Dominated perturbation preserves winding number
= Dominated perturbation preserves winding number
If closed continuous paths $\gamma,\phi$ satisfy $|\gamma(t)|>|\phi(t)|$ for every $t$, then $H(s,t)=\gamma(t)+s\phi(t)$ never vanishes. It is therefore a homotopy through closed paths in $\mathbb C\setminus\{0\}$, and
$$
\operatorname{wind}(\gamma+\phi,0)=\operatorname{wind}(\gamma,0).
$$