The order structure of forts iterated by quadratic polynomials

High complexity may possibly occur through the iterations of simple mappings, where the representative of complexity considered here is nonmonotonicity—specifically we investigate how non-monotone points, namely forts, occur by iteration. We introduce a partial order, actually a tree structure, on the set of all forts iterated by a quadratic polynomial, which makes every fort coded by a finite 0–1 sequence, and thus the generation of forts through iteration can be studied combinatorially. Using an operation called cutting to get an abstract tree, we show it is an actual set of forts iterated by a quadratic polynomial. The tree structure is highly asymmetrical, contrasting the symmetry of its iterative source—a quadratic polynomial.