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May  2012, 11(3): 1051-1062. doi: 10.3934/cpaa.2012.11.1051

A faithful symbolic extension

Jacek Serafin 1,

1. 

Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland

Received  November 2010 Revised  February 2011 Published  December 2011

We construct a symbolic extension of an aperiodic zero-dimensional topological system in such a way that the bonding map is one-to-one on the set of invariant measures.
Keywords: Topological dynamical system, symbolic extension., entropy.
Mathematics Subject Classification: Primary: 37B10, 37B4.
Citation: Jacek Serafin. A faithful symbolic extension. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1051-1062. doi: 10.3934/cpaa.2012.11.1051
References:
[1]

M. Boyle, Lower entropy factors of sofic systems, Ergodic Theory and Dynam. Systems, 3 (1983), 541-557. doi: 10.1017/S0143385700002133.  Google Scholar

[2]

M. Boyle and T. Downarowicz, The entropy theory of symbolic extensions, Invent. Math., 156 (2004), 119-161. doi: 10.1007/s00222-003-0335-2.  Google Scholar

[3]

M. Boyle, D. Fiebig and U. Fiebig, Residual entropy, conditional entropy and subshift covers, Forum Math., 14 (2002), 713-757. doi: 10.1515/form.2002.031.  Google Scholar

[4]

D. Burguet, Examples of $C^r$ interval maps with large symbolic extension entropy, Discrete Contin. Dyn. Syst., 26 (2010), 873-899. doi: 10.3934/dcds.2010.26.873.  Google Scholar

[5]

T. Downarowicz, Entropy of a symbolic extension of a totally disconnected dynamical system, Ergodic Theory and Dynam. Systems, 21 (2001), 1051-1070. doi: 10.1017/S014338570100150X.  Google Scholar

[6]

T. Downarowicz, Entropy structure, J. Anal. Math., 96 (2005), 57-116. doi: 10.1007/BF02787825.  Google Scholar

[7]

T. Downarowicz, Minimal models for noninvertible and not uniquely ergodic systems, Israel J. Math., 156 (2006), 93-110. doi: 10.1007/BF02773826.  Google Scholar

[8]

T. Downarowicz, "Entropy in Dynamical Systems," New Mathematical Monographs, No. 18, Cambridge University Press, 2011.  Google Scholar

[9]

T. Downarowicz and A. Maass, Smooth interval maps have symbolic extensions: the antarctic theorem, Invent. Math., 176 (2009), 617-636. doi: 10.1007/s00222-008-0172-4.  Google Scholar

[10]

T. Downarowicz and S. E. Newhouse, Symbolic extensions and smooth dynamical systems, Invent. Math., 160 (2005), 453-499. doi: 10.1007/s00222-004-0413-0.  Google Scholar

[11]

E. Lindenstrauss, Lowering topological entropy, J. Anal. Math., 67 (1995), 231-267. doi: 10.1007/BF02787792.  Google Scholar

[12]

J. Serafin, Universally finitary symbolic extensions, Fund. Math., 206 (2009), 281-285. doi: 10.4064/fm206-0-16.  Google Scholar

show all references

References:
[1]

M. Boyle, Lower entropy factors of sofic systems, Ergodic Theory and Dynam. Systems, 3 (1983), 541-557. doi: 10.1017/S0143385700002133.  Google Scholar

[2]

M. Boyle and T. Downarowicz, The entropy theory of symbolic extensions, Invent. Math., 156 (2004), 119-161. doi: 10.1007/s00222-003-0335-2.  Google Scholar

[3]

M. Boyle, D. Fiebig and U. Fiebig, Residual entropy, conditional entropy and subshift covers, Forum Math., 14 (2002), 713-757. doi: 10.1515/form.2002.031.  Google Scholar

[4]

D. Burguet, Examples of $C^r$ interval maps with large symbolic extension entropy, Discrete Contin. Dyn. Syst., 26 (2010), 873-899. doi: 10.3934/dcds.2010.26.873.  Google Scholar

[5]

T. Downarowicz, Entropy of a symbolic extension of a totally disconnected dynamical system, Ergodic Theory and Dynam. Systems, 21 (2001), 1051-1070. doi: 10.1017/S014338570100150X.  Google Scholar

[6]

T. Downarowicz, Entropy structure, J. Anal. Math., 96 (2005), 57-116. doi: 10.1007/BF02787825.  Google Scholar

[7]

T. Downarowicz, Minimal models for noninvertible and not uniquely ergodic systems, Israel J. Math., 156 (2006), 93-110. doi: 10.1007/BF02773826.  Google Scholar

[8]

T. Downarowicz, "Entropy in Dynamical Systems," New Mathematical Monographs, No. 18, Cambridge University Press, 2011.  Google Scholar

[9]

T. Downarowicz and A. Maass, Smooth interval maps have symbolic extensions: the antarctic theorem, Invent. Math., 176 (2009), 617-636. doi: 10.1007/s00222-008-0172-4.  Google Scholar

[10]

T. Downarowicz and S. E. Newhouse, Symbolic extensions and smooth dynamical systems, Invent. Math., 160 (2005), 453-499. doi: 10.1007/s00222-004-0413-0.  Google Scholar

[11]

E. Lindenstrauss, Lowering topological entropy, J. Anal. Math., 67 (1995), 231-267. doi: 10.1007/BF02787792.  Google Scholar

[12]

J. Serafin, Universally finitary symbolic extensions, Fund. Math., 206 (2009), 281-285. doi: 10.4064/fm206-0-16.  Google Scholar

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