Given a positive Borel measure without atoms $\tau$ on the circle $\mathbb T =\mathbb R /\mathbb Z =[0,1)$ , in https://arxiv.org/abs/0912.3423 a homeomorphism $h:[0,1)\to [0,1)$ is defined as \begin{equation} h(t)=\tau ([0,t))/\tau ([0,1) )%\qquad \mbox{for}\;\; \theta\in [0,1) \label{h1} \end{equation} Specifically they take $\tau$ formally proportional to $ e^{\beta X(t)}dt$ where $\beta\geq 0$ and $X$ is the Gaussian Free Field on the circle i.e. the random field $X$ with covariance $\mathbb E\, X(t)X(t')=-\log |e^{2\pi it}-e^{2\pi it'}|$ (see within the link and https://arxiv.org/abs/0909.1003 for all the rigorous details).
Questions:
- Is the map $\tau \mapsto h$ a homeomorphism between some space and the space of homeomorphisms of the circle?
- Can we use this map to define a measure on the space of homeomorphisms of the circle?
