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Existence of transversal loops on a vector field without singular points on $mathbbT^2$
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Let $X$ be a smooth vector field on $mathbbT^2$ without singular points and periodic orbits. Then by this theorem
Theorem:(Poincaré, Denjoy) Every non-singular $mathcalC^2$ vector field on a compact surface that has no periodic orbits is topologically equivalent to a linear flow on a torus with the irrational slope,
$X$ is topologically equivalent to the irrational flow. It is trivial that if $Y$ is a irrational flow on $mathbbT^2$ then there exists two $mathcalC^1$ loops $$alpha_1,alpha_2 : mathbbS^1to mathbbT^2$$
with $alpha_1(0) = alpha_2(0) = p$, such that $Y(alpha_i(t)),alpha_i'(t)$ is a basis of $T_alpha(t) mathbbT^2$, $forall$ $t$ $in$ $mathbbS^1$, and $[alpha_1 (t)], [alpha_2 (t)]$ are generators of the group $pi_1(mathbbT^2,p)$.
My Doubt: Using that $X$ is topologically equivalent to $Y$, is it possible to guarantee that there exists two $mathcalC^1$ loops $$beta_1,beta_2 : mathbbS^1to mathbbT^2$$
with $beta_1(0) = beta_2(0) = p$, such that $X(beta_i(t)),beta_i'(t)$ is a basis of $T_alpha(t) mathbbT^2$, $forall$ $t$ $in$ $mathbbS^1$, and $[beta_1 (t)], [beta_1 (t)]$ are generators of the group $pi_1(mathbbT^2,p)$?
differential-equations dynamical-systems smooth-manifolds
asked Aug 6 at 2:47
Matheus Manzatto
979520
add a comment |Â
up vote
0
down vote
favorite
Let $X$ be a smooth vector field on $mathbbT^2$ without singular points and periodic orbits. Then by this theorem
Theorem:(Poincaré, Denjoy) Every non-singular $mathcalC^2$ vector field on a compact surface that has no periodic orbits is topologically equivalent to a linear flow on a torus with the irrational slope,
$X$ is topologically equivalent to the irrational flow. It is trivial that if $Y$ is a irrational flow on $mathbbT^2$ then there exists two $mathcalC^1$ loops $$alpha_1,alpha_2 : mathbbS^1to mathbbT^2$$
with $alpha_1(0) = alpha_2(0) = p$, such that $Y(alpha_i(t)),alpha_i'(t)$ is a basis of $T_alpha(t) mathbbT^2$, $forall$ $t$ $in$ $mathbbS^1$, and $[alpha_1 (t)], [alpha_2 (t)]$ are generators of the group $pi_1(mathbbT^2,p)$.
My Doubt: Using that $X$ is topologically equivalent to $Y$, is it possible to guarantee that there exists two $mathcalC^1$ loops $$beta_1,beta_2 : mathbbS^1to mathbbT^2$$
with $beta_1(0) = beta_2(0) = p$, such that $X(beta_i(t)),beta_i'(t)$ is a basis of $T_alpha(t) mathbbT^2$, $forall$ $t$ $in$ $mathbbS^1$, and $[beta_1 (t)], [beta_1 (t)]$ are generators of the group $pi_1(mathbbT^2,p)$?
differential-equations dynamical-systems smooth-manifolds
asked Aug 6 at 2:47
Matheus Manzatto
979520
When you say topologically equivalent, do you mean there is a homeomorphism from $X$ to the torus taking the given flow to a standard linear flow of irrational slope?
– Max
Aug 6 at 2:53
If $Y$ is the standard linear flow of irrational slope, $phi$ and $psi$ are, respectively, the flow of the vector fields $Y$ and $X$, then there are a homeomorphism $h:mathbbT^2 to mathbbT^2$ and a continuous function $tau: mathbbRtimes mathbbT^2 to mathbbR$, crescent on the first entry ($tau(cdot, p)$ is a bijection for every $p in mathbbT^2$ fixed), such that $phi(t,h(p) ) = h circ psi (tau(t,p), p) $
– Matheus Manzatto
Aug 6 at 3:02
1
I would imagine you could use something like Stone-Weierstrass to show that the desired result holds up to an arbitrarily small homotopy of $X$, then show that for a sufficiently small $C^1$ homotopy the desired properties are preserved. There are some troublesome details to flesh out though...
– Max
Aug 6 at 3:21
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $X$ be a smooth vector field on $mathbbT^2$ without singular points and periodic orbits. Then by this theorem
Theorem:(Poincaré, Denjoy) Every non-singular $mathcalC^2$ vector field on a compact surface that has no periodic orbits is topologically equivalent to a linear flow on a torus with the irrational slope,
$X$ is topologically equivalent to the irrational flow. It is trivial that if $Y$ is a irrational flow on $mathbbT^2$ then there exists two $mathcalC^1$ loops $$alpha_1,alpha_2 : mathbbS^1to mathbbT^2$$
with $alpha_1(0) = alpha_2(0) = p$, such that $Y(alpha_i(t)),alpha_i'(t)$ is a basis of $T_alpha(t) mathbbT^2$, $forall$ $t$ $in$ $mathbbS^1$, and $[alpha_1 (t)], [alpha_2 (t)]$ are generators of the group $pi_1(mathbbT^2,p)$.
My Doubt: Using that $X$ is topologically equivalent to $Y$, is it possible to guarantee that there exists two $mathcalC^1$ loops $$beta_1,beta_2 : mathbbS^1to mathbbT^2$$
with $beta_1(0) = beta_2(0) = p$, such that $X(beta_i(t)),beta_i'(t)$ is a basis of $T_alpha(t) mathbbT^2$, $forall$ $t$ $in$ $mathbbS^1$, and $[beta_1 (t)], [beta_1 (t)]$ are generators of the group $pi_1(mathbbT^2,p)$?
differential-equations dynamical-systems smooth-manifolds
asked Aug 6 at 2:47
Matheus Manzatto
979520
Let $X$ be a smooth vector field on $mathbbT^2$ without singular points and periodic orbits. Then by this theorem
Theorem:(Poincaré, Denjoy) Every non-singular $mathcalC^2$ vector field on a compact surface that has no periodic orbits is topologically equivalent to a linear flow on a torus with the irrational slope,
$X$ is topologically equivalent to the irrational flow. It is trivial that if $Y$ is a irrational flow on $mathbbT^2$ then there exists two $mathcalC^1$ loops $$alpha_1,alpha_2 : mathbbS^1to mathbbT^2$$
with $alpha_1(0) = alpha_2(0) = p$, such that $Y(alpha_i(t)),alpha_i'(t)$ is a basis of $T_alpha(t) mathbbT^2$, $forall$ $t$ $in$ $mathbbS^1$, and $[alpha_1 (t)], [alpha_2 (t)]$ are generators of the group $pi_1(mathbbT^2,p)$.
My Doubt: Using that $X$ is topologically equivalent to $Y$, is it possible to guarantee that there exists two $mathcalC^1$ loops $$beta_1,beta_2 : mathbbS^1to mathbbT^2$$
with $beta_1(0) = beta_2(0) = p$, such that $X(beta_i(t)),beta_i'(t)$ is a basis of $T_alpha(t) mathbbT^2$, $forall$ $t$ $in$ $mathbbS^1$, and $[beta_1 (t)], [beta_1 (t)]$ are generators of the group $pi_1(mathbbT^2,p)$?
differential-equations dynamical-systems smooth-manifolds
asked Aug 6 at 2:47
Matheus Manzatto
979520
asked Aug 6 at 2:47
Matheus Manzatto
979520
asked Aug 6 at 2:47
Matheus Manzatto
979520
asked Aug 6 at 2:47
Matheus Manzatto
979520
979520
When you say topologically equivalent, do you mean there is a homeomorphism from $X$ to the torus taking the given flow to a standard linear flow of irrational slope?
– Max
Aug 6 at 2:53
If $Y$ is the standard linear flow of irrational slope, $phi$ and $psi$ are, respectively, the flow of the vector fields $Y$ and $X$, then there are a homeomorphism $h:mathbbT^2 to mathbbT^2$ and a continuous function $tau: mathbbRtimes mathbbT^2 to mathbbR$, crescent on the first entry ($tau(cdot, p)$ is a bijection for every $p in mathbbT^2$ fixed), such that $phi(t,h(p) ) = h circ psi (tau(t,p), p) $
– Matheus Manzatto
Aug 6 at 3:02
1
I would imagine you could use something like Stone-Weierstrass to show that the desired result holds up to an arbitrarily small homotopy of $X$, then show that for a sufficiently small $C^1$ homotopy the desired properties are preserved. There are some troublesome details to flesh out though...
– Max
Aug 6 at 3:21
add a comment |Â
When you say topologically equivalent, do you mean there is a homeomorphism from $X$ to the torus taking the given flow to a standard linear flow of irrational slope?
– Max
Aug 6 at 2:53
If $Y$ is the standard linear flow of irrational slope, $phi$ and $psi$ are, respectively, the flow of the vector fields $Y$ and $X$, then there are a homeomorphism $h:mathbbT^2 to mathbbT^2$ and a continuous function $tau: mathbbRtimes mathbbT^2 to mathbbR$, crescent on the first entry ($tau(cdot, p)$ is a bijection for every $p in mathbbT^2$ fixed), such that $phi(t,h(p) ) = h circ psi (tau(t,p), p) $
– Matheus Manzatto
Aug 6 at 3:02
1
I would imagine you could use something like Stone-Weierstrass to show that the desired result holds up to an arbitrarily small homotopy of $X$, then show that for a sufficiently small $C^1$ homotopy the desired properties are preserved. There are some troublesome details to flesh out though...
– Max
Aug 6 at 3:21
When you say topologically equivalent, do you mean there is a homeomorphism from $X$ to the torus taking the given flow to a standard linear flow of irrational slope?
– Max
Aug 6 at 2:53
When you say topologically equivalent, do you mean there is a homeomorphism from $X$ to the torus taking the given flow to a standard linear flow of irrational slope?
– Max
Aug 6 at 2:53
If $Y$ is the standard linear flow of irrational slope, $phi$ and $psi$ are, respectively, the flow of the vector fields $Y$ and $X$, then there are a homeomorphism $h:mathbbT^2 to mathbbT^2$ and a continuous function $tau: mathbbRtimes mathbbT^2 to mathbbR$, crescent on the first entry ($tau(cdot, p)$ is a bijection for every $p in mathbbT^2$ fixed), such that $phi(t,h(p) ) = h circ psi (tau(t,p), p) $
– Matheus Manzatto
Aug 6 at 3:02
If $Y$ is the standard linear flow of irrational slope, $phi$ and $psi$ are, respectively, the flow of the vector fields $Y$ and $X$, then there are a homeomorphism $h:mathbbT^2 to mathbbT^2$ and a continuous function $tau: mathbbRtimes mathbbT^2 to mathbbR$, crescent on the first entry ($tau(cdot, p)$ is a bijection for every $p in mathbbT^2$ fixed), such that $phi(t,h(p) ) = h circ psi (tau(t,p), p) $
– Matheus Manzatto
Aug 6 at 3:02
1
1
I would imagine you could use something like Stone-Weierstrass to show that the desired result holds up to an arbitrarily small homotopy of $X$, then show that for a sufficiently small $C^1$ homotopy the desired properties are preserved. There are some troublesome details to flesh out though...
– Max
Aug 6 at 3:21
I would imagine you could use something like Stone-Weierstrass to show that the desired result holds up to an arbitrarily small homotopy of $X$, then show that for a sufficiently small $C^1$ homotopy the desired properties are preserved. There are some troublesome details to flesh out though...
– Max
Aug 6 at 3:21
add a comment |Â
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When you say topologically equivalent, do you mean there is a homeomorphism from $X$ to the torus taking the given flow to a standard linear flow of irrational slope?
– Max
Aug 6 at 2:53
If $Y$ is the standard linear flow of irrational slope, $phi$ and $psi$ are, respectively, the flow of the vector fields $Y$ and $X$, then there are a homeomorphism $h:mathbbT^2 to mathbbT^2$ and a continuous function $tau: mathbbRtimes mathbbT^2 to mathbbR$, crescent on the first entry ($tau(cdot, p)$ is a bijection for every $p in mathbbT^2$ fixed), such that $phi(t,h(p) ) = h circ psi (tau(t,p), p) $
– Matheus Manzatto
Aug 6 at 3:02
1
I would imagine you could use something like Stone-Weierstrass to show that the desired result holds up to an arbitrarily small homotopy of $X$, then show that for a sufficiently small $C^1$ homotopy the desired properties are preserved. There are some troublesome details to flesh out though...
– Max
Aug 6 at 3:21