Laura DeMarco gave a beautiful talk at the Joint Meetings about her work (with Matt Baker) on the postcritically finite locus in the moduli space of polynomial dynamical systems. (Here are her slides for a similar talk.)
To say only a tiny bit about what that means: The dynamical systems in question are those coming from a polynomial map f: C -> C. Like, say,
for some complex number c. The set of c such that the forward orbit of 0 stays bounded is the Mandelbrot set — you know, this guy:
One way an orbit to be bounded is for it to be eventually periodic; when z^2 + c has this property, we say it is postcritically finite, or PCF. More generally, the postcritically finite polynomials are those whose critical points all have finite forward orbits. Number theorists like these because they’re the ones whose inverse iterates generate big interesting number fields with finite ramification. But that’s not what I want to mention now. DeMarco mentioned the very interesting fact (sorry, I don’t know who proved this or whether I’m stating it correctly) that as you range over PCFs with longer and longer period, the set of PCFs approaches the uniform distribution (with respect to a standard measure called bifurcation measure) on the boundary of M.
The PCFs, DeMarco told us, should be thought of as special points in the space of all polynomials — in this simple case of quadratics, the PCFs are special points in the complex plane. They’re kind of like CM points on the j-line, or torsion points on an abelian variety. The main thrust of DeMarco’s work with Baker concerns dynamical analogues of the Andre-Oort conjecture, which aims to classify those subvarieties of the moduli space of dynamical systems which contain a (Zariski-)dense set of PCF points. Their striking results demonstrate the unexpected ways in which arithmetic dynamics and complex dynamics have truly started to engage with each other, after a fairly long period of separate development.
But that’s also not what I want to mention now; I just wanted to record a simple thought that a number theorist might have while watching DeMarco’s talk. (Warning: as usual with math posts, this is not thought through carefully.)
The PCF points are perhaps sort of like torsion points in C^*, which is to say roots of unity; and just as PCFs of larger and larger period converge to uniform distribution on the Julia set, roots of unity of larger and larger order converge to uniform distribution on the unit circle. Equivalently: rational numbers of bounded denominator look roughly uniformly distributed on R/Z.
But there are lots of more refined questions one can ask about the way in which the rational numbers sit densely in R/Z. For example, one can ask about Diophantine approximation; given an irrational point alpha on R/Z, we know there are infinitely many “pretty good” rational approximants to alpha; fractions p/q such that
Are there theorems guaranteeing that any point x on the boundary of the Mandelbrot set has infinitely many PCFs which are “pretty good approximations” to x in the above sense?
What is the most badly approximable point on the Mandelbrot boundary — i.e what is the “golden dynamical system” that plays the role of (1/2)(1+sqrt(5))?
Does x have a canonical sequence of PCF approximants which play the role of continued fraction convergents?
Quomodocumque 
I’m no expert, but I think you put your finger on one of the hardest open questions in the theory of the Mandelbrot set. First, the PCF maps are naturally in the Mandelbrot set; I think you said “Julia set” several times when you meant “Mandelbrot set”, right? You probably know this, but the PCF maps are in fact in natural bijection with Q/Z. The Mandelbrot set is connected and simply connected, so there’s a Riemann map from the complement of the unit disk to the complement of the Mandelbrot set. It’s a hard and important open question whether the Mandelbrot set is locally connected; this is, I’m told, equivalent to whether the Riemann map above extends continuously. The PCF points are (more or less) the image of the roots of unity on the circle. (To get the PCF dynamics, you take a root of unity and look at its behaviour under repeated squaring.) Points in R/Z can be well-approximated by points in Q/Z, so the continuity of the map should boil down to checking whether that good approximation still holds once you push forward to the boundary of the Mandelbrot set.
Nice post, Jordan. I agree with Dylan’s comment that your questions seem to be intimately linked with whether the Mandelbrot set is locally connected. A couple of small technical remarks: only the PCF quadratic maps for which 0 is strictly preperiodic (i.e. not periodic) belong to the boundary of the Mandelbrot set — they are called Misiurewicz points. (The Wikipedia page for ‘Misiurewicz point’ says that a more appropriate term would be Misiurewicz-Thurston points — sorry, Dylan.) The PCF quadratic maps for which 0 is periodic (called hyperbolic centers) belong to the interior of the Mandelbrot set and are discrete there. Your approximation question should therefore probably be restricted to points in the *boundary* of the Mandelbrot set.
The results which Dylan mentioned are due to Douady-Hubbard and are described in slightly more precise terms in Joe Silverman’s book “The Arithmetic of Dynamical Systems”, see Theorems 4.23 and 4.25. (You can see the relevant pages on Google books.) There are also various survey articles on complex dynamics which describe these results.
Silly technical comment: the dynamical Andre-Oort conjecture aims to classify those subvarieties of the moduli space of dynamical systems which contain *a Zariski dense* set of PCF points, not infinitely many. Of course Laura and I only prove something non-trivial about *curves* in the moduli space, so for all practical purposes your description is correct.