Abstract
Two theorems are proved—the first and the more important of them due to Šarkovskii—providing complete and surprisingly simple answers to the following two questions: (i) given that a continuous mapT of an interval into itself (more generally, into the real line) has a periodic orbit of periodn, which other integers must occur as periods of the periodic orbits ofT? (ii) given thatn is the least odd integer which occurs as a period of a periodic orbit ofT, what is the “shape” of that orbit relative to its natural ordering as a finite subset of the real line? As an application, we obtain improved lower bounds for the topological entropy ofT.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Šarkovskii, A. N.: Coexistence of cycles of a continuous map of a line into itself. Ukr. Mat. Z.16, 61–71 (1964)
Bowen, R., Franks, J.: The periodic points of maps of the disk and the interval. I.H.E.S. preprint, November 1975
Guckenheimer, J.: On the bifurcation of maps of the interval. Preprint, 1976
Li, T., Yorke, J. A.: Period three implies chaos. Am. Mat. Monthly82, 985–992 (1975)
Author information
Authors and Affiliations
Additional information
Communicated by J. L. Lebowitz
Rights and permissions
About this article
Cite this article
Štefan, P. A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line. Commun.Math. Phys. 54, 237–248 (1977). https://doi.org/10.1007/BF01614086
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01614086