Empirical invariance principle for ergodic torus automorphisms; genericity.

International audienceWe consider the dynamical system given by an algebraic ergodic automorphism $T$ on a torus. We study a Central Limit Theorem for the empirical process associated to the stationary process $(f\circ T^i)_{i\in\N}$, where $f$ is a given $\R$-valued function. We give a sufficient condition on $f$ for this Central Limit Theorem to hold. In a second part, we prove that the distribution function of a Morse function is continuously differentiable if the dimension of the manifold is at least 3 and Hölder continuous if the dimension is 1 or 2. As a consequence, the Morse functions satisfy the empirical invariance principle, which is therefore generically verified