$C^1$-Smooth Skew Products on a $3D$-Torus Which Have an $\Omega$-Stable Quotient and Continuous Itinerary Auxiliary Multifunctions

This work is devoted to the description of some set in the space of $C^1$-smooth skew products of circle maps that have a $3D$-torus as phase space and possess periodic points. This set of maps consists of skew products with an $\Omega$-stable quotient, continuous itinerary auxiliary multifunctions and contains, in particular, an open, but not dense (in the distinguished set) subset of $\Omega$-stable maps (with respect to $C^1$-smooth perturbations of skew products class).
We also select here the other set of $C^1$-smooth skew products of circle maps such that $C^1$-smooth $\Omega$-stable skew products form an everywhere dense subset of this set.
Much attention is paid to maps of the second selected set admitting a maximal ``prickly'' attractor, which is presented as the union of a $2D$-torus with a family of arcs of various lengths, orthogonal to the above $2D$-torus and originating from points of an everywhere dense set of continuum cardinality on this $2D$-torus.