Critical points of one parameter families of maps of the interval

Block, Louis

doi:10.1090/S0002-9939-1983-0695273-4

Critical points of one parameter families of maps of the interval
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by Louis Block

Proc. Amer. Math. Soc. 88 (1983), 347-350

DOI: https://doi.org/10.1090/S0002-9939-1983-0695273-4

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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 ${f_s}(x) = sx(1 - x)$ (and other ${C^1}$ families with a single critical point) occurs for each critical point in ${C^1}$ 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.

References

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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