# 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

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