Mixing
$C^1$-foliation are absolutely continuous
$begingroup$
I trying to figure out why this is result is true :
If $F$ is a $C^1$-foliation with $C^1$-leaves, then it is absolutely continuous, where absolutely continuity here means that given a foliation chart $(U,h)$ and $L$ some transversal, there exists a measurable family of positive measurable functions $j_x : F_U(x) to mathbb{R}$ such that for any measurable set $A subset U$, we have that $m(A) = int_L int_{F_U(x)} mathbb{1}_A(x,y)j_x(y)dm_{F(x)}(y)dm_L(x)$ where $m$ is our volume.
I've read that this is a direct application of Fubini Theorem, but since I don't know much about foliation and differential geometry, I don't really feel confident about this...
Any help is very greatly appreciated.
EDIT : this question has been put on hold because it lacks some context. So here it is. I'm interesting in proving that ergodicity of $C^2$ Anosov on compact riemannian manifold. I'm following Brin & Stuck Introduction to dynamical systems. The key idea of the proof is to prove that the stable/unstable foliation is absolutely continuous, as defined above. At one point, authors show that absolutely continuous implies transversely absolute continuity. To do that, they introduce some auxiliary foliation that is $C^1$, while explaining that it is obviously absolutely continuous and transversely absolutely continuous. I've read this in differents books/ressources, as a direct application of Fubini theorem. The context is pretty standard in DS, so maybe I should be posting this question of MO...
EDIT 2 : Some more context here (especially the line "...basic results from calculus show that a version of Fubini’s theorem holds..." This is precisely the claim for which I'm looking a proof. Since I'm working only on the torus, it might goes like this :
Let $F$ a $C^1$ foliation with $C^1$-leaves on the torus $mathbb{T}^2$, with $lambda$ the Lebesgue measure. Let $U$ a foliation chart. Then, for any $E subset U subset mathbb{T}^2$ measurable, we have : $lambda(E) = int_L int_{F_U(x)} mathbb{1}_E(x,y)delta_x(y) dm_{F(x)(y)} dm_L(x)$ with $L$ any local transversal and $(delta_x)_x$ a family of positive measurable functions.
differential-geometry dynamical-systems
asked Jan 8 at 22:16
HermèsHermès
1,710612
$endgroup$
add a comment |
$begingroup$
I trying to figure out why this is result is true :
If $F$ is a $C^1$-foliation with $C^1$-leaves, then it is absolutely continuous, where absolutely continuity here means that given a foliation chart $(U,h)$ and $L$ some transversal, there exists a measurable family of positive measurable functions $j_x : F_U(x) to mathbb{R}$ such that for any measurable set $A subset U$, we have that $m(A) = int_L int_{F_U(x)} mathbb{1}_A(x,y)j_x(y)dm_{F(x)}(y)dm_L(x)$ where $m$ is our volume.
I've read that this is a direct application of Fubini Theorem, but since I don't know much about foliation and differential geometry, I don't really feel confident about this...
Any help is very greatly appreciated.
EDIT : this question has been put on hold because it lacks some context. So here it is. I'm interesting in proving that ergodicity of $C^2$ Anosov on compact riemannian manifold. I'm following Brin & Stuck Introduction to dynamical systems. The key idea of the proof is to prove that the stable/unstable foliation is absolutely continuous, as defined above. At one point, authors show that absolutely continuous implies transversely absolute continuity. To do that, they introduce some auxiliary foliation that is $C^1$, while explaining that it is obviously absolutely continuous and transversely absolutely continuous. I've read this in differents books/ressources, as a direct application of Fubini theorem. The context is pretty standard in DS, so maybe I should be posting this question of MO...
EDIT 2 : Some more context here (especially the line "...basic results from calculus show that a version of Fubini’s theorem holds..." This is precisely the claim for which I'm looking a proof. Since I'm working only on the torus, it might goes like this :
Let $F$ a $C^1$ foliation with $C^1$-leaves on the torus $mathbb{T}^2$, with $lambda$ the Lebesgue measure. Let $U$ a foliation chart. Then, for any $E subset U subset mathbb{T}^2$ measurable, we have : $lambda(E) = int_L int_{F_U(x)} mathbb{1}_E(x,y)delta_x(y) dm_{F(x)(y)} dm_L(x)$ with $L$ any local transversal and $(delta_x)_x$ a family of positive measurable functions.
differential-geometry dynamical-systems
asked Jan 8 at 22:16
HermèsHermès
1,710612
$endgroup$
$begingroup$
Should I consider posting this on MO ? Some professor at my lab had not really a clue on how to do that...It seems to be part of some folkore : this has been proved on rare occasions, so we just need to find the right reference...
$endgroup$
– Hermès
Jan 11 at 17:59
$begingroup$
Indeed, it is an immediate application of Fubini's theorem after making a coordinate change as in the flow box theorem (use implicit function theorem). Quite unlikely to find an explicit proof written since it is really immediate. Problems only occur when the foliation is not smooth (which is the typical case...).
$endgroup$
– John B
Jan 13 at 21:35
add a comment |
$begingroup$
I trying to figure out why this is result is true :
If $F$ is a $C^1$-foliation with $C^1$-leaves, then it is absolutely continuous, where absolutely continuity here means that given a foliation chart $(U,h)$ and $L$ some transversal, there exists a measurable family of positive measurable functions $j_x : F_U(x) to mathbb{R}$ such that for any measurable set $A subset U$, we have that $m(A) = int_L int_{F_U(x)} mathbb{1}_A(x,y)j_x(y)dm_{F(x)}(y)dm_L(x)$ where $m$ is our volume.
I've read that this is a direct application of Fubini Theorem, but since I don't know much about foliation and differential geometry, I don't really feel confident about this...
Any help is very greatly appreciated.
EDIT : this question has been put on hold because it lacks some context. So here it is. I'm interesting in proving that ergodicity of $C^2$ Anosov on compact riemannian manifold. I'm following Brin & Stuck Introduction to dynamical systems. The key idea of the proof is to prove that the stable/unstable foliation is absolutely continuous, as defined above. At one point, authors show that absolutely continuous implies transversely absolute continuity. To do that, they introduce some auxiliary foliation that is $C^1$, while explaining that it is obviously absolutely continuous and transversely absolutely continuous. I've read this in differents books/ressources, as a direct application of Fubini theorem. The context is pretty standard in DS, so maybe I should be posting this question of MO...
EDIT 2 : Some more context here (especially the line "...basic results from calculus show that a version of Fubini’s theorem holds..." This is precisely the claim for which I'm looking a proof. Since I'm working only on the torus, it might goes like this :
Let $F$ a $C^1$ foliation with $C^1$-leaves on the torus $mathbb{T}^2$, with $lambda$ the Lebesgue measure. Let $U$ a foliation chart. Then, for any $E subset U subset mathbb{T}^2$ measurable, we have : $lambda(E) = int_L int_{F_U(x)} mathbb{1}_E(x,y)delta_x(y) dm_{F(x)(y)} dm_L(x)$ with $L$ any local transversal and $(delta_x)_x$ a family of positive measurable functions.
differential-geometry dynamical-systems
asked Jan 8 at 22:16
HermèsHermès
1,710612
$endgroup$
I trying to figure out why this is result is true :
If $F$ is a $C^1$-foliation with $C^1$-leaves, then it is absolutely continuous, where absolutely continuity here means that given a foliation chart $(U,h)$ and $L$ some transversal, there exists a measurable family of positive measurable functions $j_x : F_U(x) to mathbb{R}$ such that for any measurable set $A subset U$, we have that $m(A) = int_L int_{F_U(x)} mathbb{1}_A(x,y)j_x(y)dm_{F(x)}(y)dm_L(x)$ where $m$ is our volume.
I've read that this is a direct application of Fubini Theorem, but since I don't know much about foliation and differential geometry, I don't really feel confident about this...
Any help is very greatly appreciated.
EDIT : this question has been put on hold because it lacks some context. So here it is. I'm interesting in proving that ergodicity of $C^2$ Anosov on compact riemannian manifold. I'm following Brin & Stuck Introduction to dynamical systems. The key idea of the proof is to prove that the stable/unstable foliation is absolutely continuous, as defined above. At one point, authors show that absolutely continuous implies transversely absolute continuity. To do that, they introduce some auxiliary foliation that is $C^1$, while explaining that it is obviously absolutely continuous and transversely absolutely continuous. I've read this in differents books/ressources, as a direct application of Fubini theorem. The context is pretty standard in DS, so maybe I should be posting this question of MO...
EDIT 2 : Some more context here (especially the line "...basic results from calculus show that a version of Fubini’s theorem holds..." This is precisely the claim for which I'm looking a proof. Since I'm working only on the torus, it might goes like this :
Let $F$ a $C^1$ foliation with $C^1$-leaves on the torus $mathbb{T}^2$, with $lambda$ the Lebesgue measure. Let $U$ a foliation chart. Then, for any $E subset U subset mathbb{T}^2$ measurable, we have : $lambda(E) = int_L int_{F_U(x)} mathbb{1}_E(x,y)delta_x(y) dm_{F(x)(y)} dm_L(x)$ with $L$ any local transversal and $(delta_x)_x$ a family of positive measurable functions.
differential-geometry dynamical-systems
differential-geometry dynamical-systems
asked Jan 8 at 22:16
HermèsHermès
1,710612
asked Jan 8 at 22:16
HermèsHermès
1,710612
edited Jan 10 at 17:29
Hermès
asked Jan 8 at 22:16
HermèsHermès
1,710612
asked Jan 8 at 22:16
HermèsHermès
1,710612
asked Jan 8 at 22:16
HermèsHermès
1,710612
1,710612
$begingroup$
Should I consider posting this on MO ? Some professor at my lab had not really a clue on how to do that...It seems to be part of some folkore : this has been proved on rare occasions, so we just need to find the right reference...
$endgroup$
– Hermès
Jan 11 at 17:59
$begingroup$
Indeed, it is an immediate application of Fubini's theorem after making a coordinate change as in the flow box theorem (use implicit function theorem). Quite unlikely to find an explicit proof written since it is really immediate. Problems only occur when the foliation is not smooth (which is the typical case...).
$endgroup$
– John B
Jan 13 at 21:35
add a comment |
$begingroup$
Should I consider posting this on MO ? Some professor at my lab had not really a clue on how to do that...It seems to be part of some folkore : this has been proved on rare occasions, so we just need to find the right reference...
$endgroup$
– Hermès
Jan 11 at 17:59
$begingroup$
Indeed, it is an immediate application of Fubini's theorem after making a coordinate change as in the flow box theorem (use implicit function theorem). Quite unlikely to find an explicit proof written since it is really immediate. Problems only occur when the foliation is not smooth (which is the typical case...).
$endgroup$
– John B
Jan 13 at 21:35
$begingroup$
Should I consider posting this on MO ? Some professor at my lab had not really a clue on how to do that...It seems to be part of some folkore : this has been proved on rare occasions, so we just need to find the right reference...
$endgroup$
– Hermès
Jan 11 at 17:59
$begingroup$
Should I consider posting this on MO ? Some professor at my lab had not really a clue on how to do that...It seems to be part of some folkore : this has been proved on rare occasions, so we just need to find the right reference...
$endgroup$
– Hermès
Jan 11 at 17:59
$begingroup$
Indeed, it is an immediate application of Fubini's theorem after making a coordinate change as in the flow box theorem (use implicit function theorem). Quite unlikely to find an explicit proof written since it is really immediate. Problems only occur when the foliation is not smooth (which is the typical case...).
$endgroup$
– John B
Jan 13 at 21:35
$begingroup$
Indeed, it is an immediate application of Fubini's theorem after making a coordinate change as in the flow box theorem (use implicit function theorem). Quite unlikely to find an explicit proof written since it is really immediate. Problems only occur when the foliation is not smooth (which is the typical case...).
$endgroup$
– John B
Jan 13 at 21:35
add a comment |
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$begingroup$
Should I consider posting this on MO ? Some professor at my lab had not really a clue on how to do that...It seems to be part of some folkore : this has been proved on rare occasions, so we just need to find the right reference...
$endgroup$
– Hermès
Jan 11 at 17:59
$begingroup$
Indeed, it is an immediate application of Fubini's theorem after making a coordinate change as in the flow box theorem (use implicit function theorem). Quite unlikely to find an explicit proof written since it is really immediate. Problems only occur when the foliation is not smooth (which is the typical case...).
$endgroup$
– John B
Jan 13 at 21:35