\(C^*\)-algebras associated with \({\mathcal F}_{2^n}\) unimodal dynamical systems (Q2784870)

This paper deals with a dynamical system \((f,\mathbb R)\) generated by such a unimodal continuous map \(f:[0,1] \mapsto \mathbb R\) that \(f(0)=f(1)\) and \(\text{Fix}(f^{2^{n+1}})=\text{Fix}(f^{2^{n}})\) for some natural number \(n\). For this system, the authors obtain a description of all positive orbits in the case where \(f\) has only a pair of fixed points and, besides, for any \(m\leq n\) there exists a unique cycle of period \(2^m\) which is the repeller for \(m<n\) and attractor for \(m=n\).NEWLINENEWLINENEWLINENext they introduce an enveloping \(C^*\)-algebra associated with \(f\). For this algebra, the complete classification of irreducible representations in Hilbert spaces as well as a description of corresponding dual space are obtained.