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

The $\beta$-transformation with a hole

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*}
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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