Orbit equivalence and von Neumann algebras for expansive interval maps

We study the orbit equivalence relation Rτ for dynamical systems (I, τ) arising from piecewise linear maps τ: I → I on the interval I = [0, 1]. Under regularity conditions, we prove that the crossed product von Neumann algebra L∞(I) × Rτ is the type IIIλ hyperfinite factor where λ ]0, 1] is determined by the subgroup of generated by {m(τ(Ii))/m(Ii)}, with the Ii’s being the underlying partitioning intervals for τ and m the Lebesgue measure. Thus we compute the complete invariant for the orbit structures of these maps.