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

Topological and ergodic properties of symmetric sub-shifts

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.
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
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show all references

References:
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Math. Pannon., 18 (2007), 101-124.  Google Scholar

[2]

Ergodic Theory Dynam. Systems, 29 (2009), 1055-1074. doi: 10.1017/S0143385708000746.  Google Scholar

[3]

Integers, 14 (2014) Paper No. A15, 1-28. Google Scholar

[4]

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

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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

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Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511755316.  Google Scholar

[8]

Ergodic Theory Dynam. Systems, 31 (2011), 1305-1323. doi: 10.1017/S0143385710000556.  Google Scholar

[9]

Israel J. Math., 192 (2012), 785-817. doi: 10.1007/s11856-012-0052-x.  Google Scholar

[10]

Nonlinearity, 25 (2012), 2133-2150. doi: 10.1088/0951-7715/25/7/2133.  Google Scholar

[11]

Nonlinearity, 26 (2013), 307-317. doi: 10.1088/0951-7715/26/1/307.  Google Scholar

[12]

Bull. Soc. Math. France, 118 (1990), 377-390.  Google Scholar

[13]

Math. Res. Lett., 8 (2001), 535-543. doi: 10.4310/MRL.2001.v8.n4.a12.  Google Scholar

[14]

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]

Dokl. Akad. Nauk SSSR, 204 (1972), 15-17.  Google Scholar

[16]

Discrete Contin. Dyn. Syst., 33 (2013), 1965-1973. doi: 10.3934/dcds.2013.33.1965.  Google Scholar

[17]

J. Stat. Phys., 135 (2009), 519-534. doi: 10.1007/s10955-009-9747-8.  Google Scholar

[18]

Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511626302.  Google Scholar

[19]

Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1983.  Google Scholar

[20]

Israel J. Math., 171 (2009), 93-110. doi: 10.1007/s11856-009-0042-9.  Google Scholar

[21]

ArXiv e-prints, arXiv:1002.4614, February 2010. Google Scholar

[22]

Acta Math. Acad. Sci. Hungar., 11 (1960), 401-416. doi: 10.1007/BF02020954.  Google Scholar

[23]

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

[24]

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

[25]

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]

Acta Mathematica Hungarica, pages 1-15, 2014. Google Scholar

[27]

Ergodic Theory Dynam. Systems, 7 (1987), 627-645. doi: 10.1017/S0143385700004247.  Google Scholar

[28]

Springer-Verlag, New York, 1982.  Google Scholar

[29]

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

[30]

Monatsh. Math., 77 (1973), 462-474. doi: 10.1007/BF01295322.  Google Scholar

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