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September  2016, 36(9): 5067-5096. doi: 10.3934/dcds.2016020

Correlation integral and determinism for a family of $2^\infty$ maps

Jana Majerová 1,

1. 

Slovanet a.s., Záhradnícka 151, 821 08 Bratislava, Slovak Republic

Received  June 2015 Revised  March 2016 Published  May 2016

The correlation integral and determinism are quantitative characteristics of a dynamical system based on the recurrence of orbits. For strongly non-chaotic interval maps, the determinism equals $1$ for every small enough threshold. This means that trajectories of such systems are perfectly predictable in the infinite horizon. In this paper we study the correlation integral and determinism for the family of $2^\infty$ non-chaotic maps, first considered by Delahaye in 1980. The determinism in a finite horizon equals $1$. However, the behaviour of the determinism in the infinite horizon is counter-intuitive. Sharp bounds on the determinism are provided.
Keywords: odometer, Correlation integral, predictability, distance matrix., asymptotic determinism.
Mathematics Subject Classification: Primary: 37E05, 37E15; Secondary: 37M2.
Citation: Jana Majerová. Correlation integral and determinism for a family of $2^\infty$ maps. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 5067-5096. doi: 10.3934/dcds.2016020
References:
[1]

L. S. Block and W. A. Coppel, Dynamics in One Dimension, Springer-Verlag, Berlin, 1992.  Google Scholar

[2]

L. S. Block and J. Keesling, A characterization of adding machine maps, Topology Appl., 140 (2004), 151-161. doi: 10.1016/j.topol.2003.07.006.  Google Scholar

[3]

J. P. Boroński and P. Oprocha, On indecomposability in chaotic attractors, Proc. Amer. Math. Soc., 143 (2015), 3659-3670. doi: 10.1090/S0002-9939-2015-12526-9.  Google Scholar

[4]

P. Collas and D. Klein, An ergodic adding machine on the Cantor set, Enseign. Math. (2), 40 (1994), 249-266.  Google Scholar

[5]

J.-P. Delahaye, Fonctions admettant des cycles d'ordre n'importe quelle puissance de $2$ et aucun autre cycle, C. R. Acad. Sci. Paris Sér. A-B, 291 (1980), A323-A325.  Google Scholar

[6]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems, $2^{nd}$ edition, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989.  Google Scholar

[7]

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

[8]

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

[9]

R. Hric, Topological sequence entropy for maps of the interval, Proc. Amer. Math. Soc., 127 (1999), 2045-2052. doi: 10.1090/S0002-9939-99-04799-1.  Google Scholar

[10]

H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, $2^{nd}$ edition, Cambridge University Press, Cambridge, 2004.  Google Scholar

[11]

M. Misiurewicz, Invariant measures for continuous transformations of $[0,1]$ with zero topological entropy, in Ergodic theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978), Springer, Berlin, 729 (1979), 144-152.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

S. Ruette, Chaos for continuous interval maps,, 2003. Available from: , ().   Google Scholar

[16]

J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc., 297 (1986), 269-282. doi: 10.1090/S0002-9947-1986-0849479-9.  Google Scholar

[17]

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

show all references

References:
[1]

L. S. Block and W. A. Coppel, Dynamics in One Dimension, Springer-Verlag, Berlin, 1992.  Google Scholar

[2]

L. S. Block and J. Keesling, A characterization of adding machine maps, Topology Appl., 140 (2004), 151-161. doi: 10.1016/j.topol.2003.07.006.  Google Scholar

[3]

J. P. Boroński and P. Oprocha, On indecomposability in chaotic attractors, Proc. Amer. Math. Soc., 143 (2015), 3659-3670. doi: 10.1090/S0002-9939-2015-12526-9.  Google Scholar

[4]

P. Collas and D. Klein, An ergodic adding machine on the Cantor set, Enseign. Math. (2), 40 (1994), 249-266.  Google Scholar

[5]

J.-P. Delahaye, Fonctions admettant des cycles d'ordre n'importe quelle puissance de $2$ et aucun autre cycle, C. R. Acad. Sci. Paris Sér. A-B, 291 (1980), A323-A325.  Google Scholar

[6]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems, $2^{nd}$ edition, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989.  Google Scholar

[7]

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

[8]

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

[9]

R. Hric, Topological sequence entropy for maps of the interval, Proc. Amer. Math. Soc., 127 (1999), 2045-2052. doi: 10.1090/S0002-9939-99-04799-1.  Google Scholar

[10]

H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, $2^{nd}$ edition, Cambridge University Press, Cambridge, 2004.  Google Scholar

[11]

M. Misiurewicz, Invariant measures for continuous transformations of $[0,1]$ with zero topological entropy, in Ergodic theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978), Springer, Berlin, 729 (1979), 144-152.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

S. Ruette, Chaos for continuous interval maps,, 2003. Available from: , ().   Google Scholar

[16]

J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc., 297 (1986), 269-282. doi: 10.1090/S0002-9947-1986-0849479-9.  Google Scholar

[17]

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

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