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Decay of correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations and the central limit theorem for the Teichmüller flow on the moduli space of Abelian differentials. (English) Zbl 1100.37002
Let \({\mathcal G}\) be the Rauzy-Veech-Zorich induction map G. Rauzy [Acta Arith. 34, 315–328 (1979; Zbl 0414.28018)] obtained from a permutation of \(1,\dots,m\) and \(\lambda\in\mathbb{R}^m_+\). It is shown that \({\mathcal G}^2\) is exact on invariant sets with respect to the induced Zorich measure \(\nu\) [A. Zorich, Ann. Inst. Fourier 46, 325–370 (1996; Zbl 0853.28007)] and has a speed of mixing estimated by \[ | \varphi \cdot\psi\circ{\mathcal G}^{2n}\,d\nu-\int\varphi \,d\nu\int\psi \,d\nu|\leq C (\varphi,\psi)\exp[-\delta n^{1/6}], \] where \(\delta\) and \(C(\cdot, \cdot)\) are constants only depending on the functions through \(L_p\)- and Hölder-norms. It follows from this estimate that the central limit theorem holds for \(\varphi\) provided it is not an \(L_2\)-coboundary with respect to \({\mathcal G}^2\). Denote by \({\mathcal M}_\kappa\) the moduli space of Riemann surfaces of genus \(g\) endowed with a holomorphic differential of area 1 with singularities of order \(k_i\), \(i\leq \sigma\), and let \(g_t\) denote the Teichmüller flow on this space, invariant with respect to the natural absolutely continuous measure. It is shown that for a (centered) \(L_p\)-function on a connected component of \({\mathcal M}_\kappa,p>2\), which is also Hölder in the sense of Veech and which is not a coboundary in \(L_2\), the central limit theorem holds. These results are stated in Section 1, the remaining part of the paper provides well written proofs of the claims.

MSC:
37A25 Ergodicity, mixing, rates of mixing
37F25 Renormalization of holomorphic dynamical systems
37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
37E05 Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth)
60F05 Central limit and other weak theorems
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
28D05 Measure-preserving transformations
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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