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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[21]

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

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[21]

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

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[1]

Milton Ko. Rényi entropy and recurrence. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2403-2421. doi: 10.3934/dcds.2013.33.2403

[2]

John Kieffer and En-hui Yang. Ergodic behavior of graph entropy. Electronic Research Announcements, 1997, 3: 11-16.

[3]

Katrin Gelfert. Lower bounds for the topological entropy. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555

[4]

Jaume Llibre. Brief survey on the topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3363-3374. doi: 10.3934/dcdsb.2015.20.3363

[5]

Boris Hasselblatt, Zbigniew Nitecki, James Propp. Topological entropy for nonuniformly continuous maps. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 201-213. doi: 10.3934/dcds.2008.22.201

[6]

Michał Misiurewicz. On Bowen's definition of topological entropy. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 827-833. doi: 10.3934/dcds.2004.10.827

[7]

Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 545-556. doi: 10.3934/dcds.2011.31.545

[8]

Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547

[9]

Lassi Roininen, Markku S. Lehtinen, Sari Lasanen, Mikko Orispää, Markku Markkanen. Correlation priors. Inverse Problems & Imaging, 2011, 5 (1) : 167-184. doi: 10.3934/ipi.2011.5.167

[10]

Jean René Chazottes, E. Ugalde. Entropy estimation and fluctuations of hitting and recurrence times for Gibbsian sources. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 565-586. doi: 10.3934/dcdsb.2005.5.565

[11]

Jan Philipp Schröder. Ergodicity and topological entropy of geodesic flows on surfaces. Journal of Modern Dynamics, 2015, 9: 147-167. doi: 10.3934/jmd.2015.9.147

[12]

Eva Glasmachers, Gerhard Knieper, Carlos Ogouyandjou, Jan Philipp Schröder. Topological entropy of minimal geodesics and volume growth on surfaces. Journal of Modern Dynamics, 2014, 8 (1) : 75-91. doi: 10.3934/jmd.2014.8.75

[13]

Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461

[14]

César J. Niche. Topological entropy of a magnetic flow and the growth of the number of trajectories. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 577-580. doi: 10.3934/dcds.2004.11.577

[15]

Yujun Ju, Dongkui Ma, Yupan Wang. Topological entropy of free semigroup actions for noncompact sets. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 995-1017. doi: 10.3934/dcds.2019041

[16]

Yun Zhao, Wen-Chiao Cheng, Chih-Chang Ho. Q-entropy for general topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2059-2075. doi: 10.3934/dcds.2019086

[17]

Tao Wang, Yu Huang. Weighted topological and measure-theoretic entropy. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3941-3967. doi: 10.3934/dcds.2019159

[18]

Xiaomin Zhou. Relative entropy dimension of topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6631-6642. doi: 10.3934/dcds.2019288

[19]

L'ubomír Snoha, Vladimír Špitalský. Recurrence equals uniform recurrence does not imply zero entropy for triangular maps of the square. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 821-835. doi: 10.3934/dcds.2006.14.821

[20]

Ming Su, Arne Winterhof. Hamming correlation of higher order. Advances in Mathematics of Communications, 2018, 12 (3) : 505-513. doi: 10.3934/amc.2018029

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (61)
  • HTML views (67)
  • Cited by (0)

Other articles
by authors

[Back to Top]