A decomposition theorem for the space of $C^1$-smooth skew products with complicated dynamics of the quotient map

We use the notions of the $\Omega$-function and functions suitable to it, to give a detailed proof of a decomposition theorem for the space of $C^{1}$-smooth skew products of interval maps whose quotient maps have complicated dynamics and satisfy the additional condition of $\Omega$-stability with respect to the $C^1$-norm. In our theorem, the space of $C^1$-smooth skew products is decomposed into a union of four nonempty, pairwise disjoint subspaces. We give examples of maps contained in each of the four subspaces.
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