Monodromy

Remarks

Continuing $\Phi$ along $\gamma_1$ and then along $\gamma_2$ gives $\Phi M_{\gamma_2} M_{\gamma_1}$, so $\gamma \mapsto M_\gamma$ reverses products; taking inverses makes $\rho$ a representation of $\pi_1$ in the general linear group. Changing $\Phi$ to $\Phi C$ with $C$ invertible conjugates every $M_\gamma$ by $C$, so the representation is determined up to conjugation. The same definition applies to a scalar equation of order $n$, using a fundamental system of solutions.

Example: for $y' = \frac{a}{z} y$ on $U = \C \setminus \set{0}$ with $a \in \C$, take $\Phi(z) = z^a = e^{a \log z}$ near $z_0 = 1$. Continuation once counterclockwise around $0$ adds $2 \pi i$ to $\log z$, so $M_\gamma = e^{2 \pi i a}$. The monodromy is trivial exactly when $a \in \Z$.

Non-example: for $y' = y$ on $\C$, the monodromy is trivial. The solution $e^z$ is single-valued and $U = \C$ is simply connected, so every monodromy matrix is $1$.