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

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

Received  April 2013 Revised  July 2013 Published  October 2013

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.
Citation: Alexandra Antoniouk, Karsten Keller, Sergiy Maksymenko. Kolmogorov-Sinai entropy via separation properties of order-generated $\sigma$-algebras. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

C. Bandt and B. Pompe, Permutation entropy: A natural complexity measure for time series, Phys. Rev. Lett., 88 (2002), 174102. Google Scholar

[6]

M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York, 1973.  Google Scholar

[7]

V. Guillemin and A. Polak, Differential Topology, Prentice-Hall, Englewood Cliff, NJ, 1974.  Google Scholar

[8]

M. Hirsch, Differential Topology, Graduate Texts in Mathematics, Vol. 33, Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar

[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.  Google Scholar

[10]

K. Keller, Permutations and the Kolmogorov-Sinai entropy, Discrete Contin. Dyn. Syst., 32 (2012), 891-900. doi: 10.3934/dcds.2012.32.891.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. Milnor, Differential Topology, Mimeographed notes, Princeton University, New Jersey, 1958.  Google Scholar

[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.  Google Scholar

[17]

T. Sauer, J. Yorke and M. Casdagli, Embeddology, J. Stat. Phys., 65 (1991), 579-616. doi: 10.1007/BF01053745.  Google Scholar

[18]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, Vol. 79, Springer-Verlag, New York, 1982.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

C. Bandt and B. Pompe, Permutation entropy: A natural complexity measure for time series, Phys. Rev. Lett., 88 (2002), 174102. Google Scholar

[6]

M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York, 1973.  Google Scholar

[7]

V. Guillemin and A. Polak, Differential Topology, Prentice-Hall, Englewood Cliff, NJ, 1974.  Google Scholar

[8]

M. Hirsch, Differential Topology, Graduate Texts in Mathematics, Vol. 33, Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar

[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.  Google Scholar

[10]

K. Keller, Permutations and the Kolmogorov-Sinai entropy, Discrete Contin. Dyn. Syst., 32 (2012), 891-900. doi: 10.3934/dcds.2012.32.891.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. Milnor, Differential Topology, Mimeographed notes, Princeton University, New Jersey, 1958.  Google Scholar

[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.  Google Scholar

[17]

T. Sauer, J. Yorke and M. Casdagli, Embeddology, J. Stat. Phys., 65 (1991), 579-616. doi: 10.1007/BF01053745.  Google Scholar

[18]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, Vol. 79, Springer-Verlag, New York, 1982.  Google Scholar

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