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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
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show all references

References:
[1]

Springer-Verlag, Berlin, 1992.  Google Scholar

[2]

Topology Appl., 140 (2004), 151-161. doi: 10.1016/j.topol.2003.07.006.  Google Scholar

[3]

Proc. Amer. Math. Soc., 143 (2015), 3659-3670. doi: 10.1090/S0002-9939-2015-12526-9.  Google Scholar

[4]

Enseign. Math. (2), 40 (1994), 249-266.  Google Scholar

[5]

C. R. Acad. Sci. Paris Sér. A-B, 291 (1980), A323-A325.  Google Scholar

[6]

$2^{nd}$ edition, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989.  Google Scholar

[7]

Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350147, 13 pp. doi: 10.1142/S0218127413501472.  Google Scholar

[8]

Phys. D, 9 (1983), 189-208. doi: 10.1016/0167-2789(83)90298-1.  Google Scholar

[9]

Proc. Amer. Math. Soc., 127 (1999), 2045-2052. doi: 10.1090/S0002-9939-99-04799-1.  Google Scholar

[10]

$2^{nd}$ edition, Cambridge University Press, Cambridge, 2004.  Google Scholar

[11]

in Ergodic theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978), Springer, Berlin, 729 (1979), 144-152.  Google Scholar

[12]

J. Statist. Phys., 90 (1998), 1047-1049. doi: 10.1023/A:1023253709865.  Google Scholar

[13]

J. Statist. Phys., 71 (1993), 529-547. doi: 10.1007/BF01058436.  Google Scholar

[14]

Random Comput. Dynam., 3 (1995), 137-156.  Google Scholar

[15]

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

[16]

Trans. Amer. Math. Soc., 297 (1986), 269-282. doi: 10.1090/S0002-9947-1986-0849479-9.  Google Scholar

[17]

Physics Letters A, 171 (1992), 199-203. doi: 10.1016/0375-9601(92)90426-M.  Google Scholar

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