One-parameter families of mappings of the interval with negative Schwarzian derivative, in which monotonicity of bifurcations breaks down
The paper gives examples of \(C^ 3\)-convex unimodal (i.e. having a unique extremum at a point c) functions f: \(I\to I\), where \(I=[0,1]\), \(f(0)=f(1)=0\), \(f(c)=1\) with the Schwarzian \(Sf(x)=f'''(x)/f'(x)- ((3/2)(f''(x)/f'(x))^ 2<0\) for \(x\in I\setminus \{c\}\) such that, for the family \(\{f_{\lambda}\}_{\lambda \in (0,1]}\), where \(f_{\lambda}=\lambda \cdot f\), a monotonicity of bifurcations of cycles is violated. This means that there is \(\lambda >\lambda_ f(n)=\inf \{\lambda (0,1]: f_{\lambda}\) has a cycle of the period \(n\}\) such that \(f_{\lambda}\) has no cycle of period n. Some consequences of the fact are discussed.
scientific article; zbMATH DE number 4143091