# zbMATH — the first resource for mathematics

On some properties of orientation-preserving surjections on the circle. (English) Zbl 1164.37013
The author generalizes the known fact that every two periodic points of an orientation-preserving homeomorphism of the circle have the same period (cf. Theorem 1) proving what follows. If $$B \subset S^1$$, $${\operatorname {card}}\, B\geq 3$$ and $$F\colon B\rightarrow B$$ is an orientation-preserving surjection with $${\operatorname {Per}}\, F \not = \emptyset$$, then every two periodic points of $$F$$ have the same period. He also presents some properties of orientation-preserving surjections with a finite and non-empty set of periodic points.

##### MSC:
 3.7e+11 Dynamical systems involving maps of the circle
Full Text:
##### References:
 [1] ALSEDÀ L.-LLIBRE J.-MISIUREWICZ M.: Combinatorial Dynamics and Entropy in Dimension One. Adv. Ser. Nolinear Dynam. 5, World Scientific Publishing Co., Inc, River Edge, NJ, 1993. · Zbl 0843.58034 [2] BAJGER M.: On the structure of some flows on the unit circle. Aequationes Math. 55 (1998), 106-121. · Zbl 0891.39017 [3] CIEPLINSKI K.: On the embeddability of a homeomorphism of the unit circle in disjoint iteration groups. Publ. Math. Debrecen 55 (1999), 363-383. · Zbl 0935.39010 [4] CIEPLIŃSKI K.: On conjugacy of disjoint iteration groups on the unit circle. Ann. Math. Sil. 13 (1999), 103-118. · Zbl 0945.39006 [5] CORNFELD I. R-FOMIN S. V.-SINAI Y. G.: Ergodic Theory. Grundlehren Math. Wiss. 245, Spirnger Verlag, Berlin-Heidelberg-New York, 1982. · Zbl 0493.28007 [6] de MELO W.-van STREIN S.: One-dimensional Dynamics. Ergeb. Math. Grenzegeb. (3) 25, Springer-Verlag, New York-Berlin, 1993. [7] JARCZYK W.: Babbage equation on the circle. Publ. Math. Debrecen 63 (2003), 389-400. · Zbl 1052.39019 [8] KUCZMA M.-CHOCZEWSKI B.-GER R.: Iterative Functional Equations. Encyclopaedia Math. Appl. 32, Cambridge University Press, Cambridge-New York-Port Chester-Melbourne-Sydney, 1990. · Zbl 0703.39005 [9] LLIBRE J.: Minimal periodic orbits of continuous mappings of the circle. Proc. Amer. Math. Soc. 83 (1981), 625-628. · Zbl 0469.54024 [10] WALTERS P.: An Introduction to Ergodic Theory. Grad. Text in Math. 79, Springer-Verlag, New York-Heidelberg-Berlin, 1982. · Zbl 0475.28009
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.