Earthquakes in the length-spectrum Teichmüller spaces

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 Teichmüller 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 Teichmüller space $T_{ls}(X_0)$. We find a necessary condition and several sufficient conditions on earthquake measure $μ$ such that the corresponding earthquake $E^μ$ describes the hyperbolic metric on $X_0$ which is in the length spectrum Teichmüller space. Moreover, we give examples of earthquake paths $t\mapsto E^{tμ}$, for $t\geq 0$, such that $E^{tμ}\in T_{ls}(X_0)$ for $0\leq t<t_0$, $E^{t_0μ}\notin T_{ls}(X_0)$ and $E^{tμ}\in T_{ls}(X_0)$ for $t>t_0$.

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