Renormalization-group analysis of the global structure of the period-doubling attractor

We use a recently developed renormalization-group formalism to study the global properties of the period-doubling attractor. The renormalization scheme can be written in closed form in terms of universal functions. The results of the calculation appear as a smooth spectrum of scaling indices in full agreement with direct numerical calculations. As a special result we obtain the fractal dimension of the attractor to be $D_0$=0.538 045 143 5, accurate to one part in ${10}^{\mathrm{-}10}$. The only input for the calculation is the Taylor expansion of Feigenbaum’s universal function.