We consider step skew products generated by a finite family of $C^1$ circle diffeomorphisms. We introduce an Axiomatic setting that models transitive and nonhyperbolic systems. Roughly, this implies the coexistence of contracting and expanding regions which are intermingled by the dynamics. For those systems we proved that for every nonzero Lyapunov exponent $\alpha$ there exists an ergodic measure with Lyapunov exponent $\alpha$ whose entropy is arbitrarily close to the entropy of all points with that exponent. Results of this type are usually called restricted variational principle. In this talk, we consider the case $\alpha=0$ and prove the the restricted variational principle holds.

The proof goes by construction of a nonhyperbolic ergodic measure of high entropy. For this we extend a technique of Gorodetski, Ilyashenko, Kleptsyn, and Nalski for the construction of nonhyperbolic ergodic measures as limit of periodic ones. However, by Kwietniak and Łącka, the measures obtained from this approach have zero entropy and hence are of no use in our context where the zero level set of Lyapunov exponents has positive entropy. Here, very naively, we replace periodic orbits by horseshoes. The main difficulty of this extension comes from nonuniform and not-everywhere convergence of Birkhoff averages, which we overcome by implementing a probabilistic approach.