Forcing of periodic orbits for interval maps and renormalization of piecewise affine maps
HTML articles powered by AMS MathViewer
- by Marco Martens and Charles Tresser
- Proc. Amer. Math. Soc. 124 (1996), 2863-2870
- DOI: https://doi.org/10.1090/S0002-9939-96-03508-3
- PDF | Request permission
Abstract:
We prove that for continuous maps on the interval, the existence of an $n$-cycle implies the existence of $n-1$ points which interwind the original ones and are permuted by the map. We then use this combinatorial result to show that piecewise affine maps (with no zero slope) cannot be infinitely renormalizable.References
- K. M. Brucks, M. Misiurewicz, and C. Tresser, Monotonicity properties of the family of trapezoidal maps, Comm. Math. Phys. 137 (1991), no. 1, 1–12. MR 1099253, DOI 10.1007/BF02099114
- R. Galeeva, M. Martens, C. Tresser, Inducing, Slopes, and Conjugacy Classes, preprint 1994/4 at SUNY at Stony Brook, Israel J. Math. (to appear).
- M. Martens, W. de Melo, and S. van Strien, Julia-Fatou-Sullivan theory for real one-dimensional dynamics, Acta Math. 168 (1992), no. 3-4, 273–318. MR 1161268, DOI 10.1007/BF02392981
- John Milnor and William Thurston, On iterated maps of the interval, Dynamical systems (College Park, MD, 1986–87) Lecture Notes in Math., vol. 1342, Springer, Berlin, 1988, pp. 465–563. MR 970571, DOI 10.1007/BFb0082847
- V. J. Lopez, L. Snoha, to appear.
- Dennis Sullivan, Bounds, quadratic differentials, and renormalization conjectures, American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988) Amer. Math. Soc., Providence, RI, 1992, pp. 417–466. MR 1184622
- Charles Tresser, Fine structure of universal Cantor sets, Instabilities and nonequilibrium structures, III (Valparaíso, 1989) Math. Appl., vol. 64, Kluwer Acad. Publ., Dordrecht, 1991, pp. 27–42. MR 1177838
Bibliographic Information
- Marco Martens
- Affiliation: Institute of Mathematical Sciences, SUNY at Stony Brook, Stony Brook, New York 11794-3651
- MR Author ID: 120380
- Email: marco@math.sunysb.edu
- Charles Tresser
- Affiliation: I.B.M., P.O. Box 218, Yorktown Heights, New York 10598
- MR Author ID: 174225
- Email: tresser@watson.ibm.com
- Received by editor(s): December 29, 1994
- Communicated by: Linda Keen
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 2863-2870
- MSC (1991): Primary 58F11
- DOI: https://doi.org/10.1090/S0002-9939-96-03508-3
- MathSciNet review: 1343712