March  2012, 32(3): 891-900. doi: 10.3934/dcds.2012.32.891

Permutations and the Kolmogorov-Sinai entropy

1. 

Institute of Mathematics, University of Lübeck, Wallstraße 40, D-23560 Luebeck, Germany

Received  September 2010 Revised  December 2010 Published  October 2011

This paper provides a way for determining the Kolmogorov-Sinai entropy of time-discrete dynamical systems on the base of quantifying ordinal patterns obtained from a finite set of observables. As a consequence, it is shown that the Kolmogorov-Sinai entropy is bounded from above by a quantity which generalizes the concept of permutation entropy. In this framework, the determination of the Kolmogorov-Sinai entropy of a multidimensional system by use of only a single one-dimensional observable and Takens' embedding theorem is discussed.
Citation: Karsten Keller. Permutations and the Kolmogorov-Sinai entropy. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 891-900. doi: 10.3934/dcds.2012.32.891
References:
[1]

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. doi: 10.1016/j.physd.2005.07.006. Google Scholar

[2]

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. Google Scholar

[3]

C. Bandt and B. Pompe, Permutation entropy: A natural complexity measure for time series,, Phys. Rev. Lett., 88 (2002). doi: 10.1103/PhysRevLett.88.174102. Google Scholar

[4]

M. Einsiedler and T. Ward, "Ergodic Theory With a View Towards Number Theory,", Graduate Texts in Mathematics, 259 (2011). Google Scholar

[5]

M. Einsiedler, E. Lindestrauss and T. Ward, "Entropy in Ergodic Theory and Homogeneous Dynamics.'', Available from: \url{http://www.uea.ac.uk/menu/acad\_depts/mth/entropy}., (). Google Scholar

[6]

K. Keller and M. Sinn, Kolmogorov-Sinai entropy from the ordinal viewpoint,, Physica D, 239 (2010), 997. doi: 10.1016/j.physd.2010.02.006. Google Scholar

[7]

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. Google Scholar

[8]

K. Keller, J. Emonds and M. Sinn, Time series from the ordinal viewpoint,, Stochastics and Dynamics, 2 (2007), 247. doi: 10.1142/S0219493707002025. Google Scholar

[9]

M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings,, Stud. Math., 67 (1980), 45. Google Scholar

[10]

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

[11]

F. Takens, Detecting strange attractors in turbulence,, in, 898 (1981), 366. Google Scholar

[12]

P. Walters, "An Introduction to Ergodic Theory,'', Graduate Texts in Mathematics, 79 (1982). Google Scholar

show all references

References:
[1]

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. doi: 10.1016/j.physd.2005.07.006. Google Scholar

[2]

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. Google Scholar

[3]

C. Bandt and B. Pompe, Permutation entropy: A natural complexity measure for time series,, Phys. Rev. Lett., 88 (2002). doi: 10.1103/PhysRevLett.88.174102. Google Scholar

[4]

M. Einsiedler and T. Ward, "Ergodic Theory With a View Towards Number Theory,", Graduate Texts in Mathematics, 259 (2011). Google Scholar

[5]

M. Einsiedler, E. Lindestrauss and T. Ward, "Entropy in Ergodic Theory and Homogeneous Dynamics.'', Available from: \url{http://www.uea.ac.uk/menu/acad\_depts/mth/entropy}., (). Google Scholar

[6]

K. Keller and M. Sinn, Kolmogorov-Sinai entropy from the ordinal viewpoint,, Physica D, 239 (2010), 997. doi: 10.1016/j.physd.2010.02.006. Google Scholar

[7]

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. Google Scholar

[8]

K. Keller, J. Emonds and M. Sinn, Time series from the ordinal viewpoint,, Stochastics and Dynamics, 2 (2007), 247. doi: 10.1142/S0219493707002025. Google Scholar

[9]

M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings,, Stud. Math., 67 (1980), 45. Google Scholar

[10]

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

[11]

F. Takens, Detecting strange attractors in turbulence,, in, 898 (1981), 366. Google Scholar

[12]

P. Walters, "An Introduction to Ergodic Theory,'', Graduate Texts in Mathematics, 79 (1982). Google Scholar

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