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 Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the family of tent maps--continuous, unimodal, piecewise linear maps of the interval with slopes , . We show that tent maps have the shadowing property (every pseudo-orbit can be approximated by an actual orbit) for almost all parameters , 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.

**[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**, 10.1017/S0143385700003473

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

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

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1988-0946440-2

Article copyright:
© Copyright 1988
American Mathematical Society