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One-way intervals of circle maps


Authors: Lauren W. Ancel and Michael W. Hero
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
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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 [Enhancements On Off] (What's this?)

  • 1. L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Mathematics, 1513, Springer-Verlag, Berlin, 1991. MR 93g:58091
  • 2. E. M. Coven and I. Mulvey, Transitivity and the center for maps of the circle, Ergodic Theory and Dynamical Systems 6 (1986), 1-8. MR 87j:58074
  • 3. 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.

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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

DOI: https://doi.org/10.1090/S0002-9939-98-04652-8
Received by editor(s): January 31, 1995
Received by editor(s) in revised form: January 10, 1996
Communicated by: James West
Article copyright: © Copyright 1998 American Mathematical Society

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