A class of twisted products of maps of an interval (Q1337870)

Let \(I\) be a closed rectangle in the plane with the product topology. The author discusses properties of \(C^ 1\)-smooth mappings \(F : I \to I\) being of the form \(F(x,y) = (f(x), g_ x(y))\) assuming some additional conditions (among them: \(\Omega\)-stability of \(g_ x\)). Dynamics of such maps over a quasiminimal set of the factor \(f\) and the structure of the set of periodic points of \(F\) and homoclinic points of \(F\) are investigated. In particular, it is proved (Theorem B) that for \(F\) being under consideration, the following three assertions are equivalent: (i) there exists a periodic point of period different from \(2^ i\), \(i = 0, 1, 2, \dots\) (ii) the set of periodic points is not a \(G_ \delta\)-set (iii) there is a homoclinic point.