Critical sets and unimodal mappings of the square

A lower estimate for the number of different invariant subsets of the set of nonwandering points for a class of unimodal mappings is given. Sufficient conditions for such a mapping to have periodic points of arbitrarily large period are described. The machinery of the appearance of such points may be of very different nature. The existence of mappings with trajectory behavior chaotic in the Li–York sense is established. Conditions for the domain of these trajectories to be arbitrary small are given. Therefore, such trajectories cannot be found by numerical methods.