Weakly non-wandering points in the dynamics of skew products in high dimensions

We consider continuous skew products on n-dimensional (n > 1) manifolds such as cells, cylinders, and tori, and introduce the notion of weakly nonwandering points with respect to the family of mappings in layers over wandering points of the factor mapping. We prove a criterion for the existence of such points in terms of the $?$-explosion (in the $C^0$-norm) in the mappings in layers over the limit (for a wandering set) non-wandering points of the factor mapping. Using the obtained results, we describe the structure of the nonwandering set of skew products on $n$-dimensional cells, cylinders and tori.