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.