A parameter renormalization for orbit-splittings and band-mergings of iterated maps of an interval

A renormalization scheme based directly on a spatial scaling is presented for a parameter which shows the location of the maximum in the “confining square” of iterated maps of an interval into itself. The cause and the properties of band-mergings are made especially clear in the geometrical set-up which allows one to see the symmetry between the forward and the reverse bifurcations (the orbit-splittings and the band-mergings) in an obvious way and yields an economical estimate of Feigenbaum's ratios. The method is applicable to any sequences and to arbitrary maxima of the mapping function. If the maximum is a cusp, a pseudo fixed “point” causes a finite number of pairwise bandmergings which follow the same number of foregoingband-splittings.