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Distortion of S-unimodal maps. (English) Zbl 0708.58007
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.
Reviewer: M.Bartušek

##### MSC:
 37E99 Low-dimensional dynamical systems 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 37C75 Stability theory for smooth dynamical systems 34A40 Differential inequalities involving functions of a single real variable
##### Keywords:
S-unimodal maps; Schwarzian derivative
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