A criterion for zero averages and full support of ergodic measures

Consider a homeomorphism f defined on a compact metric space X and a continuous map φ: X → ℝ. We provide an abstract criterion, called control at any scale with a long sparse tail for a point x ∈ X and the map φ, which guarantees that any weak* limit measure µ of the Birkhoff average of Dirac measures (Formula Presented) is such that µ-almost every point y has a dense orbit in X and the Birkhoff average of φ along the orbit of y is zero. As an illustration of the strength of this criterion, we prove that the diffeomorphisms with nonhyperbolic ergodic measures form a C¹-open and dense subset of the set of robustly transitive partially hyperbolic diffeomorphisms with one dimensional nonhyperbolic central direction. We also obtain applications for nonhyperbolic homoclinic classes.