Critical points of one parameter families of maps of the interval

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.