# Differences between homeomorphic and topologically conjugate dynamical systems.

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

• 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 '18 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 '18 at 12:51