I am confused by the proof of Lemma 3.6 in Kerckhoff's "The Nielsen Realization Problem." Let $\gamma \in S$ ($S$ is the set of isotopy classes of simple closed curves) and $\bar{\gamma}(t)$ be the image of $\gamma(0)$ (understood as the lift of some representative of the class; we only care about its endpoints as these will be the same for every element in the class by compactness so the particular representative doesn't matter) under the earthquake a distance $t$ along $\mu$ a measured lamination in the universal cover. Then for $x$ an isolated point in the intersection of $\gamma$ and $\mu$, the endpoints of $\bar{\gamma}(t)$ are strictly to the left the endpoints of $\gamma(0)$ when viewed from $x$ (which we keep fixed under all quakes.)
The proof in the discrete case is straightforward. However, I am confused by the general case. Let $l$ denote the lead of $\mu$ that $x$ is in. I believe that for any geodesic in a compact neighborhood $N$ of $l$ (in the space of all geodesics with the topology of an open mobius band) that there is some constant $C$ such that a quake along any element $\lambda$ of $N$ decreases $\chi$ by at least $Ct$ for any $t$ less than some number $\tau_0$. This should follow by a straightforward enough calculation of the movement of the endpoints under $l$ when taken as the line from $0$ to $\infty$, taylor expanding, and then applying compactness (though Kerckhoff actually changes from saying $\chi$ is decreasing to increasing, which confuses me.) It's also not clear to me what the significance of the bounded ratios are.
Then comes the issue of going from these calculations along any given leaf in $N$ to shearing a total distance $t$ about $\mu$ in $N^\circ$ the union of the supports of all the elements of $N$. I presume the argument being made is something like that if we take a sequence $(\varphi_n,s_n)$ in $S\times \mathbb{R}_{>0}$ that converges to $\mu$ then these will intersect the interval $I$ of $\gamma$ lying in $N^\circ$ in some number of points so that it subdivides the interval into subintervals $I_{-k_n},...,I_0,...,I_{j_n}$. We take $I_0$ to be the interval containing $x$ where the quake acts simply as the identity. For large enough $n$ as the $(\varphi_n,s_n)$ converge to $\mu$ we know that along this interval we should get that the difference each of the average angle and total mass between $\mu$ and $(\varphi_n,s_n)$ is bounded by $\epsilon$ for any $n$ sufficiently large. The idea would be then to use lemma 2.2 in the paper then to give a bound on the earthquake difference between $(\varphi_n,s_n)$ and $\mu$ at each intersection point, but these details don't seem to be working out for me. How do I do these details? Or is this the completely wrong way of going about it?
Thanks much for any help.