November 2018, 38(11): 5711-5733. doi: 10.3934/dcds.2018249

Local correlation entropy

1. 

Slovanet a.s., Záhradnícka 151, Bratislava, Slovakia

2. 

Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, Banská Bystrica, Slovakia

Received  December 2017 Revised  June 2018 Published  August 2018

Local correlation entropy, introduced by Takens in 1983, represents the exponential decay rate of the relative frequency of recurrences in the trajectory of a point, as the embedding dimension grows to infinity. In this paper we study relationship between the supremum of local correlation entropies and the topological entropy. For dynamical systems on topological graphs we prove that the two quantities coincide. Moreover, there is an uncountable set of points with local correlation entropy arbitrarily close to the topological entropy. On the other hand, we construct a strictly ergodic subshift with positive topological entropy having all local correlation entropies equal to zero. As a necessary tool, we derive an expected relationship between the local correlation entropies of a system and those of its iterates.

Citation: Vladimír Špitalský. Local correlation entropy. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5711-5733. doi: 10.3934/dcds.2018249
References:
[1]

J. AaronsonR. BurtonH. DehlingD. GilatT. Hill and B. Weiss, Strong laws for L- and U-statistics, Trans. Amer. Math. Soc., 348 (1996), 2845-2866. doi: 10.1090/S0002-9947-96-01681-9.

[2]

H. Broer, F. Takens and B. Hasselblatt, Handbook of Dynamical Systems, vol. 3, Elsevier, 2010. doi: 10.1016/C2009-0-18143-6.

[3]

P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Reprint of the 1980 edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2009. doi: 10.1007/978-0-8176-4927-2.

[4]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, vol. 527 of Lecture Notes in Mathematics, Springer, 1976. doi: 10.1007/BFB0082364.

[5]

M. Denker and G. Keller, Rigorous statistical procedures for data from dynamical systems, J. Stat. Phys., 44 (1986), 67-93. doi: 10.1007/BF01010905.

[6]

J. P. EckmannS. O. Kamphorst and D. Ruelle, Recurrence plots of dynamical systems, Europhys. Lett., 4 (1987), 973-977. doi: 10.1142/9789812833709_0030.

[7]

M. Einsiedler and T. Ward, Ergodic Theory with a view towards Number Theory, vol. 259 of Graduate Texts in Mathematics, Springer, 2011. doi: 10.1007/978-0-85729-021-2.

[8]

K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, 3rd edition, John Wiley & Sons, Ltd., Chichester, 2014. doi: 10.1002/0470013850.

[9]

P. Grassberger and I. Procaccia, Characterization of strange attractors, Phys. Rev. Lett., 50 (1983), 346-349. doi: 10.1103/PhysRevLett.50.346.

[10]

P. Grassberger and I. Procaccia, Measuring the strangeness of strange attractors, Phys. D, 9 (1983), 189-208. doi: 10.1016/0167-2789(83)90298-1.

[11]

M. Grendár, J. Majerová and V. Špitalský, Strong laws for recurrence quantification analysis, Internat. J. Bifur. Chaos, 23 (2013), 1350147, 13pp. doi: 10.1142/S0218127413501472.

[12]

C. Grillenberger, Constructions of strictly ergodic systems Ⅰ. Given entropy, Probab. Theory Related Fields, 25 (1973), 323-334. doi: 10.1007/BF00537161.

[13]

F. Hahn and Y. Katznelson, On the entropy of uniquely ergodic transformations, Trans. Amer. Math. Soc., 126 (1967), 335-360. doi: 10.1090/S0002-9947-1967-0207959-1.

[14]

J. Llibre and M. Misiurewicz, Horseshoes, entropy and periods for graph maps, Topology, 32 (1993), 649-664. doi: 10.1016/0040-9383(93)90014-M.

[15]

A. Manning and K. Simon, A short existence proof for correlation dimension, J. Stat. Phys., 90 (1998), 1047-1049. doi: 10.1023/A:1023253709865.

[16]

N. MarwanM. C. RomanoM. Thiel and J. Kurths, Recurrence plots for the analysis of complex systems, Phys. Rep., 438 (2007), 237-329. doi: 10.1016/j.physrep.2006.11.001.

[17]

J. C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc., 58 (1952), 116-136. doi: 10.1090/S0002-9904-1952-09580-X.

[18]

Y. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications, University of Chicago Press, 1997. doi: 10.7208/chicago/9780226662237.001.0001.

[19]

Y. B. Pesin, On rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions, J. Stat. Phys., 71 (1993), 529-547. doi: 10.1007/BF01058436.

[20]

Y. B. Pesin and A. Tempelman, Correlation dimension of measures invariant under group action, Random Comput. Dyn., 3 (1995), 137-156.

[21]

H. Robbins, A remark on Stirling's formula, Amer. Math. Monthly, 62 (1955), 26-29. doi: 10.2307/2308012.

[22]

R. J. Serinko, Ergodic theorems arising in correlation dimension estimation, J. Stat. Phys., 85 (1996), 25-40. doi: 10.1007/BF02175554.

[23]

S. M. Srivastava, A Course on Borel Sets, vol. 180 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1998. doi: 10.1007/978-3-642-85473-6.

[24]

F. Takens, Invariants related to dimension and entropy, In Atas do 13 Colóquio Brasileiro de Mathematica, Rio de Janeiro, (1983), 353-359.

[25]

F. Takens and E. Verbitskiy, Generalized entropies: Rényi and correlation integral approach, Nonlinearity, 11 (1998), 771-782. doi: 10.1088/0951-7715/11/4/001.

[26]

F. Takens and E. Verbitskiy, Multifractal analysis of local entropies for expansive homeomorphisms with specification, Comm. Math. Phys., 203 (1999), 593-612. doi: 10.1007/S002200050627.

[27]

E. Verbitskiy, Generalized Entropies in Dynamical Systems, PhD thesis, University of Groningen, 2000, https://www.rug.nl/research/portal/files/14525487/thesis.pdf.

[28]

P. Walters, An Introduction to Ergodic Theory, vol. 79 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982.

[29]

C. L. Webber Jr and N. Marwan, Recurrence Quantification Analysis: Theory and Best Practices, Springer, 2015. doi: 10.1007/978-3-319-07155-8.

[30]

J. P. Zbilut and C. L. Webber, Embeddings and delays as derived from quantification of recurrence plots, Phys. Lett. A, 171 (1992), 199-203. doi: 10.1016/0375-9601(92)90426-M.

show all references

References:
[1]

J. AaronsonR. BurtonH. DehlingD. GilatT. Hill and B. Weiss, Strong laws for L- and U-statistics, Trans. Amer. Math. Soc., 348 (1996), 2845-2866. doi: 10.1090/S0002-9947-96-01681-9.

[2]

H. Broer, F. Takens and B. Hasselblatt, Handbook of Dynamical Systems, vol. 3, Elsevier, 2010. doi: 10.1016/C2009-0-18143-6.

[3]

P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Reprint of the 1980 edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2009. doi: 10.1007/978-0-8176-4927-2.

[4]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, vol. 527 of Lecture Notes in Mathematics, Springer, 1976. doi: 10.1007/BFB0082364.

[5]

M. Denker and G. Keller, Rigorous statistical procedures for data from dynamical systems, J. Stat. Phys., 44 (1986), 67-93. doi: 10.1007/BF01010905.

[6]

J. P. EckmannS. O. Kamphorst and D. Ruelle, Recurrence plots of dynamical systems, Europhys. Lett., 4 (1987), 973-977. doi: 10.1142/9789812833709_0030.

[7]

M. Einsiedler and T. Ward, Ergodic Theory with a view towards Number Theory, vol. 259 of Graduate Texts in Mathematics, Springer, 2011. doi: 10.1007/978-0-85729-021-2.

[8]

K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, 3rd edition, John Wiley & Sons, Ltd., Chichester, 2014. doi: 10.1002/0470013850.

[9]

P. Grassberger and I. Procaccia, Characterization of strange attractors, Phys. Rev. Lett., 50 (1983), 346-349. doi: 10.1103/PhysRevLett.50.346.

[10]

P. Grassberger and I. Procaccia, Measuring the strangeness of strange attractors, Phys. D, 9 (1983), 189-208. doi: 10.1016/0167-2789(83)90298-1.

[11]

M. Grendár, J. Majerová and V. Špitalský, Strong laws for recurrence quantification analysis, Internat. J. Bifur. Chaos, 23 (2013), 1350147, 13pp. doi: 10.1142/S0218127413501472.

[12]

C. Grillenberger, Constructions of strictly ergodic systems Ⅰ. Given entropy, Probab. Theory Related Fields, 25 (1973), 323-334. doi: 10.1007/BF00537161.

[13]

F. Hahn and Y. Katznelson, On the entropy of uniquely ergodic transformations, Trans. Amer. Math. Soc., 126 (1967), 335-360. doi: 10.1090/S0002-9947-1967-0207959-1.

[14]

J. Llibre and M. Misiurewicz, Horseshoes, entropy and periods for graph maps, Topology, 32 (1993), 649-664. doi: 10.1016/0040-9383(93)90014-M.

[15]

A. Manning and K. Simon, A short existence proof for correlation dimension, J. Stat. Phys., 90 (1998), 1047-1049. doi: 10.1023/A:1023253709865.

[16]

N. MarwanM. C. RomanoM. Thiel and J. Kurths, Recurrence plots for the analysis of complex systems, Phys. Rep., 438 (2007), 237-329. doi: 10.1016/j.physrep.2006.11.001.

[17]

J. C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc., 58 (1952), 116-136. doi: 10.1090/S0002-9904-1952-09580-X.

[18]

Y. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications, University of Chicago Press, 1997. doi: 10.7208/chicago/9780226662237.001.0001.

[19]

Y. B. Pesin, On rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions, J. Stat. Phys., 71 (1993), 529-547. doi: 10.1007/BF01058436.

[20]

Y. B. Pesin and A. Tempelman, Correlation dimension of measures invariant under group action, Random Comput. Dyn., 3 (1995), 137-156.

[21]

H. Robbins, A remark on Stirling's formula, Amer. Math. Monthly, 62 (1955), 26-29. doi: 10.2307/2308012.

[22]

R. J. Serinko, Ergodic theorems arising in correlation dimension estimation, J. Stat. Phys., 85 (1996), 25-40. doi: 10.1007/BF02175554.

[23]

S. M. Srivastava, A Course on Borel Sets, vol. 180 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1998. doi: 10.1007/978-3-642-85473-6.

[24]

F. Takens, Invariants related to dimension and entropy, In Atas do 13 Colóquio Brasileiro de Mathematica, Rio de Janeiro, (1983), 353-359.

[25]

F. Takens and E. Verbitskiy, Generalized entropies: Rényi and correlation integral approach, Nonlinearity, 11 (1998), 771-782. doi: 10.1088/0951-7715/11/4/001.

[26]

F. Takens and E. Verbitskiy, Multifractal analysis of local entropies for expansive homeomorphisms with specification, Comm. Math. Phys., 203 (1999), 593-612. doi: 10.1007/S002200050627.

[27]

E. Verbitskiy, Generalized Entropies in Dynamical Systems, PhD thesis, University of Groningen, 2000, https://www.rug.nl/research/portal/files/14525487/thesis.pdf.

[28]

P. Walters, An Introduction to Ergodic Theory, vol. 79 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982.

[29]

C. L. Webber Jr and N. Marwan, Recurrence Quantification Analysis: Theory and Best Practices, Springer, 2015. doi: 10.1007/978-3-319-07155-8.

[30]

J. P. Zbilut and C. L. Webber, Embeddings and delays as derived from quantification of recurrence plots, Phys. Lett. A, 171 (1992), 199-203. doi: 10.1016/0375-9601(92)90426-M.

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