Forcing of periodic orbits for interval maps

and renormalization of piecewise affine maps

Authors:
Marco Martens and Charles Tresser

Journal:
Proc. Amer. Math. Soc. **124** (1996), 2863-2870

MSC (1991):
Primary 58F11

MathSciNet review:
1343712

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that for continuous maps on the interval, the existence of an -cycle implies the existence of 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.

**[BMT]**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****[GMT]**R. Galeeva, M. Martens, C. Tresser,*Inducing, Slopes, and Conjugacy Classes*, preprint 1994/4 at SUNY at Stony Brook, Israel J. Math. (to appear).**[MMS]**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**, 10.1007/BF02392981**[MT]**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**, 10.1007/BFb0082847**[LS]**V. J. Lopez, L. Snoha, to appear.**[S]**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****[T]**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**

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Additional Information

**Marco Martens**

Affiliation:
Institute of Mathematical Sciences, SUNY at Stony Brook, Stony Brook, New York 11794-3651

Email:
marco@math.sunysb.edu

**Charles Tresser**

Affiliation:
I.B.M., P.O. Box 218, Yorktown Heights, New York 10598

Email:
tresser@watson.ibm.com

DOI:
http://dx.doi.org/10.1090/S0002-9939-96-03508-3

Received by editor(s):
December 29, 1994

Communicated by:
Linda Keen

Article copyright:
© Copyright 1996
American Mathematical Society