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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 $f_n$.

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 $f_n$.

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?

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 $f_n$.

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?

For a concrete example, consider the sequence $f_n = t/n$.This doesn't converge to $0$ because the distance of any non constant function from any constant function is the diameter of $\mathbb P^1$. Nevertheless, does $f_n$ (or a subsequence of it) converge to any rational function? I believe $f_n$ is a Cauchy sequence and if I did the calculation right $\mu(f_n,f_{n+1}) = \tan^{-1}(\frac{n}{n+1}) - \tan^{-1}(\frac{n+1}{n})$ which goes to $0$ as $n \to \infty$. So is there a limit? I suspect not.

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?