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

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

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]**K. T. Alligood, J. Mallet-Paret and J. A. Yorke,*An index for the global continuation of relatively isolated sets of periodic orbits*, Geometric Dynamics, Lecture Notes in Math., vol. 1007, Springer-Verlag, New York and Berlin, pp. 1-21. MR**730259 (85b:58108)****[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]**J. Franks,*Period doubling and the Lefschetz formula*, Trans. Amer. Math. Soc.**287**(1985), 275-283. MR**766219 (86d:58093)****[MY]**J. Mallet-Paret and J. A. Yorke,*Snakes*:*oriented families of periodic orbits, their sources, sinks, and continuation*, J. Differential Equations**43**(1982), 419-450. MR**649847 (84a:58071a)****[S]**S. Smale,*Differentiable dynamical systems*, Bull. Amer. Math. Soc.**73**(1967), 747-817. MR**0228014 (37:3598)****[YA]**J. A. Yorke and K. T. Alligood,*Period doubling cascades of attractors*:*a prerequisite for horseshoes*, Comm. Math. Phys. (to appear). MR**815187 (87k:58210)**

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

Article copyright:
© Copyright 1985
American Mathematical Society