Proposition 1.1 in the following paper gives the existance of a unique $(\eta^{\chi^{(\beta})},R_\alpha)$ conformal measure $\mu_\alpha$ on $S^1$ where $\eta,\beta>0$ and $\alpha\notin \mathbb{Q}$ and $R_\alpha$ the irrational rotation on $S^1$. https://link.springer.com/article/10.1007/BF02785420.
Now suppose the Ergodic systems $(S^1, \mu_\alpha,R_\alpha)$ and $(S^1, \mu_{\alpha'},R_{\alpha'})$ for $\alpha, \alpha'\notin \mathbb{Q}$ are measure theoretically isomorphic. Can we say anything about $\alpha, \alpha'$?
Like if the ergodic systems $(S^1, m_\alpha,R_\alpha)$ and $(S^1, m_{\alpha'},R_{\alpha'})$ are isomorphic where $m_\alpha$ and $m_{\alpha'}$ are Lebesgue measure on $S^1$, $R_\alpha,R_{\alpha'}$ are irrational rotations , then $\alpha=\pm \alpha'$. Can I get any reference like this for $\mu_\alpha$ and $\mu_{\alpha'}$ above?