Let f be a smooth self-map of a compact manifold. Shub and Sullivan conjectured in seventies that under the assumption that f has unbounded Lefschetz numbers of iterations, the growth of the number of periodic points is at least (asymptotically) exponential. We "weakly" disprove this conjecture for simply connected manifolds, by showing that in the homotopy class of f there is a C¹ map with less than r periodic points, up to any given fixed period r.