This question is from here at the time around 44'.
I wonder why would a sequence of simple closed geodesics converging to a set of disjoint curves (lamination). Intuitively, wouldn't a sequence of simple closed geodesics converge to a curve? Is the lamination (to be convergent to) a single curve too?
In the video starting from 44'45, Minsky said we can take a lamination $\lambda$ from a pseudo-Anosov, take one curve and apply the pseudo-Anosov, and we can get a sequence to converge to a lamination. Did he mean the lamination to be convergent to is $\lambda$ itself? If so, how could a curve be a foliation (I think the $\lambda$ from a pseudo-Anosov means it's an associated foliation of the pseudo-Anosov element)? (Don't they have different dimensions?)
I would be very grateful if you could give an easy example to visualize a sequence of curves converging to a lamination.
There is a simple reason that a sequence of simple closed geodesics $\gamma_i$ might not converge to a single simple closed geodesic: for each $L > 0$, the set of simple closed geodesics of length $\le L$ is a finite, discrete set (in the Hausdorff metric), and therefore $\gamma_i$ converges to another simple closed geodesic $\gamma$ if and only if $\gamma_i=\gamma$ for all sufficiently large $i$.
So if I simply choose the sequence so that $\text{Length}(\gamma_i) \to \infty$ then it does not converge to a single simple closed geodesic.
But on the other hand, fixing a closed hyperbolic surface $S$, the set of closed subsets is compact (in the Hausdorff metric), and so some subsequence of $\gamma_i$ must converge to something.
Here's an example for you to ponder. Suppose that $\gamma$ and $\delta$ are simple closed curves intersecting transversely in a single point. Let $\tau$ be the Dehn twist around $\delta$. Let $\gamma_i$ be the simple closed geodesic that is homotopic to $\tau^i(\gamma)$. Then $\gamma_i$ converges to a lamination with exactly two leaves: one compact leaf, namely $\delta$ itself, and one noncompact leaf that I will denote $\ell$. To describe the appearance of $\ell$, it is a line with two ends, and each end spirals infinitely around and around $\delta$, getting closer and closer to $\delta$.
If you blur your eyes somewhat, it looks like this:
In this image, $\delta$ is the curve that "goes around the hole", and the original $\gamma$ is a curve that "goes through the hole". As $i$ gets larger and larger, $\tau^i(\gamma)$ starts by following $\gamma$ into the hole, and around the back, and then back to the front on the leftmost contour of the surface, and then spins $i$ times around $\delta$ before closing up.