Topological entropy of fixed-point free flows

Thomas, Romeo F.

doi:10.1090/S0002-9947-1990-1010414-5

Topological entropy of fixed-point free flows
HTML articles powered by AMS MathViewer

by Romeo F. Thomas PDF

Trans. Amer. Math. Soc. 319 (1990), 601-618 Request permission

Abstract:

Topological entropy was introduced as an invariant of topological conjugacy and also as an analogue of measure theoretic entropy. Topological entropy for one parameter flows on a compact metric spaces is defined by Bowen. General statements are proved about this entropy, but it is not easy to calculate the topological entropy, and to show it is invariant under conjugacy. For all this I would like to try to pose a new direction and study a definition for the topological entropy that involves handling the technical difficulties that arise from allowing reparametrizations of orbits. Some well-known results are proved as well using this definition. These results enable us to prove some results which seem difficult to prove using Bowen’s definition. Also we show here that this definition is equivalent to Bowen’s definition for any flow without fixed points on a compact metric space. Finally, it is shown that the topological entropy of an expansive flow can be defined globally on a local cross sections.

References

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, Periodic orbits for hyperbolic flows, Amer. J. Math. 94 (1972), 1–30. MR 298700, DOI 10.2307/2373590
Rufus Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc. 164 (1972), 323–331. MR 285689, DOI 10.1090/S0002-9947-1972-0285689-X

Topological entropy and Axiom

Rufus Bowen and Peter Walters, Expansive one-parameter flows, J. Differential Equations 12 (1972), 180–193. MR 341451, DOI 10.1016/0022-0396(72)90013-7
Rufus Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math. 95 (1973), 429–460. MR 339281, DOI 10.2307/2373793
John E. Franke and James F. Selgrade, Hyperbolicity and chain recurrence, J. Differential Equations 26 (1977), no. 1, 27–36. MR 467834, DOI 10.1016/0022-0396(77)90096-1
John E. Franke and James F. Selgrade, Abstract $\omega$-limit sets, chain recurrent sets, and basic sets for flows, Proc. Amer. Math. Soc. 60 (1976), 309–316 (1977). MR 423423, DOI 10.1090/S0002-9939-1976-0423423-X
John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144
H. B. Keynes and M. Sears, Real-expansive flows and topological dimension, Ergodic Theory Dynam. Systems 1 (1981), no. 2, 179–195. MR 661818, DOI 10.1017/s0143385700009214
William L. Reddy, Expansive canonical coordinates are hyperbolic, Topology Appl. 15 (1983), no. 2, 205–210. MR 686097, DOI 10.1016/0166-8641(83)90038-X
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
Romeo F. Thomas, Entropy of expansive flows, Ergodic Theory Dynam. Systems 7 (1987), no. 4, 611–625. MR 922368, DOI 10.1017/S0143385700004235
Romeo F. Thomas, Topological stability: some fundamental properties, J. Differential Equations 59 (1985), no. 1, 103–122. MR 803089, DOI 10.1016/0022-0396(85)90140-8
Romeo F. Thomas, Stability properties of one-parameter flows, Proc. London Math. Soc. (3) 45 (1982), no. 3, 479–505. MR 675418, DOI 10.1112/plms/s3-45.3.479
Hassler Whitney, Regular families of curves, Ann. of Math. (2) 34 (1933), no. 2, 244–270. MR 1503106, DOI 10.2307/1968202