A conjecture by Shub states that the asymptotic exponential growth rate of the number of periodic points of a C¹ map f : M → M is bounded from below by an algebraic topological quantity: the exponential growth rate of Lefschetz numbers L(f ⁿ). In particular, if M is a sphere the conjecture states that # Fix(f ⁿ) grows asymptotically at least as d ⁿ, where d denotes the degree of f. The conjecture is wide open in general, even in S². In the talk, we will review some results in very particular cases: maps on S³ leaving invariant a circle and maps preserving a singular foliation on a closed surface. This is joint work with H. Barge, A. Moreno, J. Sanchez-Gabites (Madrid).