Earthquakes in the length-spectrum Teichm\"uller spaces

Abstract

Let $X_0$ be a complete hyperbolic surface of infinite type that has a geodesic pants decomposition with cuff lengths bounded above. The length spectrum Teichm\"uller space $T_{ls}(X_0)$ consists of homotopy classes of hyperbolic metrics on $X_0$ such that the ratios of the corresponding simple closed geodesic for the hyperbolic metric on $X_0$ and for the other hyperbolic metric are bounded from the below away from 0 and from the above away from $\infty$ (cf. \cite{ALPS}). This paper studies earthquakes in the length spectrum Teichm\"uller space $T_{ls}(X_0)$. We find a necessary condition and several sufficient conditions on earthquake measure $\mu$ such that the corresponding earthquake $E^{\mu}$ describes the hyperbolic metric on $X_0$ which is in the length spectrum Teichm\"uller space. Moreover, we give examples of earthquake paths $t\mapsto E^{t\mu}$, for $t\geq 0$, such that $E^{t\mu}\in T_{ls}(X_0)$ for $0\leq t<t_0$, $E^{t_0\mu}\notin T_{ls}(X_0)$ and $E^{t\mu}\in T_{ls}(X_0)$ for $t>t_0$.