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 and Continuous Dynamical Systems, 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), 11-22.

[2]

M. F. Barnsley and N. Mihalache, Symmetric itinerary sets, preprint, arXiv:1110.2817v1.

[3]

M. Barnsley, W. Steiner and A. Vince, A combinatorial characterization of the critical itineraries of an overlapping dynamical system, preprint, arXiv:1205.5902v3.

[4]

M. F. Barnsley, B. Harding and A. Vince, The entropy of a special overlapping dynamical system, Ergodic Theory and Dynamical Systems, 34 (2014), 483-500. doi: 10.1017/etds.2012.140.

[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), 271-323.

[6]

F. Blanchard, $\beta$-expansions and symbolic dynamics, Theoret. Comput. Sci., 65 (1989), 131-141. doi: 10.1016/0304-3975(89)90038-8.

[7]

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

[8]

K. Dajani and C. Kraaikamp, Ergodic Theory of Numbers, Carus Mathematical Monographs, 29. Mathematical Association of America, Washington, DC, 2002.

[9]

K. Dajani and C. Kraaikamp, From greedy to lazy expansions and their driving dynamics, Expo. Math., 20 (2002), 315-327. doi: 10.1016/S0723-0869(02)80010-X.

[10]

K. Dajani and M. deVries, Measures of maximal entropy for random $\beta$-expansions, J. Eur. Math. Soc., 7 (2005), 51-68. doi: 10.4171/JEMS/21.

[11]

K. Dajani and M. deVries, Invariant densities for random $\beta$-expansions, J. Eur. Math. Soc., 9 (2007), 157-176. doi: 10.4171/JEMS/76.

[12]

I. Daubechies, R. DeVore, S. Güntürk and V. Vaishampayan, A/D conversion with imperfect quantizers, IEEE Trans. Inform. Theory, 52 (2006), 874-885. doi: 10.1109/TIT.2005.864430.

[13]

B. Eckhardt and G. Ott, Periodic orbit analysis of the Lorenz attractor, Zeit. Phys. B, 93 (1994), 259-266. doi: 10.1007/BF01316970.

[14]

K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Third edition. John Wiley & Sons, Ltd., Chichester, 2014.

[15]

A.-H. Fan and B.-W. Wang, On the lengths of basic intervals in beta expansions, Nonlinearity, 25 (2012), 1329-1343. doi: 10.1088/0951-7715/25/5/1329.

[16]

C. Frougny and A. C. Lai, On negative bases, Proceedings of DLT 09, Lecture Notes in Comput. Sci., Springer, Berlin, 5583 (2009), 252-263. doi: 10.1007/978-3-642-02737-6_20.

[17]

P. Glendinning, Topological conjugation of Lorenz maps by $\beta$-transformations, Math. Proc. Camb. Phil. Soc., 107 (1990), 401-413. doi: 10.1017/S0305004100068675.

[18]

T. Hejda, Z. Masáková and E. Pelantová, Greedy and lazy representations in negative base systems. Kybernetika, 49 (2013), 258-279.

[19]

F. Hofbauer, Maximal measures for piecewise monotonically increasing transformations on $[0, 1]$, Ergodic Theory Lecture Notes in Mathematics, 729 (1979), 66-77.

[20]

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.

[21]

S. Ito and T. Sadahiro, Beta-Expansions with negative bases, Integers, 9 (2009), 239-259. doi: 10.1515/INTEG.2009.023.

[22]

C. Kalle and W. Steiner, Beta-expansions, natural extensions and multiple tilings associated with Pisot units, Trans. Amer. Math. Soc., 364 (2012), 2281-2318. doi: 10.1090/S0002-9947-2012-05362-1.

[23]

V. Komornik and P. Loreti, Unique developments in non-integer bases, Amer. Math. Monthly, 105 (1998), 636-639. doi: 10.2307/2589246.

[24]

L. Liao and W. Steiner, Dynamical properties of the negative beta-transformation, Ergodic Theory Dyn. Sys., 32 (2012), 1673-1690. doi: 10.1017/S0143385711000514.

[25]

D. Lind, The entropies of topological Markov shifts and a related class of algebraic integers, Ergodic Theory Dyn. Sys., 4 (1984), 283-300. doi: 10.1017/S0143385700002443.

[26]

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

[27]

E. N. Lorenz, Deterministic nonperiodic flow, The Theory of Chaotic Attractors, (2004), 25-36. doi: 10.1007/978-0-387-21830-4_2.

[28]

M. R. Palmer, On the Classification of Measure Preserving Transformations of Lebesgue Spaces, Ph. D. thesis, University of Warwick, 1979.

[29]

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

[30]

W. Parry, Representations for real numbers, Acta Math. Acad. Sci. Hungar., 15 (1964), 95-105. doi: 10.1007/BF01897025.

[31]

W. Parry, Symbolic dynamics and transformations of the unit interval, Trans. Amer. Math. Soc., 122 (1966), 368-378. doi: 10.1090/S0002-9947-1966-0197683-5.

[32]

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

[33]

N. Sidorov, Arithmetic dynamics, Topics in Dynamics and Ergodic Theory, LMS Lecture Notes Ser., 310 (2003), 145-189. doi: 10.1017/CBO9780511546716.010.

[34]

N. Sidorov, Almost every number has a continuum of $\beta$-expansions, Amer. Math. Monthly, 110 (2003), 838-842. doi: 10.2307/3647804.

[35]

D. Viswanath, Symbolic dynamics and periodic orbits of the Lorenz attractor, Nonlinearity, 16 (2003), 1035-1056. doi: 10.1088/0951-7715/16/3/314.

[36]

K. M. Wilkinson, Ergodic properties of a class of piecewise linear transformations, Z. Wahrscheinlickeitstheorie verw. Gebiete, 31 (1975), 303-328.

[37]

R. F. Williams, Structure of Lorenz attractors, Publ. Math. IHES, 50 (1979), 73-99.

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), 11-22.

[2]

M. F. Barnsley and N. Mihalache, Symmetric itinerary sets, preprint, arXiv:1110.2817v1.

[3]

M. Barnsley, W. Steiner and A. Vince, A combinatorial characterization of the critical itineraries of an overlapping dynamical system, preprint, arXiv:1205.5902v3.

[4]

M. F. Barnsley, B. Harding and A. Vince, The entropy of a special overlapping dynamical system, Ergodic Theory and Dynamical Systems, 34 (2014), 483-500. doi: 10.1017/etds.2012.140.

[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), 271-323.

[6]

F. Blanchard, $\beta$-expansions and symbolic dynamics, Theoret. Comput. Sci., 65 (1989), 131-141. doi: 10.1016/0304-3975(89)90038-8.

[7]

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

[8]

K. Dajani and C. Kraaikamp, Ergodic Theory of Numbers, Carus Mathematical Monographs, 29. Mathematical Association of America, Washington, DC, 2002.

[9]

K. Dajani and C. Kraaikamp, From greedy to lazy expansions and their driving dynamics, Expo. Math., 20 (2002), 315-327. doi: 10.1016/S0723-0869(02)80010-X.

[10]

K. Dajani and M. deVries, Measures of maximal entropy for random $\beta$-expansions, J. Eur. Math. Soc., 7 (2005), 51-68. doi: 10.4171/JEMS/21.

[11]

K. Dajani and M. deVries, Invariant densities for random $\beta$-expansions, J. Eur. Math. Soc., 9 (2007), 157-176. doi: 10.4171/JEMS/76.

[12]

I. Daubechies, R. DeVore, S. Güntürk and V. Vaishampayan, A/D conversion with imperfect quantizers, IEEE Trans. Inform. Theory, 52 (2006), 874-885. doi: 10.1109/TIT.2005.864430.

[13]

B. Eckhardt and G. Ott, Periodic orbit analysis of the Lorenz attractor, Zeit. Phys. B, 93 (1994), 259-266. doi: 10.1007/BF01316970.

[14]

K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Third edition. John Wiley & Sons, Ltd., Chichester, 2014.

[15]

A.-H. Fan and B.-W. Wang, On the lengths of basic intervals in beta expansions, Nonlinearity, 25 (2012), 1329-1343. doi: 10.1088/0951-7715/25/5/1329.

[16]

C. Frougny and A. C. Lai, On negative bases, Proceedings of DLT 09, Lecture Notes in Comput. Sci., Springer, Berlin, 5583 (2009), 252-263. doi: 10.1007/978-3-642-02737-6_20.

[17]

P. Glendinning, Topological conjugation of Lorenz maps by $\beta$-transformations, Math. Proc. Camb. Phil. Soc., 107 (1990), 401-413. doi: 10.1017/S0305004100068675.

[18]

T. Hejda, Z. Masáková and E. Pelantová, Greedy and lazy representations in negative base systems. Kybernetika, 49 (2013), 258-279.

[19]

F. Hofbauer, Maximal measures for piecewise monotonically increasing transformations on $[0, 1]$, Ergodic Theory Lecture Notes in Mathematics, 729 (1979), 66-77.

[20]

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.

[21]

S. Ito and T. Sadahiro, Beta-Expansions with negative bases, Integers, 9 (2009), 239-259. doi: 10.1515/INTEG.2009.023.

[22]

C. Kalle and W. Steiner, Beta-expansions, natural extensions and multiple tilings associated with Pisot units, Trans. Amer. Math. Soc., 364 (2012), 2281-2318. doi: 10.1090/S0002-9947-2012-05362-1.

[23]

V. Komornik and P. Loreti, Unique developments in non-integer bases, Amer. Math. Monthly, 105 (1998), 636-639. doi: 10.2307/2589246.

[24]

L. Liao and W. Steiner, Dynamical properties of the negative beta-transformation, Ergodic Theory Dyn. Sys., 32 (2012), 1673-1690. doi: 10.1017/S0143385711000514.

[25]

D. Lind, The entropies of topological Markov shifts and a related class of algebraic integers, Ergodic Theory Dyn. Sys., 4 (1984), 283-300. doi: 10.1017/S0143385700002443.

[26]

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

[27]

E. N. Lorenz, Deterministic nonperiodic flow, The Theory of Chaotic Attractors, (2004), 25-36. doi: 10.1007/978-0-387-21830-4_2.

[28]

M. R. Palmer, On the Classification of Measure Preserving Transformations of Lebesgue Spaces, Ph. D. thesis, University of Warwick, 1979.

[29]

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

[30]

W. Parry, Representations for real numbers, Acta Math. Acad. Sci. Hungar., 15 (1964), 95-105. doi: 10.1007/BF01897025.

[31]

W. Parry, Symbolic dynamics and transformations of the unit interval, Trans. Amer. Math. Soc., 122 (1966), 368-378. doi: 10.1090/S0002-9947-1966-0197683-5.

[32]

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

[33]

N. Sidorov, Arithmetic dynamics, Topics in Dynamics and Ergodic Theory, LMS Lecture Notes Ser., 310 (2003), 145-189. doi: 10.1017/CBO9780511546716.010.

[34]

N. Sidorov, Almost every number has a continuum of $\beta$-expansions, Amer. Math. Monthly, 110 (2003), 838-842. doi: 10.2307/3647804.

[35]

D. Viswanath, Symbolic dynamics and periodic orbits of the Lorenz attractor, Nonlinearity, 16 (2003), 1035-1056. doi: 10.1088/0951-7715/16/3/314.

[36]

K. M. Wilkinson, Ergodic properties of a class of piecewise linear transformations, Z. Wahrscheinlickeitstheorie verw. Gebiete, 31 (1975), 303-328.

[37]

R. F. Williams, Structure of Lorenz attractors, Publ. Math. IHES, 50 (1979), 73-99.

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