Hyperbolicity of the invariant set for the logistic map with μ>4

Abstract

Classic results due to Guckenheimer and Misiurewicz imply that the invariant set of the logistic map with µ in (4,2+\sqrt{5}] is hyperbolic. This is well known, but the only obvious reference in the literature uses relatively sophisticated ideas from complex variable theory. This pedagogical note provides a brief, self-contained account of this result using only elementary real analysis. The method also gives a good estimate of the expansion rate on the invariant set.