A canonical partition of the periodic orbits of chaotic maps

Author:
Kathleen T. Alligood

Journal:
Trans. Amer. Math. Soc. **292** (1985), 713-719

MSC:
Primary 58F12; Secondary 34C35, 58F08, 58F13, 58F22

MathSciNet review:
808749

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Abstract: We show that the periodic orbits of an area-contracting horseshoe map can be partitioned into subsets of orbits of minimum period , for some positive integer . This partition is natural in the following sense: for any parametrized area-contracting map which forms a horseshoe, the orbits in one subset of the partition are contained in a single component of orbits in the full parameter space. Furthermore, prior to the formation of the horseshoe, this component contains attracting orbits of minimum period , for each nonnegative integer .

**[AMY]**Kathleen T. Alligood, John Mallet-Paret, and James A. Yorke,*An index for the global continuation of relatively isolated sets of periodic orbits*, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 1–21. MR**730259**, 10.1007/BFb0061406**[CMY]**S. N. Chow, J. Mallet-Paret and J. Yorke,*A bifurcation invariant*:*degenerate orbits treated as clusters of simple orbits*, Geometric Dynamics, Lecture Notes in Math., vol. 1007, Springer-Verlag, Berlin and New York.**[F]**John Franks,*Period doubling and the Lefschetz formula*, Trans. Amer. Math. Soc.**287**(1985), no. 1, 275–283. MR**766219**, 10.1090/S0002-9947-1985-0766219-1**[MY]**John Mallet-Paret and James A. Yorke,*Snakes: oriented families of periodic orbits, their sources, sinks, and continuation*, J. Differential Equations**43**(1982), no. 3, 419–450. MR**649847**, 10.1016/0022-0396(82)90085-7**[S]**S. Smale,*Differentiable dynamical systems*, Bull. Amer. Math. Soc.**73**(1967), 747–817. MR**0228014**, 10.1090/S0002-9904-1967-11798-1**[YA]**James A. Yorke and Kathleen T. Alligood,*Period doubling cascades of attractors: a prerequisite for horseshoes*, Comm. Math. Phys.**101**(1985), no. 3, 305–321. MR**815187**

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DOI:
https://doi.org/10.1090/S0002-9947-1985-0808749-X

Article copyright:
© Copyright 1985
American Mathematical Society