Let $T_t$ be the geodesic flow on a surface $S$ of constant negative curvature, and let $M(f,t) := \langle \bar f \cdot (f \circ T_t) \rangle$, where $\langle f \rangle := \int_S f(x) d\mu(x)$ and where $\mu$ is the natural invariant measure.

A 1984 paper by Collet, Epstein, and Gallavotti (PDF here) shows (prop. 5, p. 90) that for $f$ nice (in a sense defined at the bottom of page 71), $\lvert M(f,t) \rvert < \lVert f \rVert^2_\xi C( \xi) \cdot t^{b(\xi)}\exp(-t/2)$, where $\lVert \cdot \rVert^2_\xi$ is a certain rather complicated norm (defined in equation 3.5 of the paper) and $C$, $b$ do not depend on $f$.

I have two related questions about this result which hopefully someone here already knows (the paper is quite technical and I really don't need to know its details if I can get a bit of clarification here):

This result seems to imply that the rate of mixing is 1/2. How can this be? (see also this question)

How does this result (for which the "decay is not exponential") square with the results of Dolgopyat and Liverani that give exponential decay of correlations for reasonably nice Anosov flows?