Quasi-invariant measures on sets of piecewise smooth homeomorphisms of closed intervals and circles and representations of diffeomorphism groups (Q455588)

From the introduction: We construct two families of measures defined on some subsets of the set of piecewise smooth homeomorphisms of a closed interval. The measures in the first family are quasi-invariant with respect to the action of \(C^1\)-diffeomorphisms of the closed interval that have bounded Borel-measurable second derivative, whereas the measures in the other family are quasi-invariant with respect to the action of the diffeomorphisms of the circle of the same smoothness. We further introduce a series of representations of the group of \(C^3\)-diffeomorphisms of the circle on the space of functions square integrable against these measures. We present a complete proof of the quasi-invariance of the measures in question and prove that the representations thus constructed are irreducible and pairwise inequivalent.