given probability space $ (\Omega, T, \mu), \mu$ is ergodic and $ T $ is invertible ( can regard $T$ as two sides shift)

define random $\beta$-transformations: random variable $\beta:\Omega \to (1,\infty)$. random $\beta$-transformations $ f_{\omega}: S^1\to S^1: x\to \beta_{\omega}\cdot x \mod(1) $.( it is in fact random Lasota Yorke maps)

let $ f_{\omega}^n=f_{T^{n-1}\omega} \circ f_{T^{n-2}\omega}\circ \circ \circ f_{T^{1}\omega}\circ f_{\omega}$, $ \phi \in BV[0,1]$.

consider the limit behavior of $ \frac{\sum_{i \le n} \phi \circ f_{\omega}^i}{n^{1/2}}$, can we prove it converges(in distribution) to Gaussian distribution a.s. $ \omega$ ( it is called quenched Central Limit Theorem)?

if $ \inf_{\omega}{\beta_{\omega}} >1$, then it is uniformly hyberbolic, the quenched Central Limit Theorem has been understood very well.

The idea of proof is as follow: we can find global invariant density $h_{\omega }$ ( i.e. $ h_{T \omega}= P_{\omega} h_{\omega}, P_{\omega} \text{ is transfer operator of } f_{\omega}$) from skew product, then we can prove for any $ s>0, \exists C_{\omega, s}, \text{ s.t.} ||h_{T^n \omega}||_{BV} \le C_{\omega,s} \cdot e^{sn}$ ( see Buzzi 1999). If $ \inf_{\omega}{\beta_{\omega}} >1$, I can further show $ \sup_{n,\omega} ||h_{T^n \omega}||_{BV} < \infty $. To prove Central Limit Theorem a.s $ \omega$, decay of correlation a.s $ \omega$ plays an important role, and uniformly bounded invariant density helps us to give a good control of decay of correlation independent of $ \omega$.

However if $ \inf_{\omega}{\beta_{\omega}}=1$, we do not have uniformaly bounded invariant density :$ \sup_{n,\omega} ||h_{T^n \omega}||_{BV} = \infty $, this makes it hard to control Decay of Correlation due to random term $ \omega$. In this case, quench Central Limit Theorem is still true?

Many Thanks!