Topological and ergodic properties of symmetric sub-shifts

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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

Abstract
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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 and 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.
[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.
[3]	S. Baker, Generalised golden ratios over integer alphabets, Integers, 14 (2014) Paper No. A15, 1-28.
[4]	R. Bowen, Periodic points and measures for Axiom $A$ diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.
[5]	R. Bowen, Some systems with unique equilibrium states, Math. Systems Theory, 8 (1974/75), 193-202. doi: 10.1007/BF01762666.
[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.
[7]	M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511755316.
[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.
[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.
[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.
[11]	C. Dettmann, Open circle maps: Small hole asymptotics, Nonlinearity, 26 (2013), 307-317. doi: 10.1088/0951-7715/26/1/307.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511626302.
[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.
[20]	J. Nilsson, On numbers badly approximable by dyadic rationals, Israel J. Math., 171 (2009), 93-110. doi: 10.1007/s11856-009-0042-9.
[21]	J. Nilsson, The fine structure of Dyadically badly approximable numbers, ArXiv e-prints, arXiv:1002.4614, February 2010.
[22]	W. Parry, On the $\beta $-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11 (1960), 401-416. doi: 10.1007/BF02020954.
[23]	W. Parry, Intrinsic Markov chains, Trans. Amer. Math. Soc., 112 (1964), 55-66. doi: 10.1090/S0002-9947-1964-0161372-1.
[24]	K. Petersen, Chains, entropy, coding, Ergodic Theory Dynam. Systems, 6 (1986), 415-448. doi: 10.1017/S014338570000359X.
[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.
[26]	N. Sidorov, Supercritical holes for the doubling map, Acta Mathematica Hungarica, pages 1-15, 2014.
[27]	M. Urbański, Invariant subsets of expanding mappings of the circle, Ergodic Theory Dynam. Systems, 7 (1987), 627-645. doi: 10.1017/S0143385700004247.
[28]	P. Walters, An Introduction to Ergodic Theory, volume 79 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1982.
[29]	B. Weiss, Intrinsically ergodic systems, Bull. Amer. Math. Soc., 76 (1970), 1266-1269. doi: 10.1090/S0002-9904-1970-12632-5.
[30]	B. Weiss, Subshifts of finite type and sofic systems, Monatsh. Math., 77 (1973), 462-474. doi: 10.1007/BF01295322.

show all references

References:

[1]	S. Akiyama and K. Scheicher, Symmetric shift radix systems and finite expansions, Math. Pannon., 18 (2007), 101-124.
[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.
[3]	S. Baker, Generalised golden ratios over integer alphabets, Integers, 14 (2014) Paper No. A15, 1-28.
[4]	R. Bowen, Periodic points and measures for Axiom $A$ diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.
[5]	R. Bowen, Some systems with unique equilibrium states, Math. Systems Theory, 8 (1974/75), 193-202. doi: 10.1007/BF01762666.
[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.
[7]	M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511755316.
[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.
[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.
[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.
[11]	C. Dettmann, Open circle maps: Small hole asymptotics, Nonlinearity, 26 (2013), 307-317. doi: 10.1088/0951-7715/26/1/307.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511626302.
[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.
[20]	J. Nilsson, On numbers badly approximable by dyadic rationals, Israel J. Math., 171 (2009), 93-110. doi: 10.1007/s11856-009-0042-9.
[21]	J. Nilsson, The fine structure of Dyadically badly approximable numbers, ArXiv e-prints, arXiv:1002.4614, February 2010.
[22]	W. Parry, On the $\beta $-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11 (1960), 401-416. doi: 10.1007/BF02020954.
[23]	W. Parry, Intrinsic Markov chains, Trans. Amer. Math. Soc., 112 (1964), 55-66. doi: 10.1090/S0002-9947-1964-0161372-1.
[24]	K. Petersen, Chains, entropy, coding, Ergodic Theory Dynam. Systems, 6 (1986), 415-448. doi: 10.1017/S014338570000359X.
[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.
[26]	N. Sidorov, Supercritical holes for the doubling map, Acta Mathematica Hungarica, pages 1-15, 2014.
[27]	M. Urbański, Invariant subsets of expanding mappings of the circle, Ergodic Theory Dynam. Systems, 7 (1987), 627-645. doi: 10.1017/S0143385700004247.
[28]	P. Walters, An Introduction to Ergodic Theory, volume 79 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1982.
[29]	B. Weiss, Intrinsically ergodic systems, Bull. Amer. Math. Soc., 76 (1970), 1266-1269. doi: 10.1090/S0002-9904-1970-12632-5.
[30]	B. Weiss, Subshifts of finite type and sofic systems, Monatsh. Math., 77 (1973), 462-474. doi: 10.1007/BF01295322.

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