In this video (starting at 1'30), Etienne Ghys is refering to the following result wrt to the instability of analytic continuation (on the disk unity):
Notations:
Let $\mathbb{D}$ be the disk unity in $\mathbb{C}$ and $H(\mathbb{D}):=\{f:\mathbb{D}\to\mathbb{D} | f \mathrm{~holomorphic} \}$ the space of holomorphic function with norm $\le 1$.
For $r>0$, let $D_r$ the disk of center $0$ et radius $\tanh(r)$. For $f,g\in H(\mathbb{D})$, define $d_r(f,g)=\sup_{z\le \tan(r)} |f(z)-g(z)|$.
This defines a family of metric on $H(\mathbb{D})$ which is btw compact (Montel theorem).
For $\epsilon>0$, let $N(\epsilon,r)$ the minimal number of balls of radius $\epsilon$ for the $d_r$ metric we need to cover the whole space $H(\mathbb{D})$.
Claim : $N(\epsilon,r)$ grows very very fast, namely : $N(\epsilon,r) \sim C(\epsilon)\exp(\exp(r))$.
This gives a sense of how "instable" analytic continuation is : the fact that we can approximate an holomorphic function only up to $\epsilon$ on $\mathbb{D}_r$ isn't helpful at all to approximate its analytic continuation on a greater disk.
Ghys claims that this observation was (first ?) made by Kolmogorov.
Question : what paper of kolmogorov (if any) is he referencing here ? where can i find a proper proof of this fact ?