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Pseudo-orbit tracing property and structural stability of expanding maps of the interval

Published online by Cambridge University Press:  19 September 2008

S. V. Šlačkov
S. V. Šlačkov
Affiliation:
Obninsk Institute of Atomic Power Engineering, 249020 Obninsk, USSR
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Abstract

A version of pseudo-orbit tracing property for piecewise-continuous piecewise-expanding maps of the interval is proved. It is shown that the typical map of such a kind can be included in a 2m-parameter structurally stable family where m is the number of critical points.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 12 , Issue 3 , September 1992 , pp. 573 - 587
Copyright
Copyright © Cambridge University Press 1992

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References

REFERENCES

[ABS]Afraimovich, V. S., Bykov, V. V. & Shil'nikov, L. P.. On structurally unstable attracting limit sets of Lorenz attractor type. Trans. Moscow Math. Soc. 2 (1983), 153–216.Google Scholar
[Al]Alekseev, V. M.. Symbolic dynamics. Eleventh Math. Summer School. Naukova Dumka: Kiev, 1976 (Russian).Google Scholar
[An1]Anosov, D. V.. Geodesic flows on closed Riemannian manifolds with negative curvature. Proc Steklov Inst. Math. 90 (1967).Google Scholar
[An2]Anosov, D. V.. On certain class of invariant sets of smooth dynamical systems. Proc. 5th Int. Conf. on Nonlinear Oscillations. Vol. 2. pp. 39–45.Google Scholar
[B]Bowen, R.. On Axiom A diffeomorphisms. CMBS Regional Conf. Ser. Math., 35. Providence, RI: Am. Math. Soc., 1978.Google Scholar
[BPSY]Bunimovich, L. A., Pesin, Ya. B., Sinai, Ya. G. & Yakobson, M. V.. Ergodic theory of smooth dynamical systems. In: Dynamical Systems II. Moskva: 1985, pp. 113–121 (Russian).Google Scholar
[CKY]Coven, E. M., Kan, I. & Yorke, J. A.. Pseudo-orbit shadowing in the family of tent maps. Trans. Amer. Math. Soc. 308 (1988), 227–241.CrossRefGoogle Scholar
[GW]Guckenheimer, J. & Williams, R. F.. Structural stability of Lorenz attractors. IHES Publ. Math. 50 (1979), 59–72.CrossRefGoogle Scholar
[K]Komuro, M.. Lorenz attractors do not have the pseudo-orbit tracing property. J. Math. Soc. Japan 37 (1985), 489–514.Google Scholar
[L]Lorenz, E. N.. Deterministic nonperiodic flow. J. Atmosph. Sci. 20 (1963), 130–141.2.0.CO;2>CrossRefGoogle Scholar
[NY]Nusse, H. E. & Yorke, J. A.. Is every approximate trajectory of some process near an exact trajectory of nearby process? Commun. Math. Phys. 114 (1988), 363–379.CrossRefGoogle Scholar
[SV]Sinai, Ya. G. & Vul, E. B.. Hyperbolicity conditions for the Lorenz model. Physica 2D (1981), 3–7.Google Scholar
[š]šlačkov, S. V.. A theorem on e-trajectories for Lorenz mappings. Funkts. Anal. Prilozh. 19 (1985), 84–85 (Russian).Google Scholar
[W]Walters, P.. On the pseudo-orbit tracing property and its relationship to stability. Springer Lecture Notes in Mathematics No. 668. Berlin: Springer, 231–244.Google Scholar