Nonmonotonicity of Kneading Invariants in a Family of Kinked Maps

We study monotonicity properties of the kneading invariant for one-parameter families of piecewise-linear unimodal maps and prove a theorem on the violation of monotonicity of the kneading invariant for maps that are symmetric and convex and consist of four linear pieces. The fact that such maps cannot be approximated by smooth mappings with negative Schwarzian is proved using a dynamics argument.