When do iterations of a polynomial converge uniformly on compact subsets of the complex plane

Solution 1:

If the degree of $P$ is at least $2$ then $|p(z)| > c |z|$ for some $c > 1$ and sufficiently large $|z|$, so that the only possible limit function is $\infty$. On the other hand, $P$ has a fixed point $z_0$, so that a limit function also has $z_0$ as a fixed point. It follows that the iterates $P^n$ cannot converge locally uniformly in this case.

If $P(z) = az+b$ with $a \ne 1$ then $P$ has the fixed point $z_0 = b/(1-a)$ and $$ P(z) - z_0 = a(z-z_0) \implies P^n(z) - z_0 = a^n(z-z_0) $$ shows that the iterates converges if and only if $|a| < 1$.

If $P(z) = z+b$ then $P^n(z) = z+nb $ converges locally uniformly to the identity function (if $b=0$), and to $\infty$ otherwise.

So the iterates $P^n$ converge uniformly on compact sets exactly in the following cases:

• $P(z) = az+b$ with $|a| < 1$, $P^n(z) \to z_0 = b/(1-a)$. This includes the case of constant polynomials.

• $P(z)=z$ is the identity function.

• $P(z) = z+b$ with $b \ne 0$ is a translation, $P^n(z) \to \infty$.