CONICET | Buscador de Institutos y Recursos Humanos

For a class of non-invertible hyperbolic dynamical systems have been introduced probability states describing the distribution of preimages. These states converge to a measure of maximal entropy which is also physical. Non-invertible dynamical systems present many difficulties with respect to the invertible ones, for instance the Birkhoff ergodic theorem cannot be applied directly for the inverse of the dynamical map since there are several preimages of a point and the statistical sums may have different behaviors. Other trouble in the setting of hyperbolic endomorphisms comes from the local unstable manifolds, since they depend on the whole preimages of a point they may intersect each other and also by a point there may pass infinite unstable manifolds. So that the unstable manifolds do not form a foliation. Nevertheless, for non-invertible systems the notion of hyperbolicity can be given. Here we consider Anosov endomorphisms and show that for systems semi-conjugated to them can be constructed measures describing consecutive preimages and converging to maximal entropy measures. We also analyze when the measures obtained are inverse SRB-measure.