The bifurcation locus for numbers of bounded type

TL;DR

This paper investigates the bifurcation structure of a family of sets related to numbers with bounded continued fraction digits, revealing fractal properties and continuous dimension variation similar to quadratic polynomial dynamics.

Contribution

It introduces a new family of sets generalizing bounded continued fractions and analyzes their bifurcation and fractal properties, connecting number theory and dynamical systems.

Findings

The set E of bifurcation parameters is fractal with measure zero and Hausdorff dimension 1.

The Hausdorff dimension of B(t) varies continuously with t.

Each B(t)'s dimension matches a section of the bifurcation set E.

Abstract

We define a family B(t) of compact subsets of the unit interval which generalizes the sets of numbers whose continued fraction expansion has bounded digits. We study how the set B(t) changes as one moves the parameter t, and see that the family undergoes period-doubling bifurcations and displays the same transition pattern from periodic to chaotic behavior as the usual family of quadratic polynomials. The set E of bifurcation parameters is a fractal set of measure zero and Hausdorff dimension 1. We also show that the Hausdorff dimension of B(t) varies continuously with the parameter, and the dimension of each individual set equals the dimension of a corresponding section of the bifurcation set E.