Forcing of periodic orbits for interval maps and renormalization of piecewise affine maps

Martens, Marco; Tresser, Charles

doi:10.1090/S0002-9939-96-03508-3

Forcing of periodic orbits for interval maps and renormalization of piecewise affine maps
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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

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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