Finite jet determination of CR mappings

We prove the following finite jet determination result for CR mappings: Given a smooth generic submanifold M⊂CN, N⩾2, that is essentially finite and of finite type at each of its points, for every point p∈M there exists an integer ℓp, depending upper-semicontinuously on p, such that for every smooth generic submanifold M′⊂CN of the same dimension as M, if are two germs of smooth finite CR mappings with the same ℓp jet at p, then necessarily for all positive integers k. In the hypersurface case, this result provides several new unique jet determination properties for holomorphic mappings at the boundary in the real-analytic case; in particular, it provides the finite jet determination of arbitrary real-analytic CR mappings between real-analytic hypersurfaces in CN of D'Angelo finite type. It also yields a new boundary version of H. Cartan's uniqueness theorem: if Ω,Ω′⊂CN are two bounded domains with smooth real-analytic boundary, then there exists an integer k, depending only on the boundary ∂Ω, such that if are two proper holomorphic mappings extending smoothly up to ∂Ω near some point p∈∂Ω and agreeing up to order k at p, then necessarily H1=H2.