American Institute of Mathematical Sciences

Advanced
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 - A, 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.   Google Scholar

[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.  Google Scholar

[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.   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), 57.   Google Scholar

[7]

M. Brin and G. Stuck, Introduction to Dynamical Systems,, Cambridge University Press, (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.  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.  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.  doi: 10.1088/0951-7715/25/7/2133.  Google Scholar

[11]

C. Dettmann, Open circle maps: Small hole asymptotics,, Nonlinearity, 26 (2013), 307.  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.   Google Scholar

[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.  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.  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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding,, Cambridge University Press, (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), (1983).   Google Scholar

[20]

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

[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.  Google Scholar

[23]

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

[24]

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

[25]

N. Sidorov, Arithmetic dynamics,, In Topics in dynamics and ergodic theory, (2003), 145.  doi: 10.1017/CBO9780511546716.010.  Google Scholar

[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.  Google Scholar

[28]

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

[29]

B. Weiss, Intrinsically ergodic systems,, Bull. Amer. Math. Soc., 76 (1970), 1266.  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.  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.   Google Scholar

[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.  Google Scholar

[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.   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), 57.   Google Scholar

[7]

M. Brin and G. Stuck, Introduction to Dynamical Systems,, Cambridge University Press, (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.  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.  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.  doi: 10.1088/0951-7715/25/7/2133.  Google Scholar

[11]

C. Dettmann, Open circle maps: Small hole asymptotics,, Nonlinearity, 26 (2013), 307.  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.   Google Scholar

[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.  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.  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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding,, Cambridge University Press, (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), (1983).   Google Scholar

[20]

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

[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.  Google Scholar

[23]

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

[24]

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

[25]

N. Sidorov, Arithmetic dynamics,, In Topics in dynamics and ergodic theory, (2003), 145.  doi: 10.1017/CBO9780511546716.010.  Google Scholar

[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.  Google Scholar

[28]

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

[29]

B. Weiss, Intrinsically ergodic systems,, Bull. Amer. Math. Soc., 76 (1970), 1266.  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.  doi: 10.1007/BF01295322.  Google Scholar

[1]

Khosro Sayevand, Valeyollah Moradi. A robust computational framework for analyzing fractional dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021022
[2]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087
[3]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195
[4]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094
[5]

Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133
[6]

Demetres D. Kouvatsos, Jumma S. Alanazi, Kevin Smith. A unified ME algorithm for arbitrary open QNMs with mixed blocking mechanisms. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 781-816. doi: 10.3934/naco.2011.1.781
[7]

Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225
[8]

Simone Calogero, Juan Calvo, Óscar Sánchez, Juan Soler. Dispersive behavior in galactic dynamics. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 1-16. doi: 10.3934/dcdsb.2010.14.1
[9]

Juan Manuel Pastor, Javier García-Algarra, Javier Galeano, José María Iriondo, José J. Ramasco. A simple and bounded model of population dynamics for mutualistic networks. Networks & Heterogeneous Media, 2015, 10 (1) : 53-70. doi: 10.3934/nhm.2015.10.53
[10]

Xu Zhang, Xiang Li. Modeling and identification of dynamical system with Genetic Regulation in batch fermentation of glycerol. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 393-403. doi: 10.3934/naco.2015.5.393
[11]

Samir Adly, Oanh Chau, Mohamed Rochdi. Solvability of a class of thermal dynamical contact problems with subdifferential conditions. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 91-104. doi: 10.3934/naco.2012.2.91
[12]

Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4329-4346. doi: 10.3934/dcds.2017185
[13]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183
[14]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301
[15]

Mingxin Wang, Qianying Zhang. Dynamics for the diffusive Leslie-Gower model with double free boundaries. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2591-2607. doi: 10.3934/dcds.2018109
[16]

Wen-Bin Yang, Yan-Ling Li, Jianhua Wu, Hai-Xia Li. Dynamics of a food chain model with ratio-dependent and modified Leslie-Gower functional responses. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2269-2290. doi: 10.3934/dcdsb.2015.20.2269
[17]

Mats Gyllenberg, Jifa Jiang, Lei Niu, Ping Yan. On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 615-650. doi: 10.3934/dcds.2018027
[18]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810
[19]

Tuvi Etzion, Alexander Vardy. On $q$-analogs of Steiner systems and covering designs. Advances in Mathematics of Communications, 2011, 5 (2) : 161-176. doi: 10.3934/amc.2011.5.161
[20]

Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences