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

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.  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

[1]

Bo Tan, Bao-Wei Wang, Jun Wu, Jian Xu. Localized Birkhoff average in beta dynamical systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2547-2564. doi: 10.3934/dcds.2013.33.2547

[2]

Anushaya Mohapatra, William Ott. Memory loss for nonequilibrium open dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3747-3759. doi: 10.3934/dcds.2014.34.3747

[3]

David Ralston. Heaviness in symbolic dynamics: Substitution and Sturmian systems. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 287-300. doi: 10.3934/dcdss.2009.2.287

[4]

Michael Dellnitz, Christian Horenkamp. The efficient approximation of coherent pairs in non-autonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3029-3042. doi: 10.3934/dcds.2012.32.3029

[5]

Wen-Guei Hu, Song-Sun Lin. On spatial entropy of multi-dimensional symbolic dynamical systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3705-3717. doi: 10.3934/dcds.2016.36.3705

[6]

H. M. Hastings, S. Silberger, M. T. Weiss, Y. Wu. A twisted tensor product on symbolic dynamical systems and the Ashley's problem. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 549-558. doi: 10.3934/dcds.2003.9.549

[7]

Gary Froyland, Ognjen Stancevic. Escape rates and Perron-Frobenius operators: Open and closed dynamical systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 457-472. doi: 10.3934/dcdsb.2010.14.457

[8]

Gary Froyland, Philip K. Pollett, Robyn M. Stuart. A closing scheme for finding almost-invariant sets in open dynamical systems. Journal of Computational Dynamics, 2014, 1 (1) : 135-162. doi: 10.3934/jcd.2014.1.135

[9]

Steven T. Piantadosi. Symbolic dynamics on free groups. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 725-738. doi: 10.3934/dcds.2008.20.725

[10]

David Burguet, Todd Fisher. Symbolic extensionsfor partially hyperbolic dynamical systems with 2-dimensional center bundle. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2253-2270. doi: 10.3934/dcds.2013.33.2253

[11]

Jim Wiseman. Symbolic dynamics from signed matrices. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 621-638. doi: 10.3934/dcds.2004.11.621

[12]

George Osipenko, Stephen Campbell. Applied symbolic dynamics: attractors and filtrations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 43-60. doi: 10.3934/dcds.1999.5.43

[13]

Michael Hochman. A note on universality in multidimensional symbolic dynamics. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 301-314. doi: 10.3934/dcdss.2009.2.301

[14]

Alexander Gorodnik. Open problems in dynamics and related fields. Journal of Modern Dynamics, 2007, 1 (1) : 1-35. doi: 10.3934/jmd.2007.1.1

[15]

Élise Janvresse, Benoît Rittaud, Thierry de la Rue. Dynamics of $\lambda$-continued fractions and $\beta$-shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1477-1498. doi: 10.3934/dcds.2013.33.1477

[16]

Jose S. Cánovas, Tönu Puu, Manuel Ruiz Marín. Detecting chaos in a duopoly model via symbolic dynamics. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 269-278. doi: 10.3934/dcdsb.2010.13.269

[17]

Nicola Soave, Susanna Terracini. Symbolic dynamics for the $N$-centre problem at negative energies. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3245-3301. doi: 10.3934/dcds.2012.32.3245

[18]

Dieter Mayer, Fredrik Strömberg. Symbolic dynamics for the geodesic flow on Hecke surfaces. Journal of Modern Dynamics, 2008, 2 (4) : 581-627. doi: 10.3934/jmd.2008.2.581

[19]

Frédéric Naud. Birkhoff cones, symbolic dynamics and spectrum of transfer operators. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 581-598. doi: 10.3934/dcds.2004.11.581

[20]

Fryderyk Falniowski, Marcin Kulczycki, Dominik Kwietniak, Jian Li. Two results on entropy, chaos and independence in symbolic dynamics. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3487-3505. doi: 10.3934/dcdsb.2015.20.3487

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]