Simple \(2\)-endomorphisms whose set of nonwandering points has a complicated structure (Q1595706)

The author studies the properties of the set of nonwandering points of the family of endomorphisms \(F:(x,y)\to(A[\varphi(x)+B]\cdot f(y),x)\) where \(A>0\), \((x,y)\in\mathbb{R}^2\), \(\varphi(t)\), \(f(t)\) are \({\mathcal C}^2\) smooth functions on \(\mathbb{R}^1\) such that \(\varphi(0)=0\), \(\varphi(1)=1\), \(\varphi'(t)>0\), \(f(0)=f(1)=0\); also there exists \(q\in(0,1)\) such that \(f(q)=1\), \((t-q)f'(t)<0\) and it is assumed that \(f''(t)<0\), \(\varphi''<0\) on \(\mathbb{R}^1\).