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Let $T_t$ be the geodesic flow on a surface $M$ S$of constant negative curvature, and let$M(A,t) M(f,t) := \langle \bar f \cdot (f \circ T_t) \rangle$, where$\langle f \rangle := \int_M 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$. I have two related questions about this result which hopefully someone here already knows (the paper is quite technical and I really don't need to know its details if I can get a bit of clarification here): • This result seems to imply that the rate of mixing is 1/2. How can this be? (see also this question) • How does this result (for which the "decay is not exponential") square with the results of Dolgopyat and Liverani that give exponential decay of correlations for reasonably nice Anosov flows? 1 # Question about an early result on the mixing of geodesic flows Let$T_t$be the geodesic flow on a surface$M$of constant negative curvature, and let$M(A,t) := \langle \bar f \cdot (f \circ T_t) \rangle$, where$\langle f \rangle := \int_M 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?