The \(\alpha \)-limit sets of a unimodal map without homtervals (Q1041640)

For an interval map \(f:I \rightarrow I\), \(J \subseteq I\) is a homterval if \(f^n|_J\) is monotone for each \(n \geq 0\). The aim of this paper is to characterize the \(\alpha\)-limit set of each \(x \in [0,1]\) for a unimodal map \(f:[0,1] \rightarrow [0,1]\) when \(f\) has no homtervals. To begin, the authors find a one-to-one correspondence between renormalizations of \(f\) and proper bi-invariant closed sets of \(f\). The renormalizability of \(f\) is studied and the consecutive renormalization process is used to characterize the \(\alpha\)-limit set of each \(x \in [0,1]\). As a corollary, each proper \(\alpha\)-limit set is a closed isolated invariant set of \(f\).