American Institute of Mathematical Sciences

Advanced
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.  doi: 10.1017/S0143385700002133.  Google Scholar

[2]

M. Boyle and T. Downarowicz, The entropy theory of symbolic extensions,, Invent. Math., 156 (2004), 119.  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.  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.  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.  doi: 10.1017/S014338570100150X.  Google Scholar

[6]

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

[7]

T. Downarowicz, Minimal models for noninvertible and not uniquely ergodic systems,, Israel J. Math., 156 (2006), 93.  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.  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.  doi: 10.1007/s00222-004-0413-0.  Google Scholar

[11]

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

[12]

J. Serafin, Universally finitary symbolic extensions,, Fund. Math., 206 (2009), 281.  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.  doi: 10.1017/S0143385700002133.  Google Scholar

[2]

M. Boyle and T. Downarowicz, The entropy theory of symbolic extensions,, Invent. Math., 156 (2004), 119.  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.  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.  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.  doi: 10.1017/S014338570100150X.  Google Scholar

[6]

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

[7]

T. Downarowicz, Minimal models for noninvertible and not uniquely ergodic systems,, Israel J. Math., 156 (2006), 93.  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.  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.  doi: 10.1007/s00222-004-0413-0.  Google Scholar

[11]

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

[12]

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

[1]

David Burguet. Examples of $\mathcal{C}^r$ interval map with large symbolic extension entropy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 873-899. doi: 10.3934/dcds.2010.26.873
[2]

Mike Boyle, Tomasz Downarowicz. Symbolic extension entropy: $c^r$ examples, products and flows. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 329-341. doi: 10.3934/dcds.2006.16.329
[3]

Wen-Guei Hu, Song-Sun Lin. On spatial entropy of multi-dimensional symbolic dynamical systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3705-3717. doi: 10.3934/dcds.2016.36.3705
[4]

Yun Zhao, Wen-Chiao Cheng, Chih-Chang Ho. Q-entropy for general topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2059-2075. doi: 10.3934/dcds.2019086
[5]

Xiaomin Zhou. Relative entropy dimension of topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6631-6642. doi: 10.3934/dcds.2019288
[6]

João Ferreira Alves, Michal Málek. Zeta functions and topological entropy of periodic nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 465-482. doi: 10.3934/dcds.2013.33.465
[7]

Karsten Keller, Sergiy Maksymenko, Inga Stolz. Entropy determination based on the ordinal structure of a dynamical system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3507-3524. doi: 10.3934/dcdsb.2015.20.3507
[8]

Jakub Šotola. Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5119-5128. doi: 10.3934/dcds.2018225
[9]

Katrin Gelfert. Lower bounds for the topological entropy. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555
[10]

Jaume Llibre. Brief survey on the topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3363-3374. doi: 10.3934/dcdsb.2015.20.3363
[11]

Fryderyk Falniowski, Marcin Kulczycki, Dominik Kwietniak, Jian Li. Two results on entropy, chaos and independence in symbolic dynamics. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3487-3505. doi: 10.3934/dcdsb.2015.20.3487
[12]

Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 545-556. doi: 10.3934/dcds.2011.31.545
[13]

Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547
[14]

Boris Hasselblatt, Zbigniew Nitecki, James Propp. Topological entropy for nonuniformly continuous maps. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 201-213. doi: 10.3934/dcds.2008.22.201
[15]

Michał Misiurewicz. On Bowen's definition of topological entropy. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 827-833. doi: 10.3934/dcds.2004.10.827
[16]

Alfredo Marzocchi, Sara Zandonella Necca. Attractors for dynamical systems in topological spaces. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 585-597. doi: 10.3934/dcds.2002.8.585
[17]

Jan Philipp Schröder. Ergodicity and topological entropy of geodesic flows on surfaces. Journal of Modern Dynamics, 2015, 9: 147-167. doi: 10.3934/jmd.2015.9.147
[18]

Eva Glasmachers, Gerhard Knieper, Carlos Ogouyandjou, Jan Philipp Schröder. Topological entropy of minimal geodesics and volume growth on surfaces. Journal of Modern Dynamics, 2014, 8 (1) : 75-91. doi: 10.3934/jmd.2014.8.75
[19]

Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461
[20]

César J. Niche. Topological entropy of a magnetic flow and the growth of the number of trajectories. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 577-580. doi: 10.3934/dcds.2004.11.577

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences