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July  2019, 39(7): 4207-4224. doi: 10.3934/dcds.2019170

Equality of Kolmogorov-Sinai and permutation entropy for one-dimensional maps consisting of countably many monotone parts

Institut für Mathematik, Ratzeburger Allee 160, D-23562 Lübeck, Germany

* Corresponding author: Tim Gutjahr

Received  October 2018 Revised  February 2019 Published  April 2019

In this paper, we show that, under some technical assumptions, the Kolmogorov-Sinai entropy and the permutation entropy are equal for one-dimensional maps if there exists a countable partition of the domain of definition into intervals such that the considered map is monotone on each of those intervals. This is a generalization of a result by Bandt, Pompe and G. Keller, who showed that the above holds true under the additional assumptions that the number of intervals on which the map is monotone is finite and that the map is continuous on each of those intervals.

Citation: Tim Gutjahr, Karsten Keller. Equality of Kolmogorov-Sinai and permutation entropy for one-dimensional maps consisting of countably many monotone parts. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4207-4224. doi: 10.3934/dcds.2019170
References:
[1]

J. M. AmigóM. B. Kennel and L. Kocarev, The permutation entropy rate equals the metric entropy rate for ergodic information sources and ergodic dynamical systems, Physica D: Nonlinear Phenomena, 210 (2005), 77-95.  doi: 10.1016/j.physd.2005.07.006.

[2]

C. BandtG. 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]

K. Dajani, C. Kraaikamp and M. A. of America, Ergodic Theory of Numbers, no. Bd. 29 in Carus Mathematical Monographs, Mathematical Association of America, 2002. doi: 10.5948/UPO9781614440277.

[5]

M. Einsiedler and T. Ward, Ergodic Theory: With a View Towards Number Theory, Graduate Texts in Mathematics, 259. Springer-Verlag London, Ltd., London, 2011. doi: 10.1007/978-0-85729-021-2.

[6]

M. Einsiedler, E. Lindenstrauss and T. Ward, Entropy in ergodic theory and homogeneous dynamics, 2017., Available from: https://tbward0.wixsite.com/books/entropy.

[7]

S. Heinemann and O. Schmitt, Rokhlin's Lemma for Non-invertible Maps, Mathematica Gottingensis, Math. Inst., 2000.

[8]

G. Keller, Equilibrium States in Ergodic Theory, London Mathematical Society Student Texts, Cambridge University Press, 1998. doi: 10.1017/CBO9781107359987.

[9]

K. KellerA. M. Unakafov and V. A. Unakafova, On the relation of ks entropy and permutation entropy, Physica D: Nonlinear Phenomena, 241 (2012), 1477-1481.  doi: 10.1016/j.physd.2012.05.010.

[10]

A. Klenke, Probability Theory: A Comprehensive Course, Springer, 2008. doi: 10.1007/978-1-4471-5361-0.

[11]

X. LiG. Ouyang and D. A. Richards, Predictability analysis of absence seizures with permutation entropy, Epilepsy Research, 77 (2007), 70-74.  doi: 10.1016/j.eplepsyres.2007.08.002.

[12]

M. Misiurewicz, Permutations and topological entropy for interval maps, Nonlinearity, 16 (2003), 971-976.  doi: 10.1088/0951-7715/16/3/310.

[13]

N. Nicolaou and J. Georgiou, The use of permutation entropy to characterize sleep electroencephalograms, Clinical EEG and Neuroscience, 42 (2011), 24-28.  doi: 10.1177/155005941104200107.

[14]

K. R. Parthasarathy (ed.), II - Probability Measures in A Metric Space, Probability and Mathematical Statistics: A Series of Monographs and Textbooks, Academic Press, 1967. doi: 10.1016/C2013-0-08107-8.

[15]

A. SilvaH. Cardoso-CruzF. SilvaV. Galhardo and L. Antunes, Comparison of anesthetic depth indexes based on thalamocortical local field potentials in rats, Anesthesiology, 112 (2010), 355-363.  doi: 10.1097/ALN.0b013e3181ca3196.

[16]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, Springer New York, 2000.

show all references

References:
[1]

J. M. AmigóM. B. Kennel and L. Kocarev, The permutation entropy rate equals the metric entropy rate for ergodic information sources and ergodic dynamical systems, Physica D: Nonlinear Phenomena, 210 (2005), 77-95.  doi: 10.1016/j.physd.2005.07.006.

[2]

C. BandtG. 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]

K. Dajani, C. Kraaikamp and M. A. of America, Ergodic Theory of Numbers, no. Bd. 29 in Carus Mathematical Monographs, Mathematical Association of America, 2002. doi: 10.5948/UPO9781614440277.

[5]

M. Einsiedler and T. Ward, Ergodic Theory: With a View Towards Number Theory, Graduate Texts in Mathematics, 259. Springer-Verlag London, Ltd., London, 2011. doi: 10.1007/978-0-85729-021-2.

[6]

M. Einsiedler, E. Lindenstrauss and T. Ward, Entropy in ergodic theory and homogeneous dynamics, 2017., Available from: https://tbward0.wixsite.com/books/entropy.

[7]

S. Heinemann and O. Schmitt, Rokhlin's Lemma for Non-invertible Maps, Mathematica Gottingensis, Math. Inst., 2000.

[8]

G. Keller, Equilibrium States in Ergodic Theory, London Mathematical Society Student Texts, Cambridge University Press, 1998. doi: 10.1017/CBO9781107359987.

[9]

K. KellerA. M. Unakafov and V. A. Unakafova, On the relation of ks entropy and permutation entropy, Physica D: Nonlinear Phenomena, 241 (2012), 1477-1481.  doi: 10.1016/j.physd.2012.05.010.

[10]

A. Klenke, Probability Theory: A Comprehensive Course, Springer, 2008. doi: 10.1007/978-1-4471-5361-0.

[11]

X. LiG. Ouyang and D. A. Richards, Predictability analysis of absence seizures with permutation entropy, Epilepsy Research, 77 (2007), 70-74.  doi: 10.1016/j.eplepsyres.2007.08.002.

[12]

M. Misiurewicz, Permutations and topological entropy for interval maps, Nonlinearity, 16 (2003), 971-976.  doi: 10.1088/0951-7715/16/3/310.

[13]

N. Nicolaou and J. Georgiou, The use of permutation entropy to characterize sleep electroencephalograms, Clinical EEG and Neuroscience, 42 (2011), 24-28.  doi: 10.1177/155005941104200107.

[14]

K. R. Parthasarathy (ed.), II - Probability Measures in A Metric Space, Probability and Mathematical Statistics: A Series of Monographs and Textbooks, Academic Press, 1967. doi: 10.1016/C2013-0-08107-8.

[15]

A. SilvaH. Cardoso-CruzF. SilvaV. Galhardo and L. Antunes, Comparison of anesthetic depth indexes based on thalamocortical local field potentials in rats, Anesthesiology, 112 (2010), 355-363.  doi: 10.1097/ALN.0b013e3181ca3196.

[16]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, Springer New York, 2000.

Figure 1.  Graph of the Gauss function T
Figure 2.  The striped area corresponds to the set $R = \{(\omega_1, \omega_2)\in\Omega^2|~\omega_1\leq \omega_2\}$ and the gray area to $(T\times T)^{-1}(R)$ for the Gauss function $T$
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