Kolmogorov-Sinai entropy via separation properties of order-generated \sigma-algebras

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May 2014, 34(5): 1793-1809. doi: 10.3934/dcds.2014.34.1793

Kolmogorov-Sinai entropy via separation properties of order-generated $\sigma$-algebras

Alexandra Antoniouk ^1, , Karsten Keller ^2, and Sergiy Maksymenko ^1,

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

Received April 2013 Revised July 2013 Published October 2013

Abstract
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In a recent paper, K. Keller has given a characterization of the Kolmogorov-Sinai entropy of a discrete-time measure-preserving dynamical system on the base of an increasing sequence of special partitions. These partitions are constructed from order relations obtained via a given real-valued random vector, which can be interpreted as a collection of observables on the system and is assumed to separate points of it. In the present paper we relax the separation condition in order to generalize the given characterization of Kolmogorov-Sinai entropy, providing a statement on equivalence of $\sigma$-algebras. On its base we show that in the case that a dynamical system is living on an $m$-dimensional smooth manifold and the underlying measure is Lebesgue absolute continuous, the set of smooth random vectors of dimension $n>m$ with given characterization of Kolmogorov-Sinai entropy is large in a certain sense.

Keywords: multijets transversality theorem, permutation entropy, prevalence., Kolmogorov-Sinai entropy, residuality.

Mathematics Subject Classification: Primary: 37A35, 54E52; Secondary: 58A2.

Citation: Alexandra Antoniouk, Karsten Keller, Sergiy Maksymenko. Kolmogorov-Sinai entropy via separation properties of order-generated $\sigma$-algebras. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1793-1809. doi: 10.3934/dcds.2014.34.1793

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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