On the structure of quasi-invariant measures for non-discrete subgroups of \(\text{Diff}^{\omega}(S^{1})\) (Q2856121)

Consider a group \(G\) of diffeomorphisms acting on the circle \(S^1\). A probability measure \(\mu\) on \(S^1\) is said to be quasi-invariant by \(G\) if, for every element \(g\in G\), the measure \(g^*\mu\) has the same null-sets as \(\mu\). Recall that a group \(G\subset\mathrm{Diff}^{\omega}(S^1)\) is said to be non-discrete if it contains a sequence of elements \(\{g_n\} \subset G\) converging to the identity (say in the \(C^\infty\)-topology) and such that \(g_n \neq\mathrm{id}\). The purpose of this paper is to study the nature of quasi-invariant measures \(\mu\) for finitely generated non-discrete subgroups of \(\mathrm{Diff}^{\omega}(S^1)\). In this paper the author proves some interesting results concerning an absolute continuity of the \(\mu\). In particular the following is proved. Let \(G\subset \mathrm{Diff}^{\omega}(S^1)\) be a finitely generated non-solvable and non-discrete group. Assume also that \(G\) has no finite orbit and that \(\mu\) is a probability measure quasi-invariant under \(G\). Then \(\mu\) is absolutely continuous. Concerning singular quasi-invariant measures, it is also proved that their associated Hausdorff measures must either be zero or of infinite mass, a result contrasting with the case of dynamically defined Cantor sets and also applicable to the examples of singular stationary measures constructed by Kaimanovich and Le Prince. As a further application of this method, a rigidity theorem for measurable conjugates between groups as above is obtained. The author notices that most results are valid in the \(C^\infty\) category and, in fact, part of them works also for diffeomorphisms of class \(C^2\).