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Kolmogorov-Sinai entropy via separation properties of order-generated $\sigma$-algebras
1. | Institute of Mathematics of NAS of Ukraine, Tereshchenkivs'ka str., 3, 01601 Kyiv, Ukraine, Ukraine |
2. | Universität zu Lübeck, Institut für Mathematik, Ratzeburger Allee 160, 23562 Lübeck, Germany |
References:
[1] |
J. M. Amigó, The equality of Kolmogorov-Sinai entropy and metric permutation entropy generalized, Physica D, 241 (2012), 789-793.
doi: 10.1016/j.physd.2012.01.004. |
[2] |
J. M. Amigó, Permutation Complexity in Dynamical Systems. Ordinal Patterns, Permutation Entropy and All That, Springer Series in Synergetics, Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-04084-9. |
[3] |
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, 210 (2005), 77-95.
doi: 10.1016/j.physd.2005.07.006. |
[4] |
C. Bandt, G. Keller and B. Pompe, Entropy of Interval Maps Via Permutations, Nonlinearity, 15 (2002), 1595-1602.
doi: 10.1088/0951-7715/15/5/312. |
[5] |
C. Bandt and B. Pompe, Permutation entropy: A natural complexity measure for time series, Phys. Rev. Lett., 88 (2002), 174102. |
[6] |
M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York, 1973. |
[7] |
V. Guillemin and A. Polak, Differential Topology, Prentice-Hall, Englewood Cliff, NJ, 1974. |
[8] |
M. Hirsch, Differential Topology, Graduate Texts in Mathematics, Vol. 33, Springer-Verlag, New York-Heidelberg, 1976. |
[9] |
B. R. Hunt, T. Sauer and J. A. Yourke, Prevalence: A translation-invariant "almost every'' on infinite-dimensional spaces, Bull. Amer. Math. Soc., 27 (1992), 217-238.
doi: 10.1090/S0273-0979-1992-00328-2. |
[10] |
K. Keller, Permutations and the Kolmogorov-Sinai entropy, Discrete Contin. Dyn. Syst., 32 (2012), 891-900.
doi: 10.3934/dcds.2012.32.891. |
[11] |
K. Keller, A. Unakafov and V. Unakafova, On the Relation of KS Entropy and Permutation Entropy, Physica D, 241 (2012), 1477-1481.
doi: 10.1016/j.physd.2012.05.010. |
[12] |
K. Keller and M. Sinn, Kolmogorov-Sinai entropy from the ordinal viewpoint, Physica D, 239 (2010), 997-1000.
doi: 10.1016/j.physd.2010.02.006. |
[13] |
K. Keller and M. Sinn, A standardized approach to the Kolmogorov-Sinai entropy, Nonlinearity, 22 (2009), 2417-2422.
doi: 10.1088/0951-7715/22/10/006. |
[14] |
A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, Vol. 156, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-4190-4. |
[15] |
J. Milnor, Differential Topology, Mimeographed notes, Princeton University, New Jersey, 1958. |
[16] |
F. Takens, Detecting Strange Attractors in Turbulence, in: Dynamical Systems and Turbulence (eds. D. A. Rand, L. S. Young), Lecture Notes in Mathematics 898, Springer-Verlag, Berlin-New York, 1981, 366-381. |
[17] |
T. Sauer, J. Yorke and M. Casdagli, Embeddology, J. Stat. Phys., 65 (1991), 579-616.
doi: 10.1007/BF01053745. |
[18] |
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, Vol. 79, Springer-Verlag, New York, 1982. |
show all references
References:
[1] |
J. M. Amigó, The equality of Kolmogorov-Sinai entropy and metric permutation entropy generalized, Physica D, 241 (2012), 789-793.
doi: 10.1016/j.physd.2012.01.004. |
[2] |
J. M. Amigó, Permutation Complexity in Dynamical Systems. Ordinal Patterns, Permutation Entropy and All That, Springer Series in Synergetics, Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-04084-9. |
[3] |
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, 210 (2005), 77-95.
doi: 10.1016/j.physd.2005.07.006. |
[4] |
C. Bandt, G. Keller and B. Pompe, Entropy of Interval Maps Via Permutations, Nonlinearity, 15 (2002), 1595-1602.
doi: 10.1088/0951-7715/15/5/312. |
[5] |
C. Bandt and B. Pompe, Permutation entropy: A natural complexity measure for time series, Phys. Rev. Lett., 88 (2002), 174102. |
[6] |
M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York, 1973. |
[7] |
V. Guillemin and A. Polak, Differential Topology, Prentice-Hall, Englewood Cliff, NJ, 1974. |
[8] |
M. Hirsch, Differential Topology, Graduate Texts in Mathematics, Vol. 33, Springer-Verlag, New York-Heidelberg, 1976. |
[9] |
B. R. Hunt, T. Sauer and J. A. Yourke, Prevalence: A translation-invariant "almost every'' on infinite-dimensional spaces, Bull. Amer. Math. Soc., 27 (1992), 217-238.
doi: 10.1090/S0273-0979-1992-00328-2. |
[10] |
K. Keller, Permutations and the Kolmogorov-Sinai entropy, Discrete Contin. Dyn. Syst., 32 (2012), 891-900.
doi: 10.3934/dcds.2012.32.891. |
[11] |
K. Keller, A. Unakafov and V. Unakafova, On the Relation of KS Entropy and Permutation Entropy, Physica D, 241 (2012), 1477-1481.
doi: 10.1016/j.physd.2012.05.010. |
[12] |
K. Keller and M. Sinn, Kolmogorov-Sinai entropy from the ordinal viewpoint, Physica D, 239 (2010), 997-1000.
doi: 10.1016/j.physd.2010.02.006. |
[13] |
K. Keller and M. Sinn, A standardized approach to the Kolmogorov-Sinai entropy, Nonlinearity, 22 (2009), 2417-2422.
doi: 10.1088/0951-7715/22/10/006. |
[14] |
A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, Vol. 156, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-4190-4. |
[15] |
J. Milnor, Differential Topology, Mimeographed notes, Princeton University, New Jersey, 1958. |
[16] |
F. Takens, Detecting Strange Attractors in Turbulence, in: Dynamical Systems and Turbulence (eds. D. A. Rand, L. S. Young), Lecture Notes in Mathematics 898, Springer-Verlag, Berlin-New York, 1981, 366-381. |
[17] |
T. Sauer, J. Yorke and M. Casdagli, Embeddology, J. Stat. Phys., 65 (1991), 579-616.
doi: 10.1007/BF01053745. |
[18] |
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, Vol. 79, Springer-Verlag, New York, 1982. |
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