Nonhyperbolic step skew-products: Ergodic approximation

Focusing on the nonhyperbolic ergodic measures (with zero fiber exponent) of these systems, we prove that such measures are approximated in the weak⁎ topology and in entropy by hyperbolic ones. We also prove that they are in the intersection of the convex hulls of the measures with positive fiber exponent and with negative fiber exponent. Our methods also allow us to perturb hyperbolic measures. We can perturb a measure with negative exponent directly to a measure with positive exponent (and vice-versa), however we lose some amount of entropy in this process. The loss of entropy is determined by the difference between the Lyapunov exponents of the measures.