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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Pseudo-orbit shadowing in the family of tent maps

Authors: Ethan M. Coven, Ittai Kan and James A. Yorke
Journal: Trans. Amer. Math. Soc. 308 (1988), 227-241
MSC: Primary 58F30; Secondary 34C35
MathSciNet review: 946440
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Abstract: We study the family of tent maps--continuous, unimodal, piecewise linear maps of the interval with slopes $ \pm s$, $ \sqrt 2 \leqslant s \leqslant 2$. We show that tent maps have the shadowing property (every pseudo-orbit can be approximated by an actual orbit) for almost all parameters $ s$, although they fail to have the shadowing property for an uncountable, dense set of parameters. We also show that for any tent map, every pseudo-orbit can be approximated by an actual orbit of a tent map with a perhaps slightly larger slope.

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  • [A] D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature, Proc. Steklov Inst. Math. 90 (1967). MR 0224110 (36:7157)
  • [Bi] G. D. Birkhoff, An extension of Poincarés last geometric theorem, Acta Math. 47 (1925), 297-311.
  • [Bo] R. Bowen, On Axiom A diffeomorphisms, CBMS Regional Conf. Ser. in Math., no. 35, Amer. Math. Soc., Providence, R.I., 1978. MR 0482842 (58:2888)
  • [CE] P. Collet and J.-P. Eckmann, Iterated maps on the interval as dynamical systems, Progress in Physics 1, Birkhäuser, Boston, Mass., 1980. MR 613981 (82j:58078)
  • [DGP] B. Derrid, A. Gervois, and Y. Pomeau, Iterations of endomorphisms on the real axis and representation of numbers, Ann. Inst. Henri Poincaré 29 (1978), 305-356. MR 519698 (80g:10056)
  • [MT] J. Milnor and W. Thurston, On iterated maps of the interval, mimeographed notes, 1977.
  • [NY] J. Yorke and H. Nusse, Is every trajectory of some process near an exact trajectory of a nearby process?, preprint, 1986.
  • [W] P. Walters, On the pseudo-orbit tracing property and its relationship to stability, Lecture Notes in Math., vol. 668, Springer-Verlag, Berlin and New York, 1978, pp. 191-210. MR 518563 (80d:58055)
  • [Y] L.-S. Young, Stochastic stability of hyperbolic attractors, preprint, 1985. MR 857204 (88a:58160)

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Article copyright: © Copyright 1988 American Mathematical Society

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