Abstract: Supposant connue l'existence des representations galoisiennes associees aux formes modulaires de Siegel (elle ne l'est qu'en genre ≤ 2 pour le moment), on etudie la cohomologie des varietes de Siegel a coefficients entiers p-adiques localisee en un ideal maximal non-Eisenstein de l'algebre de Hecke, lorsque p est grand par rapport au poids du systeme de coefficients. Plus precisement, on montre qu'elle est sans torsion, concentree en degre median, et qu'elle coincide avec la cohomologie d'intersection et avec la cohomologie interieure. On utilise pour cela la theorie de Hodge p-adique et le complexe BGG dual modulo p qui calcule les poids de Hodge-Tate de la reduction modulo p de cette cohomologie. On applique ce resultat a la construction de familles de Hida p-ordinaires pour les groupes symplectiques et a l'ebauche de la construction d'un systeme de Taylor-Wiles pour ces groupes.

Abstract: — In this paper we shall consider real polynomials with one (possibly degenerate) non-escaping critical (folding) point. Necessary and sufficient conditions are given for the total disconnectedness of the Julia set of such polynomials. Also we prove that the Julia sets of such polynomials do not carry invariant linefields. In the real case, this generalises the results by Branner and Hubbard for cubic polynomials and by McMullen on absence of invariant linefields.

Abstract: We prove the existence of Sinai-Ruelle-Bowen measures for a class of C2 self-mappings of a rectangle with unbounded derivatives. The results can be regarded as a generalization of a well-known one dimensional Folklore Theorem on the existence of absolutely continuous invariant measures. In an earlier paper [8] analogous results were stated and the proofs were sketched for the case of invertible systems. Here we give complete proofs in the more general case of noninvertible systems, and, in particular, develop the theory of stable and unstable manifolds for maps with unbounded derivatives.