The local deformations of a submanifold under the effect of a smooth dynamical system are studied with the help of Oseledets' multiplicative ergodic theorem. Equivalence es of submanifolds, called jets, are introduced in order to describe these local deformations. They identify submanifolds having the same approximations up to some order at a given point. For every order $k$, a condition on the Lyapunov exponents of the dynamical system is established insuring the convergence of the $k$-jet of a submanifold evolving under the action of the dynamical system. This condition can be satisfied even by stable dynamical systems. The limit is a $k$-jet which is invariant by the dynamical system.