Chain-Recurrent $C^0$- $\Omega$-Blowup in $C^1$-Smooth Simplest Skew Products on Multidimensional Cells

In this paper we prove criteria of a $C^0$- $\Omega$-blowup in $C^1$-smooth skew products with a closed set of periodic points on multidimensional cells and give examples of maps that admit such a $\Omega$-blowup. Our method is based on the study of the properties of the set of chain-recurrent points. We also prove that the set of weakly nonwandering points of maps under consideration coincides with the chain-recurrent set, investigate the approximation (in the $C^0$-norm) and entropy properties of $C^1$-smooth skew products with a closed set of periodic points.