## Institute for Mathematical Sciences

## Preprint ims01-08

** J. Hu**
* Earthquake Measure and Cross-ratio Distortion.*

Abstract: Given an orientation-preserving circle homeomorphism $h$, let $(E, \mathcal{L})$ denote a Thurston's left or right earthquake
representation of $h$ and $\sigma $ the transversal shearing
measure induced by $(E, \mathcal{L})$. We first show that the
Thurston norm $||\cdot ||_{Th}$ of $\sigma $ is equivalent to
the cross-ratio distortion norm $||\cdot ||_{cr}$ of $h$, i.e.,
there exists a constant $C>0$ such that
$$\frac{1}{C}||h||_{cr}\le ||\sigma ||_{Th} \le C||h||_{cr}$$
for any $h$. Secondly we introduce two new norms on the
cross-ratio distortion of $h$ and show they are equivalent to
the Thurston norms of the measures of the left and right
earthquakes of $h$. Together it concludes that the Thurston
norms of the measures of the left and right earthquakes of $h$
and the three norms on the cross-ratio distortion of $h$ are
all equivalent. Furthermore, we give necessary and sufficient
conditions for the measures of the left and right earthquakes
to vanish in different orders near the boundary of the
hyperbolic plane. Vanishing conditions on either measure imply
that the homeomorphism $h$ belongs to certain classes of circle
diffeomorphisms classified by Sullivan in \cite{Sullivan}.

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