Period 3 and chaos for unimodal maps (Q379853)

For the logistic map \(f(x)=\mu x (1-x)\) with \(\mu=3.839\), \textit{R. L. Devaney} [An introduction to chaotic dynamical systems. 2nd ed. Redwood City, CA etc.: Addison-Wesley Publishing Company, Inc. (1989; Zbl 0695.58002)] showed that all points in \((0,1)\) are either in the basin of attraction of a stable period-3 orbit or iterate to an invariant Cantor set which is topologically conjugate to a subshift of finite type. The paper reviewed here shows that similar behavior occurs for a more general class of unimodal maps through a careful analysis of the ordering of fixed points and critical points and study of the basins of attraction.NEWLINENEWLINETo be more specific, the considered unimodal maps have negative Schwarzian derivative, derivative more than 1 at 0, a stable period-3 orbit, and an unstable period-3 orbit. For maps in this class, the authors show that all points in \((0,1)\) are either in the basin of attraction of the stable period-3 orbit or iterate to a compact invariant set \(\Lambda\) which is semiconjugate to a subshift of finite type. Under some additional conditions, the semiconjugacy is shown to be a conjugacy by proving that \(\Lambda\) is hyperbolic.