Orbits of points under pseudo-Anosov diffeomorphisms I have no intuition about orbits of pseudo-Anosov diffeomorphisms $\phi$ of closed surfaces $S$ of genus $>1.$
I understand that there are infinitely countably many periodic points, correct?
What can be said about accumulation points of orbits of "generic" points? 
Let $M_z=\{x\in S: \lim_n \phi^n(x)=z\}$. Then $z$ is a fixed point of $\phi.$
How "large" are such sets? Do they have positive measures?
How about the set of wondering points?
Do sets $M_z$ or the set of wondering points exhibit fractal-like properties?
 A: These are not topics that are special to the study of pseudo-Anosov diffeomorphisms, instead they are general to a wide class of dynamical systems known as hyperbolic dynamical systems, and can be analyzed using symbolic dynamics. Other examples with the same dynamical properties include the Smale horseshoe, Bernoulli shifts, Anosov diffeomorphisms, and so on.
So, for example, all of these examples have a Markov partition with a transition matrix $M$, and you can literally count the number of points of period $n$ (up to a bounded additive error): it is the trace of the matrix $M^n$. And since some power of $M$ has positive integer entries, you can then easily deduce that as $n \to \infty$ the number of points of period $n$ goes to infinity. So yes, there are a countable infinity of periodic points.
As for "generic" points, you'll have to say what definition of generic you are using, but in most reasonable definitions you'll find that generic points have dense orbits.
As for sets $M_z$ where $z$ is a fixed point, that's known as the stable manifold of $z$, and it is dense but of measure zero. It's not fractal at all, it's literally the image of a dense, one-to-one, continuous map $\mathbb R \to S$ (except when $z$ is a singularity, then it's a bit more complicated).
You should be able to read about many of these things in the book "Thurston's work on surfaces" by Fathi, Laudenback and Poenaru, but I also suggest learning about hyperbolic dynamics and symbolic dynamics.
