Def: Let $X\subset Y$ with $\phi_t:Y\to Y$ a semiflow. $X$ is called a Poincaré section of $\phi_t$ if the first return time $\tau(x):=$inf$\{t>0:\phi_t(x)\in X\}\in\mathbb{R}^+$ for every $x\in X$, with the convention inf$\{\}=+\infty$.
We then define a discrete dynamical system from the map $f:X\to X$ by $f(x)=\phi_{\tau(x)}(x)$.
Questions:
1.) I take this definition of a Poincaré section to mean that every point of $X$ eventually returns to $X$ (infinitely often) and the flow must not be parallel along $X$ (otherwise, $\tau(x)=0\notin\mathbb{R}^+$ for every $x\in X$). In another text I have (Strogatz), it says that if $Y$ has dimension $n$, then the Poincaré section $X$ has dimension $n-1$. I'm not sure why this is the true though. Does this follow from the definition above, or is it an extra condition? Why, for example, couldn't you have a line in $\mathbb{R}^3$ be a Poincaré section?
2.) I'm not sure why $\phi_{\tau(x)}(x)\in X$. $\tau(x)$ is the greatest lower bound of all times $t$ such that $\phi_t(x)\in X$, but why does this mean that $\tau(x)$ is also such a time? It would seem to me that $\tau$ would have to be defined in terms of a strict minimum instead of an infimum.