Maximal entropy odd orbit types

Geller, William; Tolosa, Juán

doi:10.1090/S0002-9947-1992-1020040-1

Maximal entropy odd orbit types
HTML articles powered by AMS MathViewer

by William Geller and Juán Tolosa PDF

Trans. Amer. Math. Soc. 329 (1992), 161-171 Request permission

Abstract:

A periodic orbit of a continuous map of an interval induces in a natural way a cyclic permutation, called its type. We consider a family of orbit types of period $n$ congruent to $1$ ($\operatorname {mod} 4$) introduced recently by Misiurewicz and Nitecki. We prove that the Misiurewicz-Nitecki orbit types and their natural generalizations to the remaining odd periods $n$ have maximal entropy among all orbit types of period $n$, and even among all $n$-permutations.

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
Stewart Baldwin, Generalizations of a theorem of Sarkovskii on orbits of continuous real-valued functions, Discrete Math. 67 (1987), no. 2, 111–127. MR 913178, DOI 10.1016/0012-365X(87)90021-5
Chris Bernhardt, The ordering on permutations induced by continuous maps of the real line, Ergodic Theory Dynam. Systems 7 (1987), no. 2, 155–160. MR 896787, DOI 10.1017/S0143385700003898
L. S. Block and W. A. Coppel, Stratification of continuous maps of an interval, Trans. Amer. Math. Soc. 297 (1986), no. 2, 587–604. MR 854086, DOI 10.1090/S0002-9947-1986-0854086-8

Approximating entropy of maps of the interval

22

Louis Block, John Guckenheimer, MichałMisiurewicz, and Lai Sang Young, Periodic points and topological entropy of one-dimensional maps, Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979) Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 18–34. MR 591173
Louis Block and David Hart, Orbit types for maps of the interval, Ergodic Theory Dynam. Systems 7 (1987), no. 2, 161–164. MR 896788, DOI 10.1017/S0143385700003904
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 and John Franks, The periodic points of maps of the disk and the interval, Topology 15 (1976), no. 4, 337–342. MR 431282, DOI 10.1016/0040-9383(76)90026-4
Irwin Jungreis, Some results on the Šarkovskiĭ partial ordering of permutations, Trans. Amer. Math. Soc. 325 (1991), no. 1, 319–344. MR 998354, DOI 10.1090/S0002-9947-1991-0998354-X
MichałMisiurewicz and Zbigniew Nitecki, Combinatorial patterns for maps of the interval, Mem. Amer. Math. Soc. 94 (1991), no. 456, vi+112. MR 1086562, DOI 10.1090/memo/0456
M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math. 67 (1980), no. 1, 45–63. MR 579440, DOI 10.4064/sm-67-1-45-63
Zbigniew Nitecki, Topological dynamics on the interval, Ergodic theory and dynamical systems, II (College Park, Md., 1979/1980), Progr. Math., vol. 21, Birkhäuser, Boston, Mass., 1982, pp. 1–73. MR 670074
Y\B{o}ichir\B{o} Takahashi, A formula for topological entropy of one-dimensional dynamics, Sci. Papers College Gen. Ed. Univ. Tokyo 30 (1980), no. 1, 11–22. MR 581221