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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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[8]

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

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[8]

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

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

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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