We discuss the entropy of nonuniformly hyperbolic measures
constructed using two methods introduced, respectively, by Gorodetski
et al. and by Bochi et al.
In a joint work with Martha Łącka, we show that measures
defined by Gorodetski et al. always have zero entropy and are Kakutani
equivalent to an ergodic group rotation.
In a joint work with Bonatti and Díaz, assuming robust
transitivity, we prove that in the partially hyperbolic setting there
robustly exists an ergodic nonhyperbolic measure with full support and
positive entropy. The novelty of this result is that we address all
four conditions (robustness, ergodicity, positive entropy, and full
support) together, while previous works dealt only with a subset of
these conditions.
For the proofs, we introduce and study a new tool: the Feldman-Katok
pseudometric fk-bar, which leads to a new notion of convergence for
invariant measures.