The structure of the space C(I, I) from the point of view of Sharkovsky stratification

We study spaces of continuous self-maps of the interval whose Sharkovsky type does not exceed a given type (as well as some similarly defined spaces). We prove that they are of second Baire category which enables us to study genericity in them. Among others we prove that type-stability is generic. The notion of intensive property is introduced and we show that maps simultaneously satisfying countably many intensive properties form a dense set in the considered spaces. One of the auxiliary results widely used in the paper says that arbitrarily close to any map there is a piecewise monotone map with the same type which is constant on an interval containing its fixed point.