Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Pseudo-orbits and topological entropy

Authors: Marcy Barge and Richard Swanson
Journal: Proc. Amer. Math. Soc. 109 (1990), 559-566
MSC: Primary 58F08; Secondary 54H20, 58F20
MathSciNet review: 1012923
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The topological entropy of a map of a compact metric space is equal to the exponential growth rate of the number of separated periodic pseudo-orbits of period $ n$ as $ n$ tends to infinity.

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

  • [AKM] R. L. Adler, A. G. Konheim, and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319. MR 0175106 (30:5291)
  • [Bl] R. Bowen, Equilibrium states and the ergodic theory of Axiom A diffeomorphisms, Lecture Notes in Math., vol. 470, Springer-Verlag, New York, 1975. MR 0442989 (56:1364)
  • [B2] -, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401-414; 181 (1973), 509-510. MR 0274707 (43:469)
  • [B3] -, Topological entropy and Axiom A, Proc. Sympos. Pure Math., vol. 14, American Mathematical Society, Providence, RI, 1970, pp. 23-42. MR 0262459 (41:7066)
  • [BS] M. Barge and R. Swanson, Rotation shadowing properties of circle and annulus maps, Ergodic Theory Dynamical Systems 8 (1988), 509-521. MR 980794 (90f:58107)
  • [C] C. Conley, Isolated invariant sets and the Morse Index, CBMS Conference Series No. 38 (1978). MR 511133 (80c:58009)
  • [DGS] M. Denker, C. Grillenberger, and K. Sigmund, Ergodic theory on compact spaces, Lecture Notes in Math., vol. 527, Springer-Verlag, New York, 1976. MR 0457675 (56:15879)
  • [E] R. Easton, Isolating blocks and epsilon chains for maps, preprint. MR 1021184 (90m:58176)
  • [F] J. Franks, Realizing rotation vectors for torus homeomorphisms, Trans. Amer. Math. Soc. 311 (1989), 107-115. MR 958891 (89k:58239)
  • [HYG1] S. M. Hammel, J. A. Yorke, and C. Grebogi, Do numerical orbits of chaotic dynamical processes represent true orbits ? J. Complexity 3 (1987), 136-145. MR 907194 (88m:58115)
  • [HYG2] -, Numerical orbits of chaotic processes represent true orbits, Bull. Amer. Math. Soc. 19 (1988), 465-469. MR 938160 (89m:58180)
  • [H] M. Hurley, Attractors in cellular automata, Ergodic Theory Dynamical Systems (to appear). MR 1053803 (91d:58139)
  • [Mc] R. McGehee, Some metric properties of attractors with applications to computer simulations of dynamical systems, preprint.
  • [Mi] M. Misiurewicz, Remark on the definition of topological entropy, Dynamical Systems and Partial Differential Equations (Caracas, 1984), Caracas, 1986, 65-67. MR 882013 (88g:58163)
  • [R] M. Rees, A minimal positive entropy homeomorphism of the $ 2$-torus, J. London Math. Soc. (2) 23 (1981), 537-550. MR 616561 (82h:58045)
  • [Ro] C. Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mountain J. Math. 7 (1977), 425-437. MR 0494300 (58:13200)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58F08, 54H20, 58F20

Retrieve articles in all journals with MSC: 58F08, 54H20, 58F20

Additional Information

Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society