DEVIATION OF ERGODIC AVERAGES 3 If CLT and LIL hold , then the stochastic process { f ◦

Introduction 1. The Kontsevich-Zorich cocycle 2. Variational formulas 3. A lower bound for the second Lyapunov exponent 4. The determinant locus 5. The Kontsevich-Zorich formula revisited and other formulas for the Lyapunov exponents 6. Basic currents for measured foliations 6.1. Basic currents and invariant distributions 6.2. Weighted Sobolev spaces of currents 6.3. A geometric estimate of the Poincaré constant 7. The structure of the space of basic currents of finite order 7.1. Basic currents with non-vanishing cohomology class 7.2. Basic currents with vanishing cohomology class 8. The non-uniform hyperbolicity of the Kontsevich-Zorich cocycle and an application to currents 8.1. The stable and unstable sub-bundles of the Kontsevich-Zorich cocycle as bundles of basic currents 8.2. The Kontsevich-Zorich cocycle is non-uniformly hyperbolic 8.3. The Oseledec's theorem for the bundle of closed currents of order 1 * This paper rests on the work of several mathematicians, H. Masur, J. Smillie, W. Veech and A. Zorich among them, and it was strongly inspired by the work of A. Zorich and M. Kontsevich-A. Zorich. I wish to thank particularly A. Eskin, J. Smillie and A. Zorich for their interest during the slow progress of this work and for discussing with me their ideas on the subject. I am grateful to R. Gunning for his encouragement and his enthusiasm during several discussions on the content of Section 4, which improved very much my understanding of moduli spaces. I am also very grateful to J. Mather for his patience in listening to a number of sometimes very confused and tentative presentations of these results and for his help in clarifying my ideas with numerous questions and suggestions.