On Entropy and Monotonicity for Real Cubic Maps

Abstract:

Consider real cubic maps of the interval onto itself, either with positive or with negative leading coefficient. This paper completes the proof of the “monotonicity conjecture”, which asserts that each locus of constant topological entropy in parameter space is a connected set. The proof makes essential use of the thesis of Christopher Heckman, and is based on the study of “bones” in the parameter triangle as defined by Tresser and R. MacKay.