Orbit equivalence and von Neumann algebras for piecewise linear unimodal maps. (English) Zbl 1131.37007
Summary: We study the orbit equivalence \(R_{f_\alpha}\) of a family of dynamical systems \((I,f_\alpha)\) arising from the tent maps \(f_\alpha\) where \(\alpha\in ]1,2]\). We prove that the associated von Neumann algebra \(L\infty(I)\rtimes R_{f_\alpha}\) is a type III factor whenever \(\alpha\in ]\sqrt 2,2]\) and the orbit of the critical point is periodic.
MSC:
| 37A20 | Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations |
| 37B10 | Symbolic dynamics |
| 37E05 | Dynamical systems involving maps of the interval |
| 46L35 | Classifications of \(C^*\)-algebras |
