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]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[2]

Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020350

[3]

Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363

[4]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[5]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[6]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (104)
  • HTML views (112)
  • Cited by (1)

Other articles
by authors

[Back to Top]