Dynamics of continued fractions and kneading sequences of unimodal maps

Claudio Bonanno, Carlo Carminati, Stefano Isola, Giulio Tiozzo

Published 2010-12-09, updated 2012-05-30Version 3

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 identifies bifurcation parameters in the two families, and allows one to transfer topological and metric properties from one setting to the other. As an application, we recover results about the real slice of the Mandelbrot set, and the set of univoque numbers.