Morphology of the period-doubling universal function

Results are presented concerning the structure on the real line of the "universal function" which is the fixed point solution of the Feigenbaum-Cvitanovic renormalization group equation associated with period-doubling chaos in quadratic maps. It is shown that the values which the function takes at its turning points can be algebraically characterized by relation to the infinite cycle associated with the original turning point of the map on the interval. These extreme become increasingly numerous as the argument increases, and their locations can be found progressively using knowledge of the previously determined extrema. As well as providing a simple understanding of the structure of the universal function these simple observations may be of assistance in investigating the convergence or asymptotic behavior of approximations to the universal function, and perhaps in providing a "corrector" step to some of these schemes.