Critical sets and unimodal mappings of the square (Q1922298)

Let \(I=[0,1]\). A mapping \(G:I^2\to I^2\) with \(G(x,y)=(\theta(x,y),x)\) is called unimodal if \(\theta\) is of class \(C^1\) and satisfies \(\theta(x,0)=\theta(x,1)=\theta(0,y)=\theta(1,y)=0\). Further it is required that there exists a point \((\overline{x},\overline{y})\) with \(\theta(\overline{x},\overline{y})>\theta(x,y)>0\) for any interior point \((x,y)\neq(\overline{x},\overline{y})\), where \((\overline{x},\overline{y})\) is unique with \(\theta_x(\overline{x},\overline{y})=\theta_y(\overline{x},\overline{y})=0\). The author considers the family of unimodal mappings of the form \[ F:I^2\to I^2 \text{ with }F(x,y)=(\lambda f(x)(g(y))^s,x)\tag{1} \] for real \(0<\lambda\leq 1\) and \(s>0\), \(f:I\to I\) of class \(C^1\) and unimodal -- defined similar as above --, and \(g=h\circ f\) for some \(C^1\) strictly increasing mapping \(h:I\to I\) with \(h(0)=0\). By writing down explicitly the critical set, i.e., the set of interior points of \(I^2\) where the Jacobian determinant of \(F\) vanishes, and the first two images and preimages of this critical set, the author is able to describe in detail the dynamical behaviour of mappings of the form (1). For example the author obtains as corollaries the following results: \(\bullet\) For a mapping of the form (1) the area of the domain with trajectory behaviour chaotic in the Li-York sense tends to zero for \(s\) to infinity. \(\bullet\) If the Schwarzian derivative of \(f\) is strictly negative on the punctured interval \(I\setminus\{q\}\) for some \(q\), then any mapping \(F\) of the form (1) has periodic points of arbitrarily large period.