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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 |
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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