# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 77-95. doi: 10.1016/j.physd.2005.07.006. [2] 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. [3] C. Bandt and B. Pompe, Permutation entropy: A natural complexity measure for time series, Phys. Rev. Lett., 88 (2002), 174102. doi: 10.1103/PhysRevLett.88.174102. [4] M. Einsiedler and T. Ward, "Ergodic Theory With a View Towards Number Theory," Graduate Texts in Mathematics, 259, Springer-Verlag London, Ltd., London, 2011. [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}., (). [6] 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. [7] 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. [8] K. Keller, J. Emonds and M. Sinn, Time series from the ordinal viewpoint, Stochastics and Dynamics, 2 (2007), 247-272. doi: 10.1142/S0219493707002025. [9] M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Stud. Math., 67 (1980), 45-63. [10] T. Sauer, J. Yorke and M. Casdagli, Embeddology, J. Stat. Phys., 65 (1991), 579-616. doi: 10.1007/BF01053745. [11] F. Takens, Detecting strange attractors in turbulence, in "Dynamical Systems and Turbulence'' (eds. D. A. Rand and L. S. Young), Lecture Notes in Mathematics, 898, Springer, Berlin-New York, (1981), 366-381. [12] P. Walters, "An Introduction to Ergodic Theory,'' Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

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), 77-95. doi: 10.1016/j.physd.2005.07.006. [2] 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. [3] C. Bandt and B. Pompe, Permutation entropy: A natural complexity measure for time series, Phys. Rev. Lett., 88 (2002), 174102. doi: 10.1103/PhysRevLett.88.174102. [4] M. Einsiedler and T. Ward, "Ergodic Theory With a View Towards Number Theory," Graduate Texts in Mathematics, 259, Springer-Verlag London, Ltd., London, 2011. [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}., (). [6] 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. [7] 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. [8] K. Keller, J. Emonds and M. Sinn, Time series from the ordinal viewpoint, Stochastics and Dynamics, 2 (2007), 247-272. doi: 10.1142/S0219493707002025. [9] M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Stud. Math., 67 (1980), 45-63. [10] T. Sauer, J. Yorke and M. Casdagli, Embeddology, J. Stat. Phys., 65 (1991), 579-616. doi: 10.1007/BF01053745. [11] F. Takens, Detecting strange attractors in turbulence, in "Dynamical Systems and Turbulence'' (eds. D. A. Rand and L. S. Young), Lecture Notes in Mathematics, 898, Springer, Berlin-New York, (1981), 366-381. [12] P. Walters, "An Introduction to Ergodic Theory,'' Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.
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