The bifurcation locus for numbers of bounded type

We define a family B(t) of compact subsets of the unit interval which provides a filtration of the set of numbers whose continued fraction expansion has bounded digits. We study how the set B(t) changes as the parameter t ranges in [0, 1], and see that the family undergoes period-doubling bifurcations and displays the same transition pattern from periodic to chaotic behaviour as the family of real quadratic polynomials. The set E of bifurcation parameters is a fractal set of measure zero and Hausdorff dimension 1. The Hausdorff dimension of B(t) varies continuously with the parameter, and we show that the dimension of each individual set equals the dimension of the corresponding section of the bifurcation set E.