Memoirs of the American Mathematical Society 2009; 128 pp; softcover Volume: 200 ISBN10: 0821843826 ISBN13: 9780821843826 List Price: US$67 Individual Members: US$40.20 Institutional Members: US$53.60 Order Code: MEMO/200/940
 The authors consider doublyperiodic travelling waves at the surface of an infinitely deep perfect fluid, only subjected to gravity \(g\) and resulting from the nonlinear interaction of two simply periodic travelling waves making an angle \(2\theta\) between them. Denoting by \(\mu =gL/c^{2}\) the dimensionless bifurcation parameter ( \(L\) is the wave length along the direction of the travelling wave and \(c\) is the velocity of the wave), bifurcation occurs for \(\mu = \cos \theta\). For nonresonant cases, we first give a large family of formal threedimensional gravity travelling waves, in the form of an expansion in powers of the amplitudes of two basic travelling waves. "Diamond waves" are a particular case of such waves, when they are symmetric with respect to the direction of propagation. The main object of the paper is the proof of existence of such symmetric waves having the above mentioned asymptotic expansion. Due to the occurence of small divisors, the main difficulty is the inversion of the linearized operator at a non trivial point, for applying the Nash Moser theorem. This operator is the sum of a second order differentiation along a certain direction, and an integrodifferential operator of first order, both depending periodically of coordinates. It is shown that for almost all angles \(\theta\), the 3dimensional travelling waves bifurcate for a set of "good" values of the bifurcation parameter having asymptotically a full measure near the bifurcation curve in the parameter plane \((\theta ,\mu ).\) Table of Contents  Introduction
 Formal solutions
 Linearized operator
 Small divisors. Estimate of \(\mathfrak{L}\) resolvent
 Descent methodinversion of the linearized operator
 Nonlinear Problem. Proof of Theorem 1.3
 Appendix A. Analytical study of \(\mathcal{G}_\eta\)
 Appendix B. Formal computation of 3dimensional waves
 Appendix C. Proof of Lemma 3.6
 Appendix D. Proofs of Lemmas 3.7 and 3.8
 Appendix E. Distribution of numbers \(\{\omega_{0}n^{2}\}\)
 Appendix F. Pseudodifferential operators
 Appendix G. DirichletNeumann operator
 Appendix H. Proof of Lemma 5.8
 Appendix I. Fluid particles dynamics
 Bibliography
