January  2016, 36(1): 323-344. doi: 10.3934/dcds.2016.36.323

Intermediate $\beta$-shifts of finite type

1. 

Department of Mathematics, South China University of Technology, Guangzhou, 510641, China

2. 

Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel

3. 

Fachbereich 3 Mathematik, Universität Bremen, 28359 Bremen, Germany

Received  March 2014 Revised  March 2015 Published  June 2015

An aim of this article is to highlight dynamical differences between the greedy, and hence the lazy, $\beta$-shift (transformation) and an intermediate $\beta$-shift (transformation), for a fixed $\beta \in (1, 2)$. Specifically, a classification in terms of the kneading invariants of the linear maps $T_{\beta,\alpha} \colon x \mapsto \beta x + \alpha \bmod 1$ for which the corresponding intermediate $\beta$-shift is of finite type is given. This characterisation is then employed to construct a class of pairs $(\beta,\alpha)$ such that the intermediate $\beta$-shift associated with $T_{\beta, \alpha}$ is a subshift of finite type. It is also proved that these maps $T_{\beta,\alpha}$ are not transitive. This is in contrast to the situation for the corresponding greedy and lazy $\beta$-shifts and $\beta$-transformations, for which both of the two properties do not hold.
Citation: Bing Li, Tuomas Sahlsten, Tony Samuel. Intermediate $\beta$-shifts of finite type. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 323-344. doi: 10.3934/dcds.2016.36.323
References:
[1]

L. Alsedá and F. Manosas, Kneading theory for a family of circle maps with one discontinuity,, Acta Math. Univ. Comenian. (N.S.), 65 (1996), 11. Google Scholar

[2]

M. F. Barnsley and N. Mihalache, Symmetric itinerary sets,, preprint, (). Google Scholar

[3]

M. Barnsley, W. Steiner and A. Vince, A combinatorial characterization of the critical itineraries of an overlapping dynamical system,, preprint, (). Google Scholar

[4]

M. F. Barnsley, B. Harding and A. Vince, The entropy of a special overlapping dynamical system,, Ergodic Theory and Dynamical Systems, 34 (2014), 483. doi: 10.1017/etds.2012.140. Google Scholar

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A. Bertrand-Mathis, Développement en base $\theta$, répartition modulo un de la suite $(x \theta^n)_{n \geq 0}$; languages codeés et $\theta$-shift,, Bull. Soc. Math. Fr., 114 (1986), 271. Google Scholar

[6]

F. Blanchard, $\beta$-expansions and symbolic dynamics,, Theoret. Comput. Sci., 65 (1989), 131. doi: 10.1016/0304-3975(89)90038-8. Google Scholar

[7]

M. Brin and G. Stuck, Introduction to Dynamical Systems,, Cambridge University Press, (2002). doi: 10.1017/CBO9780511755316. Google Scholar

[8]

K. Dajani and C. Kraaikamp, Ergodic Theory of Numbers,, Carus Mathematical Monographs, (2002). Google Scholar

[9]

K. Dajani and C. Kraaikamp, From greedy to lazy expansions and their driving dynamics,, Expo. Math., 20 (2002), 315. doi: 10.1016/S0723-0869(02)80010-X. Google Scholar

[10]

K. Dajani and M. deVries, Measures of maximal entropy for random $\beta$-expansions,, J. Eur. Math. Soc., 7 (2005), 51. doi: 10.4171/JEMS/21. Google Scholar

[11]

K. Dajani and M. deVries, Invariant densities for random $\beta$-expansions,, J. Eur. Math. Soc., 9 (2007), 157. doi: 10.4171/JEMS/76. Google Scholar

[12]

I. Daubechies, R. DeVore, S. Güntürk and V. Vaishampayan, A/D conversion with imperfect quantizers,, IEEE Trans. Inform. Theory, 52 (2006), 874. doi: 10.1109/TIT.2005.864430. Google Scholar

[13]

B. Eckhardt and G. Ott, Periodic orbit analysis of the Lorenz attractor,, Zeit. Phys. B, 93 (1994), 259. doi: 10.1007/BF01316970. Google Scholar

[14]

K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications,, Third edition. John Wiley & Sons, (2014). Google Scholar

[15]

A.-H. Fan and B.-W. Wang, On the lengths of basic intervals in beta expansions,, Nonlinearity, 25 (2012), 1329. doi: 10.1088/0951-7715/25/5/1329. Google Scholar

[16]

C. Frougny and A. C. Lai, On negative bases,, Proceedings of DLT 09, 5583 (2009), 252. doi: 10.1007/978-3-642-02737-6_20. Google Scholar

[17]

P. Glendinning, Topological conjugation of Lorenz maps by $\beta$-transformations,, Math. Proc. Camb. Phil. Soc., 107 (1990), 401. doi: 10.1017/S0305004100068675. Google Scholar

[18]

T. Hejda, Z. Masáková and E. Pelantová, Greedy and lazy representations in negative base systems., Kybernetika, 49 (2013), 258. Google Scholar

[19]

F. Hofbauer, Maximal measures for piecewise monotonically increasing transformations on $[0, 1]$,, Ergodic Theory Lecture Notes in Mathematics, 729 (1979), 66. Google Scholar

[20]

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

[21]

S. Ito and T. Sadahiro, Beta-Expansions with negative bases,, Integers, 9 (2009), 239. doi: 10.1515/INTEG.2009.023. Google Scholar

[22]

C. Kalle and W. Steiner, Beta-expansions, natural extensions and multiple tilings associated with Pisot units,, Trans. Amer. Math. Soc., 364 (2012), 2281. doi: 10.1090/S0002-9947-2012-05362-1. Google Scholar

[23]

V. Komornik and P. Loreti, Unique developments in non-integer bases,, Amer. Math. Monthly, 105 (1998), 636. doi: 10.2307/2589246. Google Scholar

[24]

L. Liao and W. Steiner, Dynamical properties of the negative beta-transformation,, Ergodic Theory Dyn. Sys., 32 (2012), 1673. doi: 10.1017/S0143385711000514. Google Scholar

[25]

D. Lind, The entropies of topological Markov shifts and a related class of algebraic integers,, Ergodic Theory Dyn. Sys., 4 (1984), 283. doi: 10.1017/S0143385700002443. Google Scholar

[26]

D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511626302. Google Scholar

[27]

E. N. Lorenz, Deterministic nonperiodic flow,, The Theory of Chaotic Attractors, (2004), 25. doi: 10.1007/978-0-387-21830-4_2. Google Scholar

[28]

M. R. Palmer, On the Classification of Measure Preserving Transformations of Lebesgue Spaces,, Ph. D. thesis, (1979). Google Scholar

[29]

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

[30]

W. Parry, Representations for real numbers,, Acta Math. Acad. Sci. Hungar., 15 (1964), 95. doi: 10.1007/BF01897025. Google Scholar

[31]

W. Parry, Symbolic dynamics and transformations of the unit interval,, Trans. Amer. Math. Soc., 122 (1966), 368. doi: 10.1090/S0002-9947-1966-0197683-5. Google Scholar

[32]

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

[33]

N. Sidorov, Arithmetic dynamics,, Topics in Dynamics and Ergodic Theory, 310 (2003), 145. doi: 10.1017/CBO9780511546716.010. Google Scholar

[34]

N. Sidorov, Almost every number has a continuum of $\beta$-expansions,, Amer. Math. Monthly, 110 (2003), 838. doi: 10.2307/3647804. Google Scholar

[35]

D. Viswanath, Symbolic dynamics and periodic orbits of the Lorenz attractor,, Nonlinearity, 16 (2003), 1035. doi: 10.1088/0951-7715/16/3/314. Google Scholar

[36]

K. M. Wilkinson, Ergodic properties of a class of piecewise linear transformations,, Z. Wahrscheinlickeitstheorie verw. Gebiete, 31 (1975), 303. Google Scholar

[37]

R. F. Williams, Structure of Lorenz attractors,, Publ. Math. IHES, 50 (1979), 73. Google Scholar

show all references

References:
[1]

L. Alsedá and F. Manosas, Kneading theory for a family of circle maps with one discontinuity,, Acta Math. Univ. Comenian. (N.S.), 65 (1996), 11. Google Scholar

[2]

M. F. Barnsley and N. Mihalache, Symmetric itinerary sets,, preprint, (). Google Scholar

[3]

M. Barnsley, W. Steiner and A. Vince, A combinatorial characterization of the critical itineraries of an overlapping dynamical system,, preprint, (). Google Scholar

[4]

M. F. Barnsley, B. Harding and A. Vince, The entropy of a special overlapping dynamical system,, Ergodic Theory and Dynamical Systems, 34 (2014), 483. doi: 10.1017/etds.2012.140. Google Scholar

[5]

A. Bertrand-Mathis, Développement en base $\theta$, répartition modulo un de la suite $(x \theta^n)_{n \geq 0}$; languages codeés et $\theta$-shift,, Bull. Soc. Math. Fr., 114 (1986), 271. Google Scholar

[6]

F. Blanchard, $\beta$-expansions and symbolic dynamics,, Theoret. Comput. Sci., 65 (1989), 131. doi: 10.1016/0304-3975(89)90038-8. Google Scholar

[7]

M. Brin and G. Stuck, Introduction to Dynamical Systems,, Cambridge University Press, (2002). doi: 10.1017/CBO9780511755316. Google Scholar

[8]

K. Dajani and C. Kraaikamp, Ergodic Theory of Numbers,, Carus Mathematical Monographs, (2002). Google Scholar

[9]

K. Dajani and C. Kraaikamp, From greedy to lazy expansions and their driving dynamics,, Expo. Math., 20 (2002), 315. doi: 10.1016/S0723-0869(02)80010-X. Google Scholar

[10]

K. Dajani and M. deVries, Measures of maximal entropy for random $\beta$-expansions,, J. Eur. Math. Soc., 7 (2005), 51. doi: 10.4171/JEMS/21. Google Scholar

[11]

K. Dajani and M. deVries, Invariant densities for random $\beta$-expansions,, J. Eur. Math. Soc., 9 (2007), 157. doi: 10.4171/JEMS/76. Google Scholar

[12]

I. Daubechies, R. DeVore, S. Güntürk and V. Vaishampayan, A/D conversion with imperfect quantizers,, IEEE Trans. Inform. Theory, 52 (2006), 874. doi: 10.1109/TIT.2005.864430. Google Scholar

[13]

B. Eckhardt and G. Ott, Periodic orbit analysis of the Lorenz attractor,, Zeit. Phys. B, 93 (1994), 259. doi: 10.1007/BF01316970. Google Scholar

[14]

K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications,, Third edition. John Wiley & Sons, (2014). Google Scholar

[15]

A.-H. Fan and B.-W. Wang, On the lengths of basic intervals in beta expansions,, Nonlinearity, 25 (2012), 1329. doi: 10.1088/0951-7715/25/5/1329. Google Scholar

[16]

C. Frougny and A. C. Lai, On negative bases,, Proceedings of DLT 09, 5583 (2009), 252. doi: 10.1007/978-3-642-02737-6_20. Google Scholar

[17]

P. Glendinning, Topological conjugation of Lorenz maps by $\beta$-transformations,, Math. Proc. Camb. Phil. Soc., 107 (1990), 401. doi: 10.1017/S0305004100068675. Google Scholar

[18]

T. Hejda, Z. Masáková and E. Pelantová, Greedy and lazy representations in negative base systems., Kybernetika, 49 (2013), 258. Google Scholar

[19]

F. Hofbauer, Maximal measures for piecewise monotonically increasing transformations on $[0, 1]$,, Ergodic Theory Lecture Notes in Mathematics, 729 (1979), 66. Google Scholar

[20]

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

[21]

S. Ito and T. Sadahiro, Beta-Expansions with negative bases,, Integers, 9 (2009), 239. doi: 10.1515/INTEG.2009.023. Google Scholar

[22]

C. Kalle and W. Steiner, Beta-expansions, natural extensions and multiple tilings associated with Pisot units,, Trans. Amer. Math. Soc., 364 (2012), 2281. doi: 10.1090/S0002-9947-2012-05362-1. Google Scholar

[23]

V. Komornik and P. Loreti, Unique developments in non-integer bases,, Amer. Math. Monthly, 105 (1998), 636. doi: 10.2307/2589246. Google Scholar

[24]

L. Liao and W. Steiner, Dynamical properties of the negative beta-transformation,, Ergodic Theory Dyn. Sys., 32 (2012), 1673. doi: 10.1017/S0143385711000514. Google Scholar

[25]

D. Lind, The entropies of topological Markov shifts and a related class of algebraic integers,, Ergodic Theory Dyn. Sys., 4 (1984), 283. doi: 10.1017/S0143385700002443. Google Scholar

[26]

D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511626302. Google Scholar

[27]

E. N. Lorenz, Deterministic nonperiodic flow,, The Theory of Chaotic Attractors, (2004), 25. doi: 10.1007/978-0-387-21830-4_2. Google Scholar

[28]

M. R. Palmer, On the Classification of Measure Preserving Transformations of Lebesgue Spaces,, Ph. D. thesis, (1979). Google Scholar

[29]

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

[30]

W. Parry, Representations for real numbers,, Acta Math. Acad. Sci. Hungar., 15 (1964), 95. doi: 10.1007/BF01897025. Google Scholar

[31]

W. Parry, Symbolic dynamics and transformations of the unit interval,, Trans. Amer. Math. Soc., 122 (1966), 368. doi: 10.1090/S0002-9947-1966-0197683-5. Google Scholar

[32]

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

[33]

N. Sidorov, Arithmetic dynamics,, Topics in Dynamics and Ergodic Theory, 310 (2003), 145. doi: 10.1017/CBO9780511546716.010. Google Scholar

[34]

N. Sidorov, Almost every number has a continuum of $\beta$-expansions,, Amer. Math. Monthly, 110 (2003), 838. doi: 10.2307/3647804. Google Scholar

[35]

D. Viswanath, Symbolic dynamics and periodic orbits of the Lorenz attractor,, Nonlinearity, 16 (2003), 1035. doi: 10.1088/0951-7715/16/3/314. Google Scholar

[36]

K. M. Wilkinson, Ergodic properties of a class of piecewise linear transformations,, Z. Wahrscheinlickeitstheorie verw. Gebiete, 31 (1975), 303. Google Scholar

[37]

R. F. Williams, Structure of Lorenz attractors,, Publ. Math. IHES, 50 (1979), 73. Google Scholar

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