Let $\Gamma $ be a pseudogroup of local holomorphic transformations of $\mathbb C^n$ fixing zero. We study the dynamics of $\Gamma $. We show that if $\Gamma $ contains two elements whose 2-jets are in “general position” and sufficiently near the identity, then : 1) $\Gamma $ acts minimally on the bundle of infinite-order jets on some pointed neighborhood $\mathcal B$ of $0$ (that is to say : for any $z_0,z_1\in \mathcal B$ and any germ $\phi :z_0\to z_1$ of biholomorphism, there exists a sequence $\gamma _n\in \Gamma $ which converges to $\phi $ uniformly on some neighborhood of $z_0$). 2) $\Gamma $ preserves no geometric structure near $0$ (this is a trivial consequence of 1). 3) For any holomorphic pseudogroup topologically conjugate to $\Gamma $, the germ of conjugacy at $0$ is either holomorphic or antiholomorphic. The main feature of the proof is to attach to any pseudogroup $\Gamma $ a sheaf $\mathfrak {g}_\Gamma $ of Lie algebrae on $\mathbb C^n$ such that $\Gamma $ is “dense” in $\mathfrak {g}_\Gamma $ in a natural sense. Then we prove that under some natural assumption on $\Gamma $, $\mathfrak {g}_\Gamma (U)$ must be the sheaf of all holomorphic vector fields for any $U$ open in $\mathcal B$, where $\mathcal B$ is the (open) complementary of $0$ in its basin of attraction.