## 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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