One-way intervals of circle maps
HTML articles powered by AMS MathViewer
- by Lauren W. Ancel and Michael W. Hero PDF
- Proc. Amer. Math. Soc. 126 (1998), 1191-1197 Request permission
Abstract:
An interval in the circle $S^1$ is one-way with respect to a map $f:S^1\to S^1$ if under repeated applications of $f$ all points of the interval move in the same direction. The main result is that every locally one-way interval is either one-way or is the union of two overlapping one-way subintervals. An example is given which illustrates that the latter case can occur; however, it is proved that the latter case cannot occur if the interval is covered by the image of the map. As a corollary, it is shown that if $f$ has periodic points, then every interval which contains no periodic points is either one-way or is the union of two overlapping one-way subintervals.References
- L. S. Block and W. A. Coppel, Dynamics in one dimension, Lecture Notes in Mathematics, vol. 1513, Springer-Verlag, Berlin, 1992. MR 1176513, DOI 10.1007/BFb0084762
- Ethan M. Coven and Irene Mulvey, Transitivity and the centre for maps of the circle, Ergodic Theory Dynam. Systems 6 (1986), no. 1, 1–8. MR 837972, DOI 10.1017/S0143385700003254
- M. W. Hero, A characterization of the attracting center for dynamical systems on the interval and circle, Ph.D. Thesis, University of Wisconsin-Milwaukee, 1990.
Additional Information
- Lauren W. Ancel
- Affiliation: Department of Biological Sciences, Stanford University, Stanford, California 94305
- Email: ancel@charles.stanford.edu
- Michael W. Hero
- Affiliation: Equable Securities Corporation, 300 N. 121 Street, Milwaukee, Wisconsin 53226
- Received by editor(s): January 31, 1995
- Received by editor(s) in revised form: January 10, 1996
- Communicated by: James West
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 1191-1197
- MSC (1991): Primary 54H20, 34C35, 58F03
- DOI: https://doi.org/10.1090/S0002-9939-98-04652-8
- MathSciNet review: 1476114