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Two results on entropy, chaos and independence in symbolic dynamics
Entropy determination based on the ordinal structure of a dynamical system
1. | Universität zu Lübeck, Institut für Mathematik, Ratzeburger Allee 160, 23562 Lübeck |
2. | Institute of Mathematics of NAS of Ukraine, Tereshchenkivs'ka str., 3, 01601 Kyiv |
3. | Universität zu Lübeck, Institut für Mathematik, Ratzeburger Allee 160, Lübeck, 23562, Germany |
References:
[1] |
J. M. Amigó, K. Keller and V. A. Unakafova, Ordinal symbolic analysis and its application to biomedical recordings,, Philosophical Transactions Royal Society A, 373 (2015). Google Scholar |
[2] |
J. M. Amigó, K. Keller and J. Kurths (eds.), Recent progess in symbolic dynamics and permutation complexity. Ten years of permutation entropy,, The European Physical Journal Special Topics, 222 (2013), 241. Google Scholar |
[3] |
J. M. Amigó, The equality of Kolmogorov-Sinai entropy and metric permutation entropy generalized,, Physica D: Nonlinear Phenomena, 241 (2012), 789.
doi: 10.1016/j.physd.2012.01.004. |
[4] |
J. M. Amigó, Permutation Complexity in Dynamical Systems: Ordinal Patterns, Permutation Entropy and All That,, Springer Series in Synergetics, (2010).
doi: 10.1007/978-3-642-04084-9. |
[5] |
J. M. Amigó, M. B. Kennel and L. Kocarev, The permutation entropy rate equals the metric entropy rate for ergodic information sources and ergodic dynamical systems,, Physica D: Nonlinear Phenomena, 210 (2005), 77.
doi: 10.1016/j.physd.2005.07.006. |
[6] |
A. Antoniouk, K. Keller and S. Maksymenko, Kolmogorov-Sinai entropy via separation properties of order-generated sigma-algebras,, Discrete and Continuous Dynamical Systems - A, 34 (2014), 1793.
|
[7] |
C. Bandt, G. Keller and B. Pompe, Entropy of interval maps via permutations,, Nonlinearity, 15 (2002), 1595.
doi: 10.1088/0951-7715/15/5/312. |
[8] |
C. Bandt and B. Pompe, Permutation entropy: A natural complexity measure for time series,, Physical Review Letters, 88 (2002). Google Scholar |
[9] |
M. Einsiedler, E. Lindenstrauss and T. Ward, Entropy in ergodic theory and homogeneous dynamics,, unpublished, (2014). Google Scholar |
[10] |
M. Einsiedler and T. Ward, Ergodic Theory With a View Towards Number Theory,, Graduate Texts in Mathematics, (2010). Google Scholar |
[11] |
W. Hurewicz and H. Wallman, Dimension Theory,, Princeton Mathematical Series, (1941).
|
[12] |
A. S. Kechris, Classical Descriptive Set Theory,, Graduate Texts in Mathematics, (1995).
doi: 10.1007/978-1-4612-4190-4. |
[13] |
K. Keller, Permutations and the Kolmogorov-Sinai entropy,, Discrete and Continuous Dynamical Systems - A, 32 (2012), 891.
doi: 10.3934/dcds.2012.32.891. |
[14] |
K. Keller and M. Sinn, Kolmogorov-Sinai entropy from the ordinal viewpoint,, Physica D: Nonlinear Phenomena, 239 (2010), 997.
doi: 10.1016/j.physd.2010.02.006. |
[15] |
K. Keller and M. Sinn, A standardized approach to the Kolmogorov-Sinai entropy,, Nonlinearity, 22 (2009), 2417.
doi: 10.1088/0951-7715/22/10/006. |
[16] |
K. Keller, M. Sinn and J. Emonds, Time series from the ordinal viewpoint,, Stochastics and Dynamics, 7 (2007), 247.
doi: 10.1142/S0219493707002025. |
[17] |
K. Keller, A. M. Unakafov and V. A. Unakafova, On the relation of KS entropy and permutation entropy,, Physica D: Nonlinear Phenomena, 241 (2012), 1477.
doi: 10.1016/j.physd.2012.05.010. |
[18] |
W. Krieger, On entropy and generators of measure-preserving transformations,, Transactions of the American Mathematical Society, 149 (1970), 453.
doi: 10.1090/S0002-9947-1970-0259068-3. |
[19] |
W. Parry, Generators and strong generators in ergodic theory,, Bulletin of the American Mathematical Society, 72 (1966), 294.
doi: 10.1090/S0002-9904-1966-11498-2. |
[20] |
M. Riedl, A. Müller and N. Wessel, Practical considerations of permutation entropy; a tutorial review,, The European Physical Journal Special Topics, 222 (2013), 249. Google Scholar |
[21] |
V. A. Rohlin, Exact endomorphisms of a Lebesgue space,, Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya, 25 (1961), 499.
|
[22] |
T. Sauer, J. A. Yorke and M. Casdagli, Embeddology,, Journal of Statistical Physics, 65 (1991), 579.
doi: 10.1007/BF01053745. |
[23] |
F. Takens, Detecting strange attractors in turbulence,, in Dynamical Systems and Turbulence (eds. D. A. Rand and L. S. Young), (1981), 366.
|
[24] |
A. M. Unakafov and K. Keller, Conditional entropy of ordinal patterns,, Physica D: Nonlinear Phenomena, 269 (2014), 94.
doi: 10.1016/j.physd.2013.11.015. |
[25] |
V. A. Unakafova and K. Keller, Efficiently measuring complexity on the basis of real-world data,, Entropy, 15 (2013), 4392.
doi: 10.3390/e15104392. |
[26] |
V. A. Unakafova, A. M. Unakafov and K. Keller, An approach to comparing Kolmogorov-Sinai and permutation entropy,, The European Physical Journal Special Topics, 222 (2013), 353.
doi: 10.1140/epjst/e2013-01846-7. |
[27] |
P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (2000).
doi: 10.1007/978-1-4612-5775-2. |
[28] |
M. Zanin, L. Zunino, O. A. Rosso and D. Papo, Permutation entropy and its main biomedical and econophysics applications: A review,, Entropy, 14 (2012), 1553.
doi: 10.3390/e14081553. |
show all references
References:
[1] |
J. M. Amigó, K. Keller and V. A. Unakafova, Ordinal symbolic analysis and its application to biomedical recordings,, Philosophical Transactions Royal Society A, 373 (2015). Google Scholar |
[2] |
J. M. Amigó, K. Keller and J. Kurths (eds.), Recent progess in symbolic dynamics and permutation complexity. Ten years of permutation entropy,, The European Physical Journal Special Topics, 222 (2013), 241. Google Scholar |
[3] |
J. M. Amigó, The equality of Kolmogorov-Sinai entropy and metric permutation entropy generalized,, Physica D: Nonlinear Phenomena, 241 (2012), 789.
doi: 10.1016/j.physd.2012.01.004. |
[4] |
J. M. Amigó, Permutation Complexity in Dynamical Systems: Ordinal Patterns, Permutation Entropy and All That,, Springer Series in Synergetics, (2010).
doi: 10.1007/978-3-642-04084-9. |
[5] |
J. M. Amigó, M. B. Kennel and L. Kocarev, The permutation entropy rate equals the metric entropy rate for ergodic information sources and ergodic dynamical systems,, Physica D: Nonlinear Phenomena, 210 (2005), 77.
doi: 10.1016/j.physd.2005.07.006. |
[6] |
A. Antoniouk, K. Keller and S. Maksymenko, Kolmogorov-Sinai entropy via separation properties of order-generated sigma-algebras,, Discrete and Continuous Dynamical Systems - A, 34 (2014), 1793.
|
[7] |
C. Bandt, G. Keller and B. Pompe, Entropy of interval maps via permutations,, Nonlinearity, 15 (2002), 1595.
doi: 10.1088/0951-7715/15/5/312. |
[8] |
C. Bandt and B. Pompe, Permutation entropy: A natural complexity measure for time series,, Physical Review Letters, 88 (2002). Google Scholar |
[9] |
M. Einsiedler, E. Lindenstrauss and T. Ward, Entropy in ergodic theory and homogeneous dynamics,, unpublished, (2014). Google Scholar |
[10] |
M. Einsiedler and T. Ward, Ergodic Theory With a View Towards Number Theory,, Graduate Texts in Mathematics, (2010). Google Scholar |
[11] |
W. Hurewicz and H. Wallman, Dimension Theory,, Princeton Mathematical Series, (1941).
|
[12] |
A. S. Kechris, Classical Descriptive Set Theory,, Graduate Texts in Mathematics, (1995).
doi: 10.1007/978-1-4612-4190-4. |
[13] |
K. Keller, Permutations and the Kolmogorov-Sinai entropy,, Discrete and Continuous Dynamical Systems - A, 32 (2012), 891.
doi: 10.3934/dcds.2012.32.891. |
[14] |
K. Keller and M. Sinn, Kolmogorov-Sinai entropy from the ordinal viewpoint,, Physica D: Nonlinear Phenomena, 239 (2010), 997.
doi: 10.1016/j.physd.2010.02.006. |
[15] |
K. Keller and M. Sinn, A standardized approach to the Kolmogorov-Sinai entropy,, Nonlinearity, 22 (2009), 2417.
doi: 10.1088/0951-7715/22/10/006. |
[16] |
K. Keller, M. Sinn and J. Emonds, Time series from the ordinal viewpoint,, Stochastics and Dynamics, 7 (2007), 247.
doi: 10.1142/S0219493707002025. |
[17] |
K. Keller, A. M. Unakafov and V. A. Unakafova, On the relation of KS entropy and permutation entropy,, Physica D: Nonlinear Phenomena, 241 (2012), 1477.
doi: 10.1016/j.physd.2012.05.010. |
[18] |
W. Krieger, On entropy and generators of measure-preserving transformations,, Transactions of the American Mathematical Society, 149 (1970), 453.
doi: 10.1090/S0002-9947-1970-0259068-3. |
[19] |
W. Parry, Generators and strong generators in ergodic theory,, Bulletin of the American Mathematical Society, 72 (1966), 294.
doi: 10.1090/S0002-9904-1966-11498-2. |
[20] |
M. Riedl, A. Müller and N. Wessel, Practical considerations of permutation entropy; a tutorial review,, The European Physical Journal Special Topics, 222 (2013), 249. Google Scholar |
[21] |
V. A. Rohlin, Exact endomorphisms of a Lebesgue space,, Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya, 25 (1961), 499.
|
[22] |
T. Sauer, J. A. Yorke and M. Casdagli, Embeddology,, Journal of Statistical Physics, 65 (1991), 579.
doi: 10.1007/BF01053745. |
[23] |
F. Takens, Detecting strange attractors in turbulence,, in Dynamical Systems and Turbulence (eds. D. A. Rand and L. S. Young), (1981), 366.
|
[24] |
A. M. Unakafov and K. Keller, Conditional entropy of ordinal patterns,, Physica D: Nonlinear Phenomena, 269 (2014), 94.
doi: 10.1016/j.physd.2013.11.015. |
[25] |
V. A. Unakafova and K. Keller, Efficiently measuring complexity on the basis of real-world data,, Entropy, 15 (2013), 4392.
doi: 10.3390/e15104392. |
[26] |
V. A. Unakafova, A. M. Unakafov and K. Keller, An approach to comparing Kolmogorov-Sinai and permutation entropy,, The European Physical Journal Special Topics, 222 (2013), 353.
doi: 10.1140/epjst/e2013-01846-7. |
[27] |
P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (2000).
doi: 10.1007/978-1-4612-5775-2. |
[28] |
M. Zanin, L. Zunino, O. A. Rosso and D. Papo, Permutation entropy and its main biomedical and econophysics applications: A review,, Entropy, 14 (2012), 1553.
doi: 10.3390/e14081553. |
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