Local power-map normal form on Riemann surfaces

(A chart expression of a nonconstant map is never locally constant.) Choose a chart ${\phi }_0$ at $x$ and a chart ${\psi }_0$ at $f(x)$ with ${\phi }_0(x)=0$, ${\psi }_0(f(x))=0$, and put $F_0={\psi }_0\circ f\circ {\phi }_0^{-1}$, holomorphic near $0$; if $F_0$ were constant on some neighbourhood of $0$, then $f$ would be constant near $x$, and the set $Z$ of points near which $f$ is locally constant would be nonempty, open by definition, and closed because a limit point of $Z$ forces the chart expression to be constant near the limit point by [F4], so $Z=X$ by connectedness of $X$ and $f$ would be constant, a contradiction.

(The local degree is chart-independent.) Let $F=\psi \circ f\circ {\phi }^{-1}$ and $G={\psi }^{\mathrm{\prime }}\circ f\circ {\phi }^{\mathrm{\prime }-1}$ be chart expressions of $f$ at $x$ with $\phi (x)=0={\phi }^{\mathrm{\prime }}(x)$ and $\psi (f(x))=0={\psi }^{\mathrm{\prime }}(f(x))=0$; writing the transitions as $z=z(u)$ and $w=w(v)$ with $z^{\mathrm{\prime }}(0)\mathrm{\ne }0\mathrm{\ne }{w}^{\mathrm{\prime }}(0)$ (they are injective holomorphic maps of complex domains), one has $G(u)=w(F(z(u)))$, and [F3] gives ${\mathrm{deg}}_{0}G={\mathrm{ord}}_0(w(F(z(u)))-w(0))={\mathrm{ord}}_0(F(z(u)))={\mathrm{ord}}_{0}F={\mathrm{deg}}_{0}F$, because a biholomorphism at $0$ differs from its derivative by a unit and preserves vanishing orders.

(Planar normal form for the chart expression.) By step 1.1 the function $F_0$ is not constant on any neighbourhood of $0$, so there is a disc $D$ around $0$ contained in its domain on which $F_0$ is nonconstant; applying [F2] to $F_0{\mathrm{\mid }}_D$ at $0$ gives a domain $V\subseteq D$, a holomorphic bijection $\theta :V\to \theta (V)$ with $\theta (0)=0$, and $F_0(z)=\theta (z{)}^m$ on $V$ with $m={\mathrm{deg}}_0{F}_0\ge 1$.

(Source coordinates putting the expression in power form.) Let $\phi :=\theta \circ {\phi }_0$ on ${\phi }_0^{-1}(V)$; it is a chart at $x$ with $\phi (x)=0$, since $\theta$ is a biholomorphic map of plane domains by [F5], and with $\psi :={\psi }_0$ the chart expression becomes $\psi \circ f\circ {\phi }^{-1}(u)=F_0({\theta }^{-1}(u))=(\theta ({\theta }^{-1}(u)){)}^m=u^m$ for $u\in \theta (V)$; hence the required coordinates exist with $e=m$.

(Uniqueness of the exponent and conclusion.) If another pair of centred coordinates exhibits $f$ as $u\mapsto u^{e^{\mathrm{\prime }}}$, then $e^{\mathrm{\prime }}={\mathrm{deg}}_{0}G$ for that chart expression $G$, which equals ${\mathrm{deg}}_0{F}_0=m$ by step 1.2, so $e^{\mathrm{\prime }}=e$: the exponent is independent of the charts. By [F3], $m\ge 1$, and $m=1$ exactly when $F_0^{\mathrm{\prime }}(0)\mathrm{\ne }0$, which is exactly the condition that $f$ is a local biholomorphism at $x$ by [F5] and [F1]; this proves the normal form and all the stated properties.