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November  2014, 34(11): 4459-4486. doi: 10.3934/dcds.2014.34.4459

Topological and ergodic properties of symmetric sub-shifts

Rafael Alcaraz Barrera 1,

1. 

School of Mathematics, The University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom

Received  July 2013 Revised  February 2014 Published  May 2014

The family of symmetric one sided sub-shifts in two symbols given by a sequence $a$ is studied. We analyse some of their topological properties such as transitivity, the specification property and intrinsic ergodicity. It is shown that almost every member of this family admits only one measure of maximal entropy. It is shown that the same results hold for attractors of the family of open dynamical systems arising from the doubling map with a centred symmetric hole depending on one parameter, and for the set of points that have unique $\beta$-expansion for $\beta \in (\varphi,2)$ where $\varphi$ is the Golden Ratio.
Keywords: symbolic dynamics, Open dynamical systems, doubling map, intrinsic ergodicity., $\beta$-expansions.
Mathematics Subject Classification: Primary: 37B10; Secondary: 28D05, 37C70, 68R1.
Citation: Rafael Alcaraz Barrera. Topological and ergodic properties of symmetric sub-shifts. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4459-4486. doi: 10.3934/dcds.2014.34.4459
References:
[1]

S. Akiyama and K. Scheicher, Symmetric shift radix systems and finite expansions, Math. Pannon., 18 (2007), 101-124.  Google Scholar

[2]

J. Allouche, M. Clarke and N. Sidorov, Periodic unique beta-expansions: The Sharkovskiĭ ordering, Ergodic Theory Dynam. Systems, 29 (2009), 1055-1074. doi: 10.1017/S0143385708000746.  Google Scholar

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S. Baker, Generalised golden ratios over integer alphabets, Integers, 14 (2014) Paper No. A15, 1-28. Google Scholar

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R. Bowen, Periodic points and measures for Axiom $A$ diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.  Google Scholar

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R. Bowen, Some systems with unique equilibrium states,, Math. Systems Theory, 8 (): 193.  doi: 10.1007/BF01762666.  Google Scholar

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

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S. Bundfuss, T. Krüger and S. Troubetzkoy, Topological and symbolic dynamics for hyperbolic systems with holes, Ergodic Theory Dynam. Systems, 31 (2011), 1305-1323. doi: 10.1017/S0143385710000556.  Google Scholar

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V. Climenhaga and D. J. Thompson, Intrinsic ergodicity beyond specification: $\beta$-shifts, $S$-gap shifts, and their factors, Israel J. Math., 192 (2012), 785-817. doi: 10.1007/s11856-012-0052-x.  Google Scholar

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C. Dettmann, Open circle maps: Small hole asymptotics, Nonlinearity, 26 (2013), 307-317. doi: 10.1088/0951-7715/26/1/307.  Google Scholar

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P. Erdös, I. Joó and V. Komornik, Characterization of the unique expansions $1=\sum$$^\infty_{i=1}q^{-n_i}$ and related problems, Bull. Soc. Math. France, 118 (1990), 377-390.  Google Scholar

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P. Glendinning and N. Sidorov, Unique representations of real numbers in non-integer bases, Math. Res. Lett., 8 (2001), 535-543. doi: 10.4310/MRL.2001.v8.n4.a12.  Google Scholar

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P. Glendinning and N. Sidorov, The doubling map with asymmetrical holes, Ergodic Theory and Dynamical Systems Ergodic Theory and Dynamical Systems, (2013), 1-21. arXiv:1302.2486. doi: 10.1017/etds.2013.98.  Google Scholar

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B. M. Gurevič, Uniqueness of the measure with maximal entropy for symbolic dynamical systems that are close to Markov ones, Dokl. Akad. Nauk SSSR, 204 (1972), 15-17.  Google Scholar

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N. T. A. Haydn, Phase transition in one-dimensional subshifts, Discrete Contin. Dyn. Syst., 33 (2013), 1965-1973. doi: 10.3934/dcds.2013.33.1965.  Google Scholar

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G. Keller and C. Liverani, Rare events, escape rates and quasistationarity: Some exact formulae, J. Stat. Phys., 135 (2009), 519-534. doi: 10.1007/s10955-009-9747-8.  Google Scholar

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

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R. Mañé, Introdução à teoria ergódica, volume 14 of Projeto Euclides [Euclid Project], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1983.  Google Scholar

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J. Nilsson, On numbers badly approximable by dyadic rationals, Israel J. Math., 171 (2009), 93-110. doi: 10.1007/s11856-009-0042-9.  Google Scholar

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J. Nilsson, The fine structure of Dyadically badly approximable numbers, ArXiv e-prints, arXiv:1002.4614, February 2010. Google Scholar

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

[23]

W. Parry, Intrinsic Markov chains, Trans. Amer. Math. Soc., 112 (1964), 55-66. doi: 10.1090/S0002-9947-1964-0161372-1.  Google Scholar

[24]

K. Petersen, Chains, entropy, coding, Ergodic Theory Dynam. Systems, 6 (1986), 415-448. doi: 10.1017/S014338570000359X.  Google Scholar

[25]

N. Sidorov, Arithmetic dynamics, In Topics in dynamics and ergodic theory, volume 310 of London Math. Soc. Lecture Note Ser., pages 145-189. Cambridge Univ. Press, Cambridge, 2003. doi: 10.1017/CBO9780511546716.010.  Google Scholar

[26]

N. Sidorov, Supercritical holes for the doubling map, Acta Mathematica Hungarica, pages 1-15, 2014. Google Scholar

[27]

M. Urbański, Invariant subsets of expanding mappings of the circle, Ergodic Theory Dynam. Systems, 7 (1987), 627-645. doi: 10.1017/S0143385700004247.  Google Scholar

[28]

P. Walters, An Introduction to Ergodic Theory, volume 79 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1982.  Google Scholar

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B. Weiss, Intrinsically ergodic systems, Bull. Amer. Math. Soc., 76 (1970), 1266-1269. doi: 10.1090/S0002-9904-1970-12632-5.  Google Scholar

[30]

B. Weiss, Subshifts of finite type and sofic systems, Monatsh. Math., 77 (1973), 462-474. doi: 10.1007/BF01295322.  Google Scholar

show all references

References:
[1]

S. Akiyama and K. Scheicher, Symmetric shift radix systems and finite expansions, Math. Pannon., 18 (2007), 101-124.  Google Scholar

[2]

J. Allouche, M. Clarke and N. Sidorov, Periodic unique beta-expansions: The Sharkovskiĭ ordering, Ergodic Theory Dynam. Systems, 29 (2009), 1055-1074. doi: 10.1017/S0143385708000746.  Google Scholar

[3]

S. Baker, Generalised golden ratios over integer alphabets, Integers, 14 (2014) Paper No. A15, 1-28. Google Scholar

[4]

R. Bowen, Periodic points and measures for Axiom $A$ diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.  Google Scholar

[5]

R. Bowen, Some systems with unique equilibrium states,, Math. Systems Theory, 8 (): 193.  doi: 10.1007/BF01762666.  Google Scholar

[6]

M. Boyle, Algebraic aspects of symbolic dynamics, In Topics in symbolic dynamics and applications (Temuco, 1997), volume 279 of London Math. Soc. Lecture Note Ser., pages 57-88. Cambridge Univ. Press, Cambridge, 2000.  Google Scholar

[7]

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

[8]

S. Bundfuss, T. Krüger and S. Troubetzkoy, Topological and symbolic dynamics for hyperbolic systems with holes, Ergodic Theory Dynam. Systems, 31 (2011), 1305-1323. doi: 10.1017/S0143385710000556.  Google Scholar

[9]

V. Climenhaga and D. J. Thompson, Intrinsic ergodicity beyond specification: $\beta$-shifts, $S$-gap shifts, and their factors, Israel J. Math., 192 (2012), 785-817. doi: 10.1007/s11856-012-0052-x.  Google Scholar

[10]

M. F. Demers and P. Wright, Behaviour of the escape rate function in hyperbolic dynamical systems, Nonlinearity, 25 (2012), 2133-2150. doi: 10.1088/0951-7715/25/7/2133.  Google Scholar

[11]

C. Dettmann, Open circle maps: Small hole asymptotics, Nonlinearity, 26 (2013), 307-317. doi: 10.1088/0951-7715/26/1/307.  Google Scholar

[12]

P. Erdös, I. Joó and V. Komornik, Characterization of the unique expansions $1=\sum$$^\infty_{i=1}q^{-n_i}$ and related problems, Bull. Soc. Math. France, 118 (1990), 377-390.  Google Scholar

[13]

P. Glendinning and N. Sidorov, Unique representations of real numbers in non-integer bases, Math. Res. Lett., 8 (2001), 535-543. doi: 10.4310/MRL.2001.v8.n4.a12.  Google Scholar

[14]

P. Glendinning and N. Sidorov, The doubling map with asymmetrical holes, Ergodic Theory and Dynamical Systems Ergodic Theory and Dynamical Systems, (2013), 1-21. arXiv:1302.2486. doi: 10.1017/etds.2013.98.  Google Scholar

[15]

B. M. Gurevič, Uniqueness of the measure with maximal entropy for symbolic dynamical systems that are close to Markov ones, Dokl. Akad. Nauk SSSR, 204 (1972), 15-17.  Google Scholar

[16]

N. T. A. Haydn, Phase transition in one-dimensional subshifts, Discrete Contin. Dyn. Syst., 33 (2013), 1965-1973. doi: 10.3934/dcds.2013.33.1965.  Google Scholar

[17]

G. Keller and C. Liverani, Rare events, escape rates and quasistationarity: Some exact formulae, J. Stat. Phys., 135 (2009), 519-534. doi: 10.1007/s10955-009-9747-8.  Google Scholar

[18]

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

[19]

R. Mañé, Introdução à teoria ergódica, volume 14 of Projeto Euclides [Euclid Project], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1983.  Google Scholar

[20]

J. Nilsson, On numbers badly approximable by dyadic rationals, Israel J. Math., 171 (2009), 93-110. doi: 10.1007/s11856-009-0042-9.  Google Scholar

[21]

J. Nilsson, The fine structure of Dyadically badly approximable numbers, ArXiv e-prints, arXiv:1002.4614, February 2010. Google Scholar

[22]

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

[23]

W. Parry, Intrinsic Markov chains, Trans. Amer. Math. Soc., 112 (1964), 55-66. doi: 10.1090/S0002-9947-1964-0161372-1.  Google Scholar

[24]

K. Petersen, Chains, entropy, coding, Ergodic Theory Dynam. Systems, 6 (1986), 415-448. doi: 10.1017/S014338570000359X.  Google Scholar

[25]

N. Sidorov, Arithmetic dynamics, In Topics in dynamics and ergodic theory, volume 310 of London Math. Soc. Lecture Note Ser., pages 145-189. Cambridge Univ. Press, Cambridge, 2003. doi: 10.1017/CBO9780511546716.010.  Google Scholar

[26]

N. Sidorov, Supercritical holes for the doubling map, Acta Mathematica Hungarica, pages 1-15, 2014. Google Scholar

[27]

M. Urbański, Invariant subsets of expanding mappings of the circle, Ergodic Theory Dynam. Systems, 7 (1987), 627-645. doi: 10.1017/S0143385700004247.  Google Scholar

[28]

P. Walters, An Introduction to Ergodic Theory, volume 79 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1982.  Google Scholar

[29]

B. Weiss, Intrinsically ergodic systems, Bull. Amer. Math. Soc., 76 (1970), 1266-1269. doi: 10.1090/S0002-9904-1970-12632-5.  Google Scholar

[30]

B. Weiss, Subshifts of finite type and sofic systems, Monatsh. Math., 77 (1973), 462-474. doi: 10.1007/BF01295322.  Google Scholar

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