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Topological and ergodic properties of symmetric sub-shifts
1. | School of Mathematics, The University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom |
References:
[1] |
S. Akiyama and K. Scheicher, Symmetric shift radix systems and finite expansions,, Math. Pannon., 18 (2007), 101.
|
[2] |
J. Allouche, M. Clarke and N. Sidorov, Periodic unique beta-expansions: The Sharkovskiĭ ordering,, Ergodic Theory Dynam. Systems, 29 (2009), 1055.
doi: 10.1017/S0143385708000746. |
[3] |
S. Baker, Generalised golden ratios over integer alphabets,, Integers, 14 (2014), 1. Google Scholar |
[4] |
R. Bowen, Periodic points and measures for Axiom $A$ diffeomorphisms,, Trans. Amer. Math. Soc., 154 (1971), 377.
|
[5] |
R. Bowen, Some systems with unique equilibrium states,, Math. Systems Theory, 8 (): 193.
doi: 10.1007/BF01762666. |
[6] |
M. Boyle, Algebraic aspects of symbolic dynamics,, In Topics in symbolic dynamics and applications (Temuco, (1997), 57.
|
[7] |
M. Brin and G. Stuck, Introduction to Dynamical Systems,, Cambridge University Press, (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.
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.
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.
doi: 10.1088/0951-7715/25/7/2133. |
[11] |
C. Dettmann, Open circle maps: Small hole asymptotics,, Nonlinearity, 26 (2013), 307.
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.
|
[13] |
P. Glendinning and N. Sidorov, Unique representations of real numbers in non-integer bases,, Math. Res. Lett., 8 (2001), 535.
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.
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.
|
[16] |
N. T. A. Haydn, Phase transition in one-dimensional subshifts,, Discrete Contin. Dyn. Syst., 33 (2013), 1965.
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.
doi: 10.1007/s10955-009-9747-8. |
[18] |
D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding,, Cambridge University Press, (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), (1983).
|
[20] |
J. Nilsson, On numbers badly approximable by dyadic rationals,, Israel J. Math., 171 (2009), 93.
doi: 10.1007/s11856-009-0042-9. |
[21] |
J. Nilsson, The fine structure of Dyadically badly approximable numbers,, ArXiv e-prints, (2010). Google Scholar |
[22] |
W. Parry, On the $\beta $-expansions of real numbers,, Acta Math. Acad. Sci. Hungar., 11 (1960), 401.
doi: 10.1007/BF02020954. |
[23] |
W. Parry, Intrinsic Markov chains,, Trans. Amer. Math. Soc., 112 (1964), 55.
doi: 10.1090/S0002-9947-1964-0161372-1. |
[24] |
K. Petersen, Chains, entropy, coding,, Ergodic Theory Dynam. Systems, 6 (1986), 415.
doi: 10.1017/S014338570000359X. |
[25] |
N. Sidorov, Arithmetic dynamics,, In Topics in dynamics and ergodic theory, (2003), 145.
doi: 10.1017/CBO9780511546716.010. |
[26] |
N. Sidorov, Supercritical holes for the doubling map,, Acta Mathematica Hungarica, (2014), 1. Google Scholar |
[27] |
M. Urbański, Invariant subsets of expanding mappings of the circle,, Ergodic Theory Dynam. Systems, 7 (1987), 627.
doi: 10.1017/S0143385700004247. |
[28] |
P. Walters, An Introduction to Ergodic Theory, volume 79 of Graduate Texts in Mathematics,, Springer-Verlag, (1982).
|
[29] |
B. Weiss, Intrinsically ergodic systems,, Bull. Amer. Math. Soc., 76 (1970), 1266.
doi: 10.1090/S0002-9904-1970-12632-5. |
[30] |
B. Weiss, Subshifts of finite type and sofic systems,, Monatsh. Math., 77 (1973), 462.
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.
|
[2] |
J. Allouche, M. Clarke and N. Sidorov, Periodic unique beta-expansions: The Sharkovskiĭ ordering,, Ergodic Theory Dynam. Systems, 29 (2009), 1055.
doi: 10.1017/S0143385708000746. |
[3] |
S. Baker, Generalised golden ratios over integer alphabets,, Integers, 14 (2014), 1. Google Scholar |
[4] |
R. Bowen, Periodic points and measures for Axiom $A$ diffeomorphisms,, Trans. Amer. Math. Soc., 154 (1971), 377.
|
[5] |
R. Bowen, Some systems with unique equilibrium states,, Math. Systems Theory, 8 (): 193.
doi: 10.1007/BF01762666. |
[6] |
M. Boyle, Algebraic aspects of symbolic dynamics,, In Topics in symbolic dynamics and applications (Temuco, (1997), 57.
|
[7] |
M. Brin and G. Stuck, Introduction to Dynamical Systems,, Cambridge University Press, (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.
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.
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.
doi: 10.1088/0951-7715/25/7/2133. |
[11] |
C. Dettmann, Open circle maps: Small hole asymptotics,, Nonlinearity, 26 (2013), 307.
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.
|
[13] |
P. Glendinning and N. Sidorov, Unique representations of real numbers in non-integer bases,, Math. Res. Lett., 8 (2001), 535.
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.
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.
|
[16] |
N. T. A. Haydn, Phase transition in one-dimensional subshifts,, Discrete Contin. Dyn. Syst., 33 (2013), 1965.
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.
doi: 10.1007/s10955-009-9747-8. |
[18] |
D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding,, Cambridge University Press, (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), (1983).
|
[20] |
J. Nilsson, On numbers badly approximable by dyadic rationals,, Israel J. Math., 171 (2009), 93.
doi: 10.1007/s11856-009-0042-9. |
[21] |
J. Nilsson, The fine structure of Dyadically badly approximable numbers,, ArXiv e-prints, (2010). Google Scholar |
[22] |
W. Parry, On the $\beta $-expansions of real numbers,, Acta Math. Acad. Sci. Hungar., 11 (1960), 401.
doi: 10.1007/BF02020954. |
[23] |
W. Parry, Intrinsic Markov chains,, Trans. Amer. Math. Soc., 112 (1964), 55.
doi: 10.1090/S0002-9947-1964-0161372-1. |
[24] |
K. Petersen, Chains, entropy, coding,, Ergodic Theory Dynam. Systems, 6 (1986), 415.
doi: 10.1017/S014338570000359X. |
[25] |
N. Sidorov, Arithmetic dynamics,, In Topics in dynamics and ergodic theory, (2003), 145.
doi: 10.1017/CBO9780511546716.010. |
[26] |
N. Sidorov, Supercritical holes for the doubling map,, Acta Mathematica Hungarica, (2014), 1. Google Scholar |
[27] |
M. Urbański, Invariant subsets of expanding mappings of the circle,, Ergodic Theory Dynam. Systems, 7 (1987), 627.
doi: 10.1017/S0143385700004247. |
[28] |
P. Walters, An Introduction to Ergodic Theory, volume 79 of Graduate Texts in Mathematics,, Springer-Verlag, (1982).
|
[29] |
B. Weiss, Intrinsically ergodic systems,, Bull. Amer. Math. Soc., 76 (1970), 1266.
doi: 10.1090/S0002-9904-1970-12632-5. |
[30] |
B. Weiss, Subshifts of finite type and sofic systems,, Monatsh. Math., 77 (1973), 462.
doi: 10.1007/BF01295322. |
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