## Institute for Mathematical Sciences

## Preprint ims01-07

** F. P. Gardiner, J. Hu and N. Lakic**
* Earthquake Curves.*

Abstract: The first two parts of this paper concern homeomorphisms of the circle, their associated earthquakes, earthquake laminations
and shearing measures. We prove a finite version of Thurston's
earthquake theorem \cite{Thurston4} and show that it implies
the existence of an earthquake realizing any homeomorphism.
Our approach gives an effective way to compute the lamination.
We then show how to recover the earthquake from the measure,
and give examples to show that locally finite measures on given
laminations do not necessarily yield homeomorphisms. One of
them also presents an example of a lamination ${\cal L}$ and
a measure $\sigma $ such that the corresponding mapping
$h_{\sigma}$ is not a homeomorphism of the circle but
$h_{2\sigma}$ is. The third part of the paper concerns the
dependence between the norm $||\sigma ||_{Th}$ of a measure
$\sigma$ and the norm $||h||_{cr}$ of its corresponding
quasisymmetric circle homeomorphism $h_{\sigma}$. We first
show that $||\sigma ||_{Th}$ is bounded by a constant multiple
of $||h||_{cr}$. Conversely, we show for any $C_0>0$, there
exists a constant $C>0$ depending on $C_0$ such that for any
$\sigma $, if $||\sigma ||_{Th}\le C_0$ then
$||h||_{cr}\le C||\sigma ||_{Th}$. The fourth part of the paper
concerns the differentiability of the earthquake curve
$h_{t\sigma }, t\ge 0,$ on the parameter $t$. We show that for
any locally finite measure $\sigma $, $h_{t\sigma }$ satisfies
the nonautonomous ordinary differential equation
$$\frac{d}{dt} h_{t\sigma}(x)=V_t(h_{t\sigma}(x)), \ t\ge 0,$$
at any point $x$ on the boundary of a stratum of the lamination
corresponding to the measure $\sigma.$ We also show that if the
norm of $\sigma $ is finite, then the differential equation
extends to every point $x$ on the boundary circle, and the
solution to the differential equation an initial condition is
unique. The fifth and last part of the paper concerns
correspondence of regularity conditions on the measure $\sigma,$
on its corresponding mapping $h_{\sigma},$ and on the tangent
vector $$V= V_0 = \frac{d}{dt}\big|_{t=0} h_{t\sigma}.$$ We give
equivalent conditions on $\sigma, h_{\sigma}$ and $V$ that
correspond to $h_{\sigma }$ being in {\em Diff}$^ {\ 1+\alpha}$
classes, where $0\le \alpha <1$.

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