# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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. 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. 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. 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. 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. 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. doi: 10.1002/cpa.3160430402. Google Scholar [8] M. Lothaire, Algebraic Combinatorics on Words,, Encyclopedia of Mathematics and its Applications, (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. 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. doi: 10.1007/BF02020331. Google Scholar [11] N. Sidorov, Supercritical holes for the doubling map,, Acta Mathematica Hungarica, 143 (2014), 298. doi: 10.1007/s10474-014-0403-7. Google Scholar [12] L. Vuillon, Balanced words,, Bull. Belg. Math. Soc. Simon Stevin, 10 (2003), 787. Google Scholar

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##### 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. 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. 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. 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. 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. 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. doi: 10.1002/cpa.3160430402. Google Scholar [8] M. Lothaire, Algebraic Combinatorics on Words,, Encyclopedia of Mathematics and its Applications, (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. 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. doi: 10.1007/BF02020331. Google Scholar [11] N. Sidorov, Supercritical holes for the doubling map,, Acta Mathematica Hungarica, 143 (2014), 298. doi: 10.1007/s10474-014-0403-7. Google Scholar [12] L. Vuillon, Balanced words,, Bull. Belg. Math. Soc. Simon Stevin, 10 (2003), 787. Google Scholar
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