Mean ergodicity for lifted vector field

Let $ X \in C^{\infty}(TM)$ be an ergodic vector field on a smooth compact manifold and $ f$ , $ \mu$ a function/measure on $ M$ satisfying $ \mathcal{L}_X \mu =f\mu$ . Consider the lifted vector field $ $ \tilde{X} = \tau X + \frac{1}{2}(1-\tau^2)f\partial_\tau $ $ on $ \tilde{M}= M \times [-1,1]_\tau$ which preserves the measure $ \mathcal{L}_{\tilde{X}} \tilde{\mu} =0$ , $ \tilde{\mu}= (1-\tau^2)d\tau \wedge \mu$ . I would like to see if the time averages of $ \tilde{X}$ satisfy mean ergodicity:

$ $ \frac{1}{T}\int_0^T dt (e^{t\tilde{X}})^* \pi_M^*(a) \rightarrow_{L^2} \int_M a \mu$ $

for all $ a\in C^{\infty}(M)$ .

The problem being that the ergodicity assumption is on $ X$ not $ \tilde{X}$ , but if it helps their flows are related by

$ $ e^{t \tilde{X}} (x,\tau) = (e^{[\int_0^t \tau(s)]X}x,\frac{1-(\frac{1-\tau}{1+\tau})e^{-2\int_0^t ds (e^{sX})^*f}}{1+(\frac{1-\tau}{1+\tau})e^{-2\int_0^t ds (e^{sX})^*f}}) .$ $