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

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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

Abstract
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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 and 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.
[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.
[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.
[4]	P. Collas and D. Klein, An ergodic adding machine on the Cantor set, Enseign. Math. (2), 40 (1994), 249-266.
[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.
[6]	R. L. Devaney, An Introduction to Chaotic Dynamical Systems, $2^{nd}$ edition, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989.
[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.
[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.
[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.
[10]	H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, $2^{nd}$ edition, Cambridge University Press, Cambridge, 2004.
[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.
[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.
[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.
[14]	Ya. B. Pesin and A. Tempelman, Correlation dimension of measures invariant under group actions, Random Comput. Dynam., 3 (1995), 137-156.
[15]	S. Ruette, Chaos for continuous interval maps,, 2003. Available from: , ().
[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.
[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.

show all references

References:

[1]	L. S. Block and W. A. Coppel, Dynamics in One Dimension, Springer-Verlag, Berlin, 1992.
[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.
[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.
[4]	P. Collas and D. Klein, An ergodic adding machine on the Cantor set, Enseign. Math. (2), 40 (1994), 249-266.
[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.
[6]	R. L. Devaney, An Introduction to Chaotic Dynamical Systems, $2^{nd}$ edition, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989.
[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.
[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.
[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.
[10]	H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, $2^{nd}$ edition, Cambridge University Press, Cambridge, 2004.
[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.
[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.
[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.
[14]	Ya. B. Pesin and A. Tempelman, Correlation dimension of measures invariant under group actions, Random Comput. Dynam., 3 (1995), 137-156.
[15]	S. Ruette, Chaos for continuous interval maps,, 2003. Available from: , ().
[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.
[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.

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