Our setting is transitive dynamics having a partially hyperbolic splitting with one-dimensional center.

Gorodetski-Ilyashenko-Kleptsyn-Nalsky (2005) introduced the method of periodic approximations (called shadowing and tailing) that allows to construct nonhyperbolic ergodic measures in several transitive and partially hyperbolic settings. This method leads to measures of zero entropy (Kwietniak-Lacka). In similar settings, in collaboration with Bochi-Bonatti we introduced a method (flip-flop) for constructing nonhyperbolic ergodic measures with positive entropy. However, in principle this method provides measures with small entropy compared with the entropy of the whole set of points having a zero exponent (the zero level set of Lyapunov exponents). 

I will explain how to show the existence of nonhyperbolic ergodic measures whose entropy is arbitrarily close to the topological entropy of the zero level set. This is a work in collaboration with Gelfert and Rams.