Mixing
Global stable manifold
$begingroup$
In Brin Stuck, the following Corollary 5.6.6 is left as an exercise, that of which I'm not sure on how to prove. The stable-manifold theorem states that there exists $epsilon >0$ small enough such that local stable manifolds $W^s_epsilon(x) = { y in M: d(f^n(x), f^n(y)) < epsilon quad forall n geq 0}$ is a $C^1$ embedded manifold for every $x in Lambda$ in the hyperbolic set. Now we're interesting in the global stable manifold (defined below). Proposition 5.6.5 is just a set theoretic result, while the Corollary that follows seems to be a differential geometry question. Note that there is a typo here, authors should have written that $W^s(x)$ is an immersed $C^1$ submanifold (not embedded). I'm trying to find a prove for this claim. As a matter of fact, one can see that $W^s(x)$ is defined as a increasing union of embedded $C^1$-manifolds ! I posted a question here where I asked whether or not an increasing sequence of embedded submanifold is still an embedded submanifold. That is not the case, but at the time I thought that the stable manifold was to be a embedded submanifold (following the typo). Therefore, the question remains open as to whether or not such an increasing sequence can end up being an immersed submanifold !
Maybe that's not the way to do it...
Anyway, thanks for the help !
dynamical-systems
asked Jan 12 at 11:44
Noam EluarNoam Eluar
227
$endgroup$
|
show 7 more comments
$begingroup$
In Brin Stuck, the following Corollary 5.6.6 is left as an exercise, that of which I'm not sure on how to prove. The stable-manifold theorem states that there exists $epsilon >0$ small enough such that local stable manifolds $W^s_epsilon(x) = { y in M: d(f^n(x), f^n(y)) < epsilon quad forall n geq 0}$ is a $C^1$ embedded manifold for every $x in Lambda$ in the hyperbolic set. Now we're interesting in the global stable manifold (defined below). Proposition 5.6.5 is just a set theoretic result, while the Corollary that follows seems to be a differential geometry question. Note that there is a typo here, authors should have written that $W^s(x)$ is an immersed $C^1$ submanifold (not embedded). I'm trying to find a prove for this claim. As a matter of fact, one can see that $W^s(x)$ is defined as a increasing union of embedded $C^1$-manifolds ! I posted a question here where I asked whether or not an increasing sequence of embedded submanifold is still an embedded submanifold. That is not the case, but at the time I thought that the stable manifold was to be a embedded submanifold (following the typo). Therefore, the question remains open as to whether or not such an increasing sequence can end up being an immersed submanifold !
Maybe that's not the way to do it...
Anyway, thanks for the help !
dynamical-systems
asked Jan 12 at 11:44
Noam EluarNoam Eluar
227
$endgroup$
$begingroup$
A duplicate of Global stable manifold always an embedded submanifold? Typo or misreading?.
$endgroup$
– user539887
Jan 12 at 12:33
$begingroup$
It's not a duplicate. I'm asking for a proof/ref of Corollary 5.6.6.
$endgroup$
– Noam Eluar
Jan 12 at 12:37
$begingroup$
Please look at the comments and the answer.
$endgroup$
– user539887
Jan 12 at 12:48
$begingroup$
It seems to me these are discussions on the notion of "$C^1$ embedded manifold" used by the authors. In the answer, it says that it means that "there exists a $C^1$-smooth injective immersion from the open disk (whose image is in the stable manifold)". If this is the case, there's nothing to prove. I'll bet that the user meant to say "...(whose image is the stable manifold)", not the "in". That's the regular definition of a immersed submanifold. We need the all image of the disk to be $W^s(x)$, not just a part of it...If you have a proof, I'll be very thankful if you could provide it !
$endgroup$
– Noam Eluar
Jan 12 at 13:05
1
$begingroup$
Regarding a reference on your last question, see Morton Brown's PAMS paper The Monotone Union of Open $n$-Cells is an Open $n$-Cell.
$endgroup$
– user539887
Jan 12 at 13:29
|
show 7 more comments
$begingroup$
In Brin Stuck, the following Corollary 5.6.6 is left as an exercise, that of which I'm not sure on how to prove. The stable-manifold theorem states that there exists $epsilon >0$ small enough such that local stable manifolds $W^s_epsilon(x) = { y in M: d(f^n(x), f^n(y)) < epsilon quad forall n geq 0}$ is a $C^1$ embedded manifold for every $x in Lambda$ in the hyperbolic set. Now we're interesting in the global stable manifold (defined below). Proposition 5.6.5 is just a set theoretic result, while the Corollary that follows seems to be a differential geometry question. Note that there is a typo here, authors should have written that $W^s(x)$ is an immersed $C^1$ submanifold (not embedded). I'm trying to find a prove for this claim. As a matter of fact, one can see that $W^s(x)$ is defined as a increasing union of embedded $C^1$-manifolds ! I posted a question here where I asked whether or not an increasing sequence of embedded submanifold is still an embedded submanifold. That is not the case, but at the time I thought that the stable manifold was to be a embedded submanifold (following the typo). Therefore, the question remains open as to whether or not such an increasing sequence can end up being an immersed submanifold !
Maybe that's not the way to do it...
Anyway, thanks for the help !
dynamical-systems
asked Jan 12 at 11:44
Noam EluarNoam Eluar
227
$endgroup$
In Brin Stuck, the following Corollary 5.6.6 is left as an exercise, that of which I'm not sure on how to prove. The stable-manifold theorem states that there exists $epsilon >0$ small enough such that local stable manifolds $W^s_epsilon(x) = { y in M: d(f^n(x), f^n(y)) < epsilon quad forall n geq 0}$ is a $C^1$ embedded manifold for every $x in Lambda$ in the hyperbolic set. Now we're interesting in the global stable manifold (defined below). Proposition 5.6.5 is just a set theoretic result, while the Corollary that follows seems to be a differential geometry question. Note that there is a typo here, authors should have written that $W^s(x)$ is an immersed $C^1$ submanifold (not embedded). I'm trying to find a prove for this claim. As a matter of fact, one can see that $W^s(x)$ is defined as a increasing union of embedded $C^1$-manifolds ! I posted a question here where I asked whether or not an increasing sequence of embedded submanifold is still an embedded submanifold. That is not the case, but at the time I thought that the stable manifold was to be a embedded submanifold (following the typo). Therefore, the question remains open as to whether or not such an increasing sequence can end up being an immersed submanifold !
Maybe that's not the way to do it...
Anyway, thanks for the help !
dynamical-systems
dynamical-systems
asked Jan 12 at 11:44
Noam EluarNoam Eluar
227
asked Jan 12 at 11:44
Noam EluarNoam Eluar
227
edited Jan 12 at 15:07
Noam Eluar
asked Jan 12 at 11:44
Noam EluarNoam Eluar
227
asked Jan 12 at 11:44
Noam EluarNoam Eluar
227
asked Jan 12 at 11:44
Noam EluarNoam Eluar
227
227
$begingroup$
A duplicate of Global stable manifold always an embedded submanifold? Typo or misreading?.
$endgroup$
– user539887
Jan 12 at 12:33
$begingroup$
It's not a duplicate. I'm asking for a proof/ref of Corollary 5.6.6.
$endgroup$
– Noam Eluar
Jan 12 at 12:37
$begingroup$
Please look at the comments and the answer.
$endgroup$
– user539887
Jan 12 at 12:48
$begingroup$
It seems to me these are discussions on the notion of "$C^1$ embedded manifold" used by the authors. In the answer, it says that it means that "there exists a $C^1$-smooth injective immersion from the open disk (whose image is in the stable manifold)". If this is the case, there's nothing to prove. I'll bet that the user meant to say "...(whose image is the stable manifold)", not the "in". That's the regular definition of a immersed submanifold. We need the all image of the disk to be $W^s(x)$, not just a part of it...If you have a proof, I'll be very thankful if you could provide it !
$endgroup$
– Noam Eluar
Jan 12 at 13:05
1
$begingroup$
Regarding a reference on your last question, see Morton Brown's PAMS paper The Monotone Union of Open $n$-Cells is an Open $n$-Cell.
$endgroup$
– user539887
Jan 12 at 13:29
|
show 7 more comments
$begingroup$
A duplicate of Global stable manifold always an embedded submanifold? Typo or misreading?.
$endgroup$
– user539887
Jan 12 at 12:33
$begingroup$
It's not a duplicate. I'm asking for a proof/ref of Corollary 5.6.6.
$endgroup$
– Noam Eluar
Jan 12 at 12:37
$begingroup$
Please look at the comments and the answer.
$endgroup$
– user539887
Jan 12 at 12:48
$begingroup$
It seems to me these are discussions on the notion of "$C^1$ embedded manifold" used by the authors. In the answer, it says that it means that "there exists a $C^1$-smooth injective immersion from the open disk (whose image is in the stable manifold)". If this is the case, there's nothing to prove. I'll bet that the user meant to say "...(whose image is the stable manifold)", not the "in". That's the regular definition of a immersed submanifold. We need the all image of the disk to be $W^s(x)$, not just a part of it...If you have a proof, I'll be very thankful if you could provide it !
$endgroup$
– Noam Eluar
Jan 12 at 13:05
1
$begingroup$
Regarding a reference on your last question, see Morton Brown's PAMS paper The Monotone Union of Open $n$-Cells is an Open $n$-Cell.
$endgroup$
– user539887
Jan 12 at 13:29
$begingroup$
A duplicate of Global stable manifold always an embedded submanifold? Typo or misreading?.
$endgroup$
– user539887
Jan 12 at 12:33
$begingroup$
A duplicate of Global stable manifold always an embedded submanifold? Typo or misreading?.
$endgroup$
– user539887
Jan 12 at 12:33
$begingroup$
It's not a duplicate. I'm asking for a proof/ref of Corollary 5.6.6.
$endgroup$
– Noam Eluar
Jan 12 at 12:37
$begingroup$
It's not a duplicate. I'm asking for a proof/ref of Corollary 5.6.6.
$endgroup$
– Noam Eluar
Jan 12 at 12:37
$begingroup$
Please look at the comments and the answer.
$endgroup$
– user539887
Jan 12 at 12:48
$begingroup$
Please look at the comments and the answer.
$endgroup$
– user539887
Jan 12 at 12:48
$begingroup$
It seems to me these are discussions on the notion of "$C^1$ embedded manifold" used by the authors. In the answer, it says that it means that "there exists a $C^1$-smooth injective immersion from the open disk (whose image is in the stable manifold)". If this is the case, there's nothing to prove. I'll bet that the user meant to say "...(whose image is the stable manifold)", not the "in". That's the regular definition of a immersed submanifold. We need the all image of the disk to be $W^s(x)$, not just a part of it...If you have a proof, I'll be very thankful if you could provide it !
$endgroup$
– Noam Eluar
Jan 12 at 13:05
$begingroup$
It seems to me these are discussions on the notion of "$C^1$ embedded manifold" used by the authors. In the answer, it says that it means that "there exists a $C^1$-smooth injective immersion from the open disk (whose image is in the stable manifold)". If this is the case, there's nothing to prove. I'll bet that the user meant to say "...(whose image is the stable manifold)", not the "in". That's the regular definition of a immersed submanifold. We need the all image of the disk to be $W^s(x)$, not just a part of it...If you have a proof, I'll be very thankful if you could provide it !
$endgroup$
– Noam Eluar
Jan 12 at 13:05
1
1
$begingroup$
Regarding a reference on your last question, see Morton Brown's PAMS paper The Monotone Union of Open $n$-Cells is an Open $n$-Cell.
$endgroup$
– user539887
Jan 12 at 13:29
$begingroup$
Regarding a reference on your last question, see Morton Brown's PAMS paper The Monotone Union of Open $n$-Cells is an Open $n$-Cell.
$endgroup$
– user539887
Jan 12 at 13:29
|
show 7 more comments
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$begingroup$
A duplicate of Global stable manifold always an embedded submanifold? Typo or misreading?.
$endgroup$
– user539887
Jan 12 at 12:33
$begingroup$
It's not a duplicate. I'm asking for a proof/ref of Corollary 5.6.6.
$endgroup$
– Noam Eluar
Jan 12 at 12:37
$begingroup$
Please look at the comments and the answer.
$endgroup$
– user539887
Jan 12 at 12:48
$begingroup$
It seems to me these are discussions on the notion of "$C^1$ embedded manifold" used by the authors. In the answer, it says that it means that "there exists a $C^1$-smooth injective immersion from the open disk (whose image is in the stable manifold)". If this is the case, there's nothing to prove. I'll bet that the user meant to say "...(whose image is the stable manifold)", not the "in". That's the regular definition of a immersed submanifold. We need the all image of the disk to be $W^s(x)$, not just a part of it...If you have a proof, I'll be very thankful if you could provide it !
$endgroup$
– Noam Eluar
Jan 12 at 13:05
1
$begingroup$
Regarding a reference on your last question, see Morton Brown's PAMS paper The Monotone Union of Open $n$-Cells is an Open $n$-Cell.
$endgroup$
– user539887
Jan 12 at 13:29