Some Measures on the Set of Piecewise Smooth Circle Homeomorphisms and Related Representations of the Circle Diffeomorphism Group

摘要

We construct a family of measures on some subsets of the set of piecewise smooth circle homeomorphisms and prove the quasi-invariance of these measures under the action of circle diffeomorphisms by composition of higher smoothness class. On the basis of this result, we obtain a series of earlier unknown unitary representations of the group of circle diffeomorphisms preserving a given point. We develop ideas of Shavgulidze [1] and Kosyak [2], who considered measures on sets of segment and circle diffeomorphisms and the corresponding representations.