on Words, Encyclopedia of Mathematics and its Applications, 90, Cambridge University Press, 2002. doi: 10.1017/CBO9781107326019.  Google Scholar

[9]

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

[10]

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

[11]

N. Sidorov, Supercritical holes for the doubling map, Acta Mathematica Hungarica, 143 (2014), 298-312. doi: 10.1007/s10474-014-0403-7.  Google Scholar

[12]

L. Vuillon, Balanced words, Bull. Belg. Math. Soc. Simon Stevin, 10 (2003), 787-805.  Google Scholar

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