Question about an early result on the mixing of geodesic flows

Question

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?

Answer 1

Dear Steve,

In general the "precise" rate of mixing depends on the class of functions one considers. In particular, the authors of the paper you quoted treat only "analytic" functions (with some fixed band of analyticity) and they show that you get a rate of $t^{b(\xi)}e^{-t/2}$. In this sense, the rate of mixing is exponential and essentially equal to 1/2: I said essentially because of the factor $t^{b(\xi)}$ can't be removed in general, so that's why the authors said that the rate of mixing is not "genuinely exponential" (i.e., the sharp bound that they obtain is not exactly an exponential function of time).

On the other hand, the results of Dolgopyat and Liverani concerns smooth but non-analytic functions and contact Anosov flows (and not only the ones coming from constant negative curvature surfaces). In this context, they show "exponential mixing" in the sense that the correlations can be bounded by some exponential function of $t$ (say $e^{-\sigma t}$ for some $\sigma>0$). Of course, they don't know whether their "rate" (i.e., the number $\sigma$ they obtain in the end of the calculations) is optimal (for instance, there is a subsequent work of Tsujii improving on Liverani's work). In particular, although Dolgopyat and Liverani obtain "exponential mixing", it is not in the same "sense" of Collet, Epstein and Gallavotti.

In resume, I guess that the confusion comes from the distinct employments of the nomenclature "exponential mixing" in the works of Collet, Epstein, Gallavotti, and Dolgopyat and Liverani (and Tsujii also).

Best,

Matheus