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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.
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    $\begingroup$ 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). $\endgroup$ – Dan Rust Jul 25 '18 at 12:51
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    $\begingroup$ 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. $\endgroup$ – Dan Rust Jul 25 '18 at 12:51
  • $\begingroup$ 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? $\endgroup$ – AzJ Jul 25 '18 at 20:33

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