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YCor
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Mixing of Geodesicgeodesic flow and rate of Mixingmixing

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}$$|(g_t\phi,\psi)-\int_{X}\psi d\mu \int_{X}\phi d\mu| \leq e^{-\gamma t}\|\phi\|_{l,2}\|\psi\|_{l,2}$ where $||h||_{l,p} =\sum_{|\alpha|\leq l}||D^{\alpha}h||_{p}$$\|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}+\}$$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.

Mixing of Geodesic flow and rate of Mixing

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.

Mixing of geodesic flow and rate of mixing

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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User1723
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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 measureSullivan measure. If yes, any reference will be very useful.

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 measure measure. If yes, any reference will be very useful.

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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User1723
User1723
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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 Patterson SullivanBowen Margulis measure measure. If yes, any reference will be very useful.

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 Patterson Sullivan measure. If yes, any reference will be very useful.

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 measure measure. If yes, any reference will be very useful.

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User1723
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