A nonhyperbolic ergodic measure is an ergodic invariant measure with one Lyapunov exponent equal zero. Gorodetski, Ilyashenko, Kleptsyn, and Nalsky constructed a nonhyperbolic ergodic measure for a skew product diffeomorphism of the three-dimensional torus. Inspired by this construction, Bonatti, Diaz and Gorodetski gave sufficient condi- tions for weak convergence of a sequence of measures supported on periodic orbits to an ergodic measure. A royal measure is a measure obtained through this scheme. Numerous authors adapted this approach to provide examples of ergodic nonhyperbolic royal measures in various settings. It was an open question whether these measures could have positive entropy.

In a joint work with Martha Łacka, we show that royal measures always have zero entropy. Furthermore, we prove that all royal measures are loosely Kronecker. In other words, every royal measure is Kakutani equivalent (Kakutani equivalence is broader than the usual isomorphism of measure-preserving systems) to an ergodic rotation of a non-discrete compact topological group. For the proof, we introduce and study a new tool: the Feldman-Katok pseudometric fk-bar.