What is the meaning of $DX_p$ for $X$ a vector field on a manifold? This is taken from Palis, Geometric Theory of Dynamical Systems, p.55: ![enter image description here]( Here $X$ is a $C^r$ vector field on $M$. What does the notation $DX_p$ mean?

This is often called the _intrinsic_ derivative. (This makes sense, more generally, for the section of any vector bundle at a zero.) It is well-defined at a zero of $X$. Think in local coordinates of $X$ as a map from $\Bbb R^n$ to $\Bbb R^n$, and compute its derivative at $0$ (corresponding to $P$). You can check that you get a well-defined map $T_PM\to T_PM$ precisely because $X(P)=0$.