I have a doubt regarding the definition of orbit equivalence as given by Fisher & Hasselblatt in their book Hyperbolic Flows.
We say that two flows $\phi, \psi$ on $X, Y$ respectively are orbit equivalent if there exists a homeomorphism $h : X \to Y$ takes orbits of $\phi$ to orbits of $\psi$. Let $x \in X$ and $O^{\phi}(x)$ its orbit under $\phi$. Is the definition of orbit equivalence $h(O^{\phi}(x)) \subseteq O^{\psi}(h(x))$ or $h(O^{\phi}(x)) = O^{\psi}(h(x))$?
It seems to me that in order for orbit equivalence to be an equivalence relation we need $h$ to take the orbits of $\phi$ onto the orbits of $\psi$. Or does this come for free given just the inclusion $\subseteq$?
Considering the case $X=Y=\mathbb{R}$, $\phi_t=\text{id}$, $\psi_t(x)=x+t$, $h=\text{id}$ ("trivial action should not be equivalent to translation action"), and the heuristic that orbit equivalence ought to be an equivalence relation on flows, it's presumably better to interpret "taking orbits to orbits" as the latter (so that the homeomorphism $h$ induces a bijection between orbit spaces $X/\phi\to Y/\psi$).
As far as I know indeed this is the more common definition, and indeed in the book you are citing the remarks right after the definition of orbit equivalence does agree with this definition (which also agrees with the definition in Katok-Hasselblatt).