Dynamics of continued fractions and kneading sequences of unimodal maps

Abstract

In this paper we construct a correspondence between the parameter spaces of two families of one-dimensional dynamical systems, the $\alpha$-continued fraction transformations $T_\alpha$ and unimodal maps. This correspondence identifiesbifurcation parameters in the two families, and allows one to transfer topological and metric propertiesfrom one setting to the other. As an application, we recover results about thereal slice of the Mandelbrot set, and the set of univoque numbers.