The deconstruction of a fractal

Abstract

In order to understand better the fractal structure attached to the bifurcation diagrams it is required to identify the fractal structures underlying that diagram. With that objective we will first construct the explicit form of the structure and morphology of MSS-sequences (shift-maximal) that appear in bifurcation diagrams of a wide class of unimodal maps. Next we will derive and prove the theorems on decomposition of MSS-sequences as compositions of other MSS-sequences in a recursive process. Those theorems allow the deconstruction of the bifurcation diagram since they permit to deduce which self-similar structure the sequences of the bifurcation diagram belong to: period doubling cascades, saddle–node bifurcation cascades or periodic window inside another periodic window.