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I know the following result which is if $X:=G/\Gamma$ where $G$ be a Lie group and $\Gamma$ is a lattice in $G$. Then geodesic floe $F=(g_t)$ is exponentially mixing i.e., there exist $\gamma > 0$ and $l \in \mathbb{Z}^{+}$ such that for any $\phi, \psi \in C_2^{\infty}(X)$ and for any $t \geq 0$ one has

$|(g_t\phi,\psi)-\int_{X}\psi d\mu \int_{X}\phi d\mu| \leq e^{-\gamma t}\|\phi\|_{l,2}\|\psi\|_{l,2}$ where $\|h\|_{l,p} =\sum_{|\alpha|\leq l}\|D^{\alpha}h\|_{p}$ and $C_2^{\infty}(X) = \{h \in C^{\infty}(X) : \|h\|_{l,2} < \infty \ \ \text{for any}\ l \in \mathbb{Z}+\}$.

My question is if there is any analogue of this mixing and rate of mixing in infinite volume surfaces other than lattices w.r.t the Bowen Margulis Sullivan measure. If yes, any reference will be very useful.

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    $\begingroup$ Do you mean the BMS measure? the PS measure lives on the boundary, not on the homogeneous space itself... $\endgroup$
    – Asaf
    Commented Mar 24 at 1:36
  • $\begingroup$ P.S. In infinite measure, every ergodic transformation is mixing (as $\mu(A\cap T^{-n}B) = o(1)$ for all $A,B$ of finite volume). One needs to consider local mixing, etc... $\endgroup$
    – Asaf
    Commented Mar 24 at 1:45
  • $\begingroup$ Right sorry BMS measure. Ok, so even for local mixing is there any kind of rate of mixing? Also, can you say some reference where infinite volume case has been considered. $\endgroup$
    – User1723
    Commented Mar 24 at 10:21
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    $\begingroup$ Babillot proved mixing of the geodesic flow for the BMS measure. Oh-Edwards proved exponential mixing for geometrically finite surfaces with high enough critical exponents. I think the general geometrically finite case was only recently announced by Khalil. [In the convex cocompact case, the symbolic techniques work well and the result is know before then]. $\endgroup$
    – Asaf
    Commented Mar 24 at 16:08
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    $\begingroup$ Your group $G$ is probably assumed to be semisimple with finite center. $\endgroup$
    – Moishe Kohan
    Commented Mar 24 at 19:11

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