TITLE: When does the sequence of iterates of a rational function converge?
QUESTION [9 upvotes]: Darsh asks at the 20-questions seminar:
Let $f:P^1 \rightarrow P^1$ be rational function.
Can you say when the sequence $\{ f^n(x)\}_n=\{ x,f(x),f(f(x)),\cdots\} $ converges? What about the sequence of averages $\left\{ \frac{1}{n+1}\sum_{i=0}^n f^i (x)\right\}_n$? Anything else?

REPLY [3 votes]: The question about convergence of the sequence of iterates is simple, as mentioned above: This happens if and only if the sequence converges to a fixed point of $f$. This means that the sequence converges if and only if one of the following holds:

$z$ belongs to the basin of attraction of an attracting or parabolic fixed point, or
$f^n(z)$ is a fixed point for some $z$.

Every rational map has a repelling fixed point in the Julia set. So the set of points in 2., while countable, is dense in the Julia set. 
Now let us turn to the convergence of averages. For points in the Fatou set, this average will always converge. Indeed, there are the following possibilities:

$z$ belongs to the basin of an attracting or parabolic periodic point, and convergence of the averages is trivial.
$z$ belongs to a rotation domain, where the map is analytically conjugate to an irrational rotation. It follows from the properties of rotations that the averages converge (to the integral of the conjugacy over the circle, with respect to Lebesgue measure). 

This leaves us with points in the Julia set. We already saw that there is a dense set of points where the orbit itself converges. On the other hand, the map is topologically chaotic on the Julia set, and it follows easily that there are orbits where the averages do not converge. Indeed, consider an orbit that follows one periodic orbit for a long time, then moves on to another, and switches back again. (I suppose that technically speaking I should justify that there are periodic orbits with different averages - this should not be difficult, but I won't go into details. It should be clear there is no reason to expect otherwise.) Again, once there is one such point, there is a dense set.
So we cannot expect any statement about all points in the Julia set. To make other statements, we should ask about ergodic invariant measures on the Julia set. This is a huge field, and in many cases the existence of such measures is known; then Birkhoff's ergodic theorem implies the convergence of the averages.
The simplest question one may ask is about convergence almost everywhere. It is known since work of Mary Rees in the 1980s that there is a positive measure set in the space of rational functions of any degree (at least $2$) where the map is ergodic with respect to Lebesgue measure on the sphere. So here the averages converge almost everywhere. It is reasonable to conjecture that the set of rational maps of any degree $\geq 2$ where the average converges for almost every point has full measure.
On the other hand, it seems reasonable to expect that there are cases where the Julia set has positive measure, but the averages do not exist almost everywhere. For the logistic family (and one-dimensional Lebesgue measure), I believe that it is known that there are maps without physical measures. I am not sure if it has been done for rational maps.
For quadratic polynomials, Julia sets of positive measure were proved to exist by Buff and Chéritat. It is considered likely that these maps are not ergodic with respect to Lebesgue measure, though as far as I know this has not yet been formally proved. Of course, this does not necessarily mean that the averages do not converge.