TITLE: Conditions for invariance under flow.
QUESTION [1 upvotes]: I am beginning to study dynamical systems.  We are given $U \subset \mathbb{R}^n$ open, a vector field $f: U \to \mathbb{R}^n$, and an associated evolution operator for fixed  $t \in \mathbb{R}$ $\Phi_t : U \to \mathbb{R}^n$ given by $\Phi_t(x_0) = x(t)$, where $x$ is the solution to the ODE 
\begin{equation}
\dot x = f(x) \\
x(0) = x_0.
\end{equation}
We say that $A \subset U$ is invariant if $\Phi_t(A) = A \ \forall t \in \mathbb{R}$.  On the other hand, we have definitions for positive (negative) invariance in which $\Phi_t(A) \subset A \ \forall t \geq 0 \ ( \leq 0)$.  I am wondering what added conditions (in addition to both positive and negative invariance) would lead to invariance.

REPLY [1 votes]: It follows from the definitions (those that you detail) that a set $A$ is invariant if and only if it is positively and negatively invariant. The reason is that since $\Phi_t$ is a flow, you have $(\Phi_t)^{-1}=\Phi_{-t}$ and so a set is negatively invariant if and only if $\Phi_t(A)\supset A$ for all $t\ge0$.
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