The Cantor set as an \(\omega\)-limit set for iterations of a smooth function on a segment (Q1893642)

In the paper the problem of the description of \(\omega\)-limit sets for smooth mappings \(f : I \to I\) of the segment \(I = [0,1]\) is considered. It is known that for a continuous \(f\), \(K\) is an \(\omega\)-limit set of \(f\) if and only if \(K\) is nonempty, compact and nowhere dense or is a finite union of segments. The result obtained in the paper is the following. Theorem. There exists a \(C^1\)-smooth mapping of the segment \(I\) into itself for which the Cantor set is an \(\omega\)-limit set.