Intermediate \beta-shifts of finite type

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

Intermediate $\beta$-shifts of finite type

Bing Li ^1, , Tuomas Sahlsten ^2, and Tony Samuel ^3,

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

Abstract
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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.

Keywords: subshifts of finite type, $\beta$-transformations, transitivity..

Mathematics Subject Classification: Primary: 37B10; Secondary: 11A67, 11R0.

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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