Dynamics of continued fractions and kneading sequences of unimodal maps

TL;DR

This paper establishes a correspondence between parameter spaces of alpha-continued fractions and unimodal maps, enabling transfer of properties and recovering results related to the Mandelbrot set and univoque numbers.

Contribution

It introduces a novel link between two dynamical systems families, facilitating the transfer of topological and metric properties between them.

Findings

Recovered results about the real slice of the Mandelbrot set.

Identified bifurcation parameters across systems.

Linked univoque numbers with dynamical system properties.

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 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.