Fernando Cardoso, Georgi Popov Quasimodes with Exponentially Small Errors Associated with Elliptic Periodic Rays (106K, LaTeX 2e) ABSTRACT. The aim of this paper is to construct compactly supported Gevrey quasimodes with exponentially small discrepancy for the Laplace operator with Dirichlet boundary conditions in a domain $X$ with analytic boundary. The quasimodes are associated with a non-degenerate elliptic closed broken geodesic $\gamma$ in $X$. We find a Cantor family $\Lambda$ of invariant tori of the corresponding Poincar map which is Gevrey smooth with respect to the transversal variables (the frequencies). Quantizing the Gevrey family $\Lambda$, we construct quasimodes with exponentially small discrepancy. As a consequence, we obtain a large amount of resonances exponentially close to the real axis for suitable compact obstacles. This is a new version of mparc 02-181, where the beginning of the proof of Proposition 6.1 is revised.