Critical points of one parameter families of maps of the interval
Proc. Amer. Math. Soc. 88 (1983), 347-350
Primary 58F14; Secondary 58F20
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Abstract: It is shown that some of the periodic phenomena which is well known to occur for the critical point of the quadratic family (and other families with a single critical point) occurs for each critical point in families with an arbitrary (possibly infinite) number of critical points. Also, some of the same behavior occurs in families of maps (which are not necessarily differentiable) where a critical point has derivative zero on either the left or the right side. A stronger condition is obtained when the derivative on the right is zero.
Collet and Jean-Pierre
Eckmann, Iterated maps on the interval as dynamical systems,
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R. May, Simple mathematical models with very complicated dynamics, Nature 261 (1976), 459-467.
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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
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