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The Teichmüller geodesic flow. (English) Zbl 0658.32016
This paper is concerned with the ergodic dynamics of the Teichmüller geodesic flow. Let $$T_{g,n}$$ be the Teichmüller space of closed Riemann surfaces of genus g with n punctures where $$3g-3+n>0.$$ The complex cotangent bundle is denoted by $$Q_{g,n}$$, and $$Q^ 1_{g,n}$$ denotes the unit cotangent bundle. Then the moduli space can be expressed $$Q^*_{g,n}=Q^ 1_{g,n}/Mod (g,n)$$ where Mod (g,n) is the modular group of $$T_{g,n}$$. The group $$G=SL(2,R)$$ acts continuously on $$Q^*$$. If we use $$T^ t$$ to denote the transformation determined by the matrix $$diag(e^{t/2},e^{-t/2})$$, then the main result of this paper is the following: There exists a unique absolutely continuous probability measure $$\mu$$ on $$Q^*_{g,n}$$ such that $$T^ t_{\mu}=\mu$$, $$t\in R$$. Moreover, $$(Q^*_{g,n},T^ t)$$ is “measurably Anosov”; $$(Q^*_{g,n},T^ t)$$ is a K-system; the metric entropy $$h_{\mu}(T)=3g-3+n;\quad \liminf_{t\to \infty} t^{-1} \log \pi (g,n,t)\geq 3g-3+n,$$ where $$\pi$$ (g,n,t) is the number of periodic orbits of $$(T^ s)$$ of least period $$s\leq t$$.
Reviewer: Z.Li

##### MSC:
 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
##### Keywords:
Teichmüller spaces; ergodic dynamics
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