Entropy-expansive maps

Bowen, Rufus

doi:10.1090/S0002-9947-1972-0285689-X

Entropy-expansive maps
HTML articles powered by AMS MathViewer

by Rufus Bowen PDF

Trans. Amer. Math. Soc. 164 (1972), 323-331 Request permission

Abstract:

Let $f:X \to X$ be a uniformly continuous map of a metric space. f is called h-expansive if there is an $\varepsilon > 0$ so that the set ${\Phi _\varepsilon }(x) = \{ y:d({f^n}(x),{f^n}(y)) \leqq \varepsilon$ for all $n \geqq 0$} has zero topological entropy for each $x \in X$. For X compact, the topological entropy of such an f is equal to its estimate using $\varepsilon :h(f) = h(f,\varepsilon )$. If X is compact finite dimensional and $\mu$ an invariant Borel measure, then ${h_\mu }(f) = {h_\mu }(f,A)$ for any finite measurable partition A of X into sets of diameter at most $\varepsilon$. A number of examples are given. No diffeomorphism of a compact manifold is known to be not h-expansive.

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
D. Z. Arov, Calculation of entropy for a class of group endomorphisms, Zap. Meh.-Mat. Fak. Har′kov. Gos. Univ. i Har′kov. Mat. Obšč. (4) 30 (1964), 48–69 (Russian). MR 0213507
Patrick Billingsley, Ergodic theory and information, John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR 0192027
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
Rufus Bowen, Periodic points and measures for Axiom $A$ diffeomorphisms, Trans. Amer. Math. Soc. 154 (1971), 377–397. MR 282372, DOI 10.1090/S0002-9947-1971-0282372-0
E. I. Dinaburg, A correlation between topological entropy and metric entropy, Dokl. Akad. Nauk SSSR 190 (1970), 19–22 (Russian). MR 0255765
L. Wayne Goodwyn, Topological entropy bounds measure-theoretic entropy, Proc. Amer. Math. Soc. 23 (1969), 679–688. MR 247030, DOI 10.1090/S0002-9939-1969-0247030-3
Charles Pugh and Michael Shub, The $\Omega$-stability theorem for flows, Invent. Math. 11 (1970), 150–158. MR 287579, DOI 10.1007/BF01404608
Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
Harvey B. Keynes and James B. Robertson, Generators for topological entropy and expansiveness, Math. Systems Theory 3 (1969), 51–59. MR 247031, DOI 10.1007/BF01695625

entropy and

capacity of sets in function spaces

14

17

22

23

William Parry, Entropy and generators in ergodic theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0262464
Ja. Sinaĭ, On the concept of entropy for a dynamic system, Dokl. Akad. Nauk SSSR 124 (1959), 768–771 (Russian). MR 0103256
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
V. A. Rohlin, Lectures on the entropy theory of transformations with invariant measure, Uspehi Mat. Nauk 22 (1967), no. 5 (137), 3–56 (Russian). MR 0217258