Let $\gamma$ be a periodic orbit of a vector field $X$. Through a point $x_{0} \in \gamma$ we consider a section $\Sigma$ transversal to the field $X$ and V a small neighbourhood which contains $x_{0}$.
Consider $P_{X}: V \subset \Sigma \rightarrow \Sigma$ the Poincaré map, which to each point $x \in V$ associates $P(x)$, the first point where the orbit of $x$ returns to intersect $\Sigma$.
If $Y$ is a vector field sufficiently near to vector field $X$, my questions are:
1) Is $\Sigma$ transversal to $Y$?
2) Is true that the orbit of $Y$ through each point of $V$ still returns to intersect $\Sigma$?
In the first question if $\Sigma$ were not transversal to $Y$ (in a point p), then $Y(p) \in T_{p}\Sigma$, since $Y$ is close to the $X$ we have $X(p) \in T_{p}\Sigma$, which contradicts transversality. (Is my argument correct? Can I take the topology over $T\Sigma$?)
Is the second question related to the structural stability? I don't know if the statement is true.
I'm self studying Dynamical Systems, so I'm having some trouble to make some stuffs precise.
Thanks in advance.