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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

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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