Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [A] D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov. 90 (1967), 209 (Russian). MR 0224110
  • [Bi] G. D. Birkhoff, An extension of Poincarés last geometric theorem, Acta Math. 47 (1925), 297-311.
  • [Bo] Rufus Bowen, On Axiom A diffeomorphisms, American Mathematical Society, Providence, R.I., 1978. Regional Conference Series in Mathematics, No. 35. MR 0482842
  • [CE] Pierre Collet and Jean-Pierre Eckmann, Iterated maps on the interval as dynamical systems, Progress in Physics, vol. 1, Birkhäuser, Boston, Mass., 1980. MR 613981
  • [DGP] B. Derrida, A. Gervois, and Y. Pomeau, Iteration of endomorphisms on the real axis and representation of numbers, Ann. Inst. H. Poincaré Sect. A (N.S.) 29 (1978), no. 3, 305–356 (English, with French summary). MR 519698
  • [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] Peter Walters, On the pseudo-orbit tracing property and its relationship to stability, The structure of attractors in dynamical systems (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977) Lecture Notes in Math., vol. 668, Springer, Berlin, 1978, pp. 231–244. MR 518563
  • [Y] Lai-Sang Young, Stochastic stability of hyperbolic attractors, Ergodic Theory Dynam. Systems 6 (1986), no. 2, 311–319. MR 857204,

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F30, 34C35

Retrieve articles in all journals with MSC: 58F30, 34C35

Additional Information

Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society