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Differences between homeomorphic and topologically conjugate dynamical systems.
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I have begun studying homeomorphisms between dynamical systems. I have a few related questions about homeomorphic and topologically conjugate dynamical systems.
Questions
- How can I determine if a homeomorphism is orientation presevering?
- If two dynamical systems are Topologically Conjugate (that is $f circ h=h circ g$), does this imply that $h$ is an orientation preserving homeomorphism? (Answered see comment below this post)
- What properties are preserved between two topologically conjugate dynamical systems, compared to two dynamical systems which are only homeomorphic to each other?
Definitions
$f:X rightarrow X$, $g:Y rightarrow Y$ and $h:Y rightarrow X$ are continuous functions on smooth orientable manifolds, $X$ and $Y$.
- Topologically Conjugate: $f circ h=h circ g$ and $h$ is homeomorphsim.
Homeomorphism- What it means for a manifold to be orientable.
Notes
- Partial answers are appreciated.
- If you need any clarification please ask.
differential-geometry differential-topology dynamical-systems
asked Jul 24 at 22:10
AzJ
257117
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up vote
4
down vote
favorite
I have begun studying homeomorphisms between dynamical systems. I have a few related questions about homeomorphic and topologically conjugate dynamical systems.
Questions
- How can I determine if a homeomorphism is orientation presevering?
- If two dynamical systems are Topologically Conjugate (that is $f circ h=h circ g$), does this imply that $h$ is an orientation preserving homeomorphism? (Answered see comment below this post)
- What properties are preserved between two topologically conjugate dynamical systems, compared to two dynamical systems which are only homeomorphic to each other?
Definitions
$f:X rightarrow X$, $g:Y rightarrow Y$ and $h:Y rightarrow X$ are continuous functions on smooth orientable manifolds, $X$ and $Y$.
- Topologically Conjugate: $f circ h=h circ g$ and $h$ is homeomorphsim.
Homeomorphism- What it means for a manifold to be orientable.
Notes
- Partial answers are appreciated.
- If you need any clarification please ask.
differential-geometry differential-topology dynamical-systems
asked Jul 24 at 22:10
AzJ
257117
2
Are you assuming that $X$ and $Y$ are orientable manifolds? Otherwise, talking about orientation preserving doesn't make sense. In any case, it feels like you're maybe asking these questions a bit too early in your study, because you very quickly encounter invariants of topological conjugacy soon after the term is introduced - similarly with homeomorphisms (which is a purely topological notion, and is divorced from any dynamics).
– Dan Rust
Jul 25 at 12:51
2
For a counterexample to your second bullet point, consider the circle $X = [0,1]/sim$ with clockwise rotation by some number $t$, so $f colon X to X colon x mapsto x+t$ and a circle $Y$ with anti-clockwise rotation $x mapsto x-t$. Clearly, the two systems are topologically conjugate by just mapping $x mapsto -x$, but this is not an orientation preserving map.
– Dan Rust
Jul 25 at 12:51
Sorry I corrected my question. I should have chosen the set of topologically spaces that I was interested in. I did not initially specify that I was looking at orientable manifolds because the literature I was looking at was interested in preserving the orientation of the flows of dynamical systems. It is clear to me now that the all the manifolds I am studying are orientable. With this clarification can you help me understand how to determine if a homeomorphism is orientation preserving?
– AzJ
Jul 25 at 20:33
add a comment |Â
up vote
4
down vote
favorite
up vote
4
down vote
favorite
I have begun studying homeomorphisms between dynamical systems. I have a few related questions about homeomorphic and topologically conjugate dynamical systems.
Questions
- How can I determine if a homeomorphism is orientation presevering?
- If two dynamical systems are Topologically Conjugate (that is $f circ h=h circ g$), does this imply that $h$ is an orientation preserving homeomorphism? (Answered see comment below this post)
- What properties are preserved between two topologically conjugate dynamical systems, compared to two dynamical systems which are only homeomorphic to each other?
Definitions
$f:X rightarrow X$, $g:Y rightarrow Y$ and $h:Y rightarrow X$ are continuous functions on smooth orientable manifolds, $X$ and $Y$.
- Topologically Conjugate: $f circ h=h circ g$ and $h$ is homeomorphsim.
Homeomorphism- What it means for a manifold to be orientable.
Notes
- Partial answers are appreciated.
- If you need any clarification please ask.
differential-geometry differential-topology dynamical-systems
asked Jul 24 at 22:10
AzJ
257117
I have begun studying homeomorphisms between dynamical systems. I have a few related questions about homeomorphic and topologically conjugate dynamical systems.
Questions
- How can I determine if a homeomorphism is orientation presevering?
- If two dynamical systems are Topologically Conjugate (that is $f circ h=h circ g$), does this imply that $h$ is an orientation preserving homeomorphism? (Answered see comment below this post)
- What properties are preserved between two topologically conjugate dynamical systems, compared to two dynamical systems which are only homeomorphic to each other?
Definitions
$f:X rightarrow X$, $g:Y rightarrow Y$ and $h:Y rightarrow X$ are continuous functions on smooth orientable manifolds, $X$ and $Y$.
- Topologically Conjugate: $f circ h=h circ g$ and $h$ is homeomorphsim.
Homeomorphism- What it means for a manifold to be orientable.
Notes
- Partial answers are appreciated.
- If you need any clarification please ask.
differential-geometry differential-topology dynamical-systems
asked Jul 24 at 22:10
AzJ
257117
edited Jul 25 at 20:35
asked Jul 24 at 22:10
AzJ
257117
asked Jul 24 at 22:10
AzJ
257117
asked Jul 24 at 22:10
AzJ
257117
257117
2
Are you assuming that $X$ and $Y$ are orientable manifolds? Otherwise, talking about orientation preserving doesn't make sense. In any case, it feels like you're maybe asking these questions a bit too early in your study, because you very quickly encounter invariants of topological conjugacy soon after the term is introduced - similarly with homeomorphisms (which is a purely topological notion, and is divorced from any dynamics).
– Dan Rust
Jul 25 at 12:51
2
For a counterexample to your second bullet point, consider the circle $X = [0,1]/sim$ with clockwise rotation by some number $t$, so $f colon X to X colon x mapsto x+t$ and a circle $Y$ with anti-clockwise rotation $x mapsto x-t$. Clearly, the two systems are topologically conjugate by just mapping $x mapsto -x$, but this is not an orientation preserving map.
– Dan Rust
Jul 25 at 12:51
Sorry I corrected my question. I should have chosen the set of topologically spaces that I was interested in. I did not initially specify that I was looking at orientable manifolds because the literature I was looking at was interested in preserving the orientation of the flows of dynamical systems. It is clear to me now that the all the manifolds I am studying are orientable. With this clarification can you help me understand how to determine if a homeomorphism is orientation preserving?
– AzJ
Jul 25 at 20:33
add a comment |Â
2
Are you assuming that $X$ and $Y$ are orientable manifolds? Otherwise, talking about orientation preserving doesn't make sense. In any case, it feels like you're maybe asking these questions a bit too early in your study, because you very quickly encounter invariants of topological conjugacy soon after the term is introduced - similarly with homeomorphisms (which is a purely topological notion, and is divorced from any dynamics).
– Dan Rust
Jul 25 at 12:51
2
For a counterexample to your second bullet point, consider the circle $X = [0,1]/sim$ with clockwise rotation by some number $t$, so $f colon X to X colon x mapsto x+t$ and a circle $Y$ with anti-clockwise rotation $x mapsto x-t$. Clearly, the two systems are topologically conjugate by just mapping $x mapsto -x$, but this is not an orientation preserving map.
– Dan Rust
Jul 25 at 12:51
Sorry I corrected my question. I should have chosen the set of topologically spaces that I was interested in. I did not initially specify that I was looking at orientable manifolds because the literature I was looking at was interested in preserving the orientation of the flows of dynamical systems. It is clear to me now that the all the manifolds I am studying are orientable. With this clarification can you help me understand how to determine if a homeomorphism is orientation preserving?
– AzJ
Jul 25 at 20:33
2
2
Are you assuming that $X$ and $Y$ are orientable manifolds? Otherwise, talking about orientation preserving doesn't make sense. In any case, it feels like you're maybe asking these questions a bit too early in your study, because you very quickly encounter invariants of topological conjugacy soon after the term is introduced - similarly with homeomorphisms (which is a purely topological notion, and is divorced from any dynamics).
– Dan Rust
Jul 25 at 12:51
Are you assuming that $X$ and $Y$ are orientable manifolds? Otherwise, talking about orientation preserving doesn't make sense. In any case, it feels like you're maybe asking these questions a bit too early in your study, because you very quickly encounter invariants of topological conjugacy soon after the term is introduced - similarly with homeomorphisms (which is a purely topological notion, and is divorced from any dynamics).
– Dan Rust
Jul 25 at 12:51
2
2
For a counterexample to your second bullet point, consider the circle $X = [0,1]/sim$ with clockwise rotation by some number $t$, so $f colon X to X colon x mapsto x+t$ and a circle $Y$ with anti-clockwise rotation $x mapsto x-t$. Clearly, the two systems are topologically conjugate by just mapping $x mapsto -x$, but this is not an orientation preserving map.
– Dan Rust
Jul 25 at 12:51
For a counterexample to your second bullet point, consider the circle $X = [0,1]/sim$ with clockwise rotation by some number $t$, so $f colon X to X colon x mapsto x+t$ and a circle $Y$ with anti-clockwise rotation $x mapsto x-t$. Clearly, the two systems are topologically conjugate by just mapping $x mapsto -x$, but this is not an orientation preserving map.
– Dan Rust
Jul 25 at 12:51
Sorry I corrected my question. I should have chosen the set of topologically spaces that I was interested in. I did not initially specify that I was looking at orientable manifolds because the literature I was looking at was interested in preserving the orientation of the flows of dynamical systems. It is clear to me now that the all the manifolds I am studying are orientable. With this clarification can you help me understand how to determine if a homeomorphism is orientation preserving?
– AzJ
Jul 25 at 20:33
Sorry I corrected my question. I should have chosen the set of topologically spaces that I was interested in. I did not initially specify that I was looking at orientable manifolds because the literature I was looking at was interested in preserving the orientation of the flows of dynamical systems. It is clear to me now that the all the manifolds I am studying are orientable. With this clarification can you help me understand how to determine if a homeomorphism is orientation preserving?
– AzJ
Jul 25 at 20:33
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2
Are you assuming that $X$ and $Y$ are orientable manifolds? Otherwise, talking about orientation preserving doesn't make sense. In any case, it feels like you're maybe asking these questions a bit too early in your study, because you very quickly encounter invariants of topological conjugacy soon after the term is introduced - similarly with homeomorphisms (which is a purely topological notion, and is divorced from any dynamics).
– Dan Rust
Jul 25 at 12:51
2
For a counterexample to your second bullet point, consider the circle $X = [0,1]/sim$ with clockwise rotation by some number $t$, so $f colon X to X colon x mapsto x+t$ and a circle $Y$ with anti-clockwise rotation $x mapsto x-t$. Clearly, the two systems are topologically conjugate by just mapping $x mapsto -x$, but this is not an orientation preserving map.
– Dan Rust
Jul 25 at 12:51
Sorry I corrected my question. I should have chosen the set of topologically spaces that I was interested in. I did not initially specify that I was looking at orientable manifolds because the literature I was looking at was interested in preserving the orientation of the flows of dynamical systems. It is clear to me now that the all the manifolds I am studying are orientable. With this clarification can you help me understand how to determine if a homeomorphism is orientation preserving?
– AzJ
Jul 25 at 20:33