Comments on the Bifurcation Structure of 1D Maps

The paper presents a complementary view on some of the phenomena related to the bifurcation structure of unimodal maps. An approximate renormalization theory for the period-doubling cascade is developed, and a mapping procedure is established that accounts directly for the box-within-a-box structure of the total bifurcation set. This presents a picture in which the homoclinic orbit bifurcations act as a skeleton for the bifurcational set. At the same time, experimental results on continued subharmonic generation for piezoelectrically amplified sound waves, predating the Feigenbaum theory, are called into attention.