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Let us consider the projective line over $\mathbb C$ equipped with a nice metric $\eta$ (like the Fubini-Study metric). We can define a metric $\mu$ on rational functions $f: \mathbb P^1 \to \mathbb P^1$ by: $$\mu(f,g) = \sup_{x \in \mathbb P^1}\eta(f(x),g(x)).$$ What can we say about $\mathbb C(t)$ as a metric space with respect to $\mu$? Is it complete? Can we classify the convergent sequences? Is this sort of construction studied more generally? Are nice conditions known for which the resulting metric space on endomorphisms is complete?

As a concrete example, if $f_n(x) = a_nx$ for $a_n \in \mathbb R$, then I believe: $$\mu(f_n,f_m) = \frac{\pi}{2} - 2\tan^{-1}\left(\sqrt{\frac{a_n}{a_m}}\right).$$ In particular, if $a_n$ converges to a non zero limit, then so does $f_n$.

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The space of continuous functions from a topological space to a complete one is always complete. The analyticity of a function in a closed condition in this topology (a uniform limit of holomorphic functions is holomorphic, and we can work locally if needed) so the space of holomorphic functions from the projective plane to itself is complete. This is precisely $\mathbb C(t)\cup\lbrace\infty\rbrace$, for $\infty$ the constant function equal to $\infty$.

I believe this argument shows that the space of holomorphic functions between two complex manifolds with a smooth metric on the target is complete for the uniform topology if and only if the target is complete.

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    $\begingroup$ Since the degree of a rational function is a continuous function on this space, it splits into infinitely many components, one for each degree. $\endgroup$
    – Alexandre Eremenko
    Commented Jul 18, 2021 at 13:10
  • $\begingroup$ @AlexandreEremenko Nice! I assume you are still talking about the Riemann sphere. I think these are connected, so we know the connected components of this space. Drifting away from the OP's question, I would be interested in more results about its homotopy type. I convinced myself, perhaps wrongly, that the degree 1 (and -1) mappings have the homotopy type of $\mathbb{RP}^3$. Of course for the degree 0 we have a copy of $\mathbb{CP}^1$. $\endgroup$
    – Pierre PC
    Commented Jul 18, 2021 at 17:38
  • $\begingroup$ yes, I am talking about the whole Riemann sphere. Yes, they are connected, though I have no ready reference. Homotopy type I do not know, but I suppose this must be known. $\endgroup$
    – Alexandre Eremenko
    Commented Jul 18, 2021 at 19:31
  • $\begingroup$ Since for every open discrete map $f$ between two spheres, there exists a homeomorphism $\phi$ such that $f\circ\phi$ is a rational function, the question about homotopy type is purely topological, not analytic. Also notice that the space is not compact, even for degree $1$. Also, notice that the space is not compact, even for degree $1$: the sequence f_n(z)=z/n$ has no limit points. $\endgroup$
    – Alexandre Eremenko
    Commented Jul 18, 2021 at 19:44
  • $\begingroup$ In degree 1, the space is $\mathrm{PGL}_2(\mathbb C)$, which I believe should be diffeomorphic to $\mathrm{SO}_3(\mathbb R)\times\mathbb R$, which indeed isn't compact. I am not sure I follow your argument for the homotopy type, though. Are you saying your argument gives a retract from the space of continuous maps to that of holomorphic ones? $\endgroup$
    – Pierre PC
    Commented Jul 18, 2021 at 20:07

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