Noncontinuity of topological entropy of maps of the Cantor set and of the interval

Block, Louis

doi:10.1090/S0002-9939-1975-0367953-7

Noncontinuity of topological entropy of maps of the Cantor set and of the interval
HTML articles powered by AMS MathViewer

by Louis Block PDF

Proc. Amer. Math. Soc. 50 (1975), 388-393 Request permission

Abstract:

We show that topological entropy, as a map on the space of continuous functions of the Cantor set into itself, is not continuous anywhere. Furthermore, topological entropy, as a map on the space of continuous functions of the interval into itself, is not continuous at any map with finite entropy.

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

Bifurcations of endomorphisms of

Louis Block and John Franke, Existence of periodic points for maps of $S^{1}$, Invent. Math. 22 (1973/74), 69–73. MR 358867, DOI 10.1007/BF01425575
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, 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
M. Misiurewicz, On non-continuity of topological entropy, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 19 (1971), 319–320 (English, with Russian summary). MR 287578
M. Shub, Dynamical systems, filtrations and entropy, Bull. Amer. Math. Soc. 80 (1974), 27–41. MR 334284, DOI 10.1090/S0002-9904-1974-13344-6
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1