Distortion of S-unimodal maps
In the paper \(C^ 3\)-transformations f: \(I\to I\), \(I=[-1,1]\) are studied such that \(f(x)=h(x^ 2)\), h is monotone and has a negative Schwarzian derivative. Two categories of f are studied: (a) There is an attracting Cantor set that attracts almost every point and upon which f is homeomorphic or (b) the map is sensitive to initial conditions. It is proved that in case (a) all homeomorphic Cantor sets have Lebesgue measure zero, and in case (b) almost every point has the same \(\omega\)- limit set which is either finite union of intervals or an absorbing non- homeomorphic Cantor set.
scientific article; zbMATH DE number 4163763