Pseudo-orbits and topological entropy

Barge, Marcy; Swanson, Richard

doi:10.1090/S0002-9939-1990-1012923-7

Pseudo-orbits and topological entropy
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by Marcy Barge and Richard Swanson PDF

Proc. Amer. Math. Soc. 109 (1990), 559-566 Request permission

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

R. L. Adler, A. G. Konheim, and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309–319. MR 175106, DOI 10.1090/S0002-9947-1965-0175106-9
Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989
Rufus Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401–414. MR 274707, DOI 10.1090/S0002-9947-1971-0274707-X
Rufus Bowen, Topological entropy and axiom $\textrm {A}$, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 23–41. MR 0262459
Marcy Barge and Richard Swanson, Rotation shadowing properties of circle and annulus maps, Ergodic Theory Dynam. Systems 8 (1988), no. 4, 509–521. MR 980794, DOI 10.1017/S0143385700004661
Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133
Manfred Denker, Christian Grillenberger, and Karl Sigmund, Ergodic theory on compact spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976. MR 0457675
Robert Easton, Isolating blocks and epsilon chains for maps, Phys. D 39 (1989), no. 1, 95–110. MR 1021184, DOI 10.1016/0167-2789(89)90041-9
John Franks, Realizing rotation vectors for torus homeomorphisms, Trans. Amer. Math. Soc. 311 (1989), no. 1, 107–115. MR 958891, DOI 10.1090/S0002-9947-1989-0958891-1
Stephen M. Hammel, James A. Yorke, and Celso Grebogi, Do numerical orbits of chaotic dynamical processes represent true orbits?, J. Complexity 3 (1987), no. 2, 136–145. MR 907194, DOI 10.1016/0885-064X(87)90024-0
Stephen M. Hammel, James A. Yorke, and Celso Grebogi, Numerical orbits of chaotic processes represent true orbits, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 465–469. MR 938160, DOI 10.1090/S0273-0979-1988-15701-1
Mike Hurley, Attractors in cellular automata, Ergodic Theory Dynam. Systems 10 (1990), no. 1, 131–140. MR 1053803, DOI 10.1017/S0143385700005435

Some metric properties of attractors with applications to computer simulations of dynamical systems

Michael Misiurewicz, Remark on the definition of topological entropy, Dynamical systems and partial differential equations (Caracas, 1984) Univ. Simon Bolivar, Caracas, 1986, pp. 65–67. MR 882013
M. Rees, A minimal positive entropy homeomorphism of the $2$-torus, J. London Math. Soc. (2) 23 (1981), no. 3, 537–550. MR 616561, DOI 10.1112/jlms/s2-23.3.537
Clark Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mountain J. Math. 7 (1977), no. 3, 425–437. MR 494300, DOI 10.1216/RMJ-1977-7-3-425