The bifurcation locus for numbers of bounded type

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