zbMATH — the first resource for mathematics

A theorem of Šarkovskii characterizing continuous maps of zero topological entropy. (English) Zbl 0686.26002
Let f be a continuous map of the compact real interval I into itself, \(f_ n\) the n-th iterate of f. For any \(x\in I\) \(\omega_ f(x)\) is the limit set of the sequence \(\{f^ n(x)\}^{\infty}_{n=1}.\)
The main aim of the paper is to give a new and short proof of the following result by A. N. Šarkovskii: f has a periodic point of period different from \(2^ n\) for any n if and only if there exists \(x\in I\) such that \(\omega_ f(x)\) is infinite and contains a periodic point.
Reviewer: K.Janková
26A18 Iteration of real functions in one variable
54H25 Fixed-point and coincidence theorems (topological aspects)
[1] BARNA B.: Über die Iteration reeller Funktionen I. Publ. Math. Debreczen, 7, 1960, 16- 40. · Zbl 0112.04301
[2] BLOCK L.: Homoclinic points of mappings of the interval. Proc Amer. Math. Soc., 72, 1978, 576-580. · Zbl 0365.58015
[3] JANKOVÁ K., SMÍTAL J.: A characterization of chaos. Bull. Austral. Math. Soc., 34, 1986, 283-292. · Zbl 0577.54041
[4] MISIUREWICZ M.: Horseshoes for mappings of the interval. Bull. Acad. Polon. Sci. Sér. Math., 27, 1979, 167-169. · Zbl 0459.54031
[5] MISIUREWICZ M., SMÍTAL J.: Smooth chaotic maps with zero topological entropy. Ergodic Th. & Dynam. Systems, to appear. · Zbl 0689.58028
[6] PREISS D., SMÍTAL J.: A characterization of non-chaotic maps of the interval stable under small perturbations. Trans. Amer. Math. Soc., to appear.
[7] ŠARKOVSKII A. N.: On cycles and the structure of continuous mappings. (Russian.) Ukrain. Mat. Ž., 17, No 3, 1965, 104-111.
[8] ŠARKOVSKII A. N.: The behavior of a map in a neighborhood of an attracting set. (Russian.) Ukrain. Mat. Ž., 18, No 2, 1966, 60-83.
[9] ŠARKOVSKII A. N.: Ona problem of isomorphism of dynamical systems. (Russian.) Proc. Internat. Conference on Nonlinear Oscillations, Vol. 2, Kiev 1970, 541-545.
[10] SMÍTAL J.: Chaotic functions with zero topological entropy. Trans. Amer. Math. Soc. 297, 1986, 269-282. · Zbl 0639.54029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.