An ergodic closing lemma in the GH^0 topology

Formulate and establish a closing lemma stated in ergodic terms for the $C^0$–Gromov–Hausdorff topology, providing an ergodic-theoretic mechanism that produces periodic orbits from recurrent or typical behavior in this metric-topological setting.

Background

The paper constructs periodic approximations using ergodic decomposition and points with dense positive orbits to obtain finite models close in the $C^0$–Gromov–Hausdorff topology. This echoes the spirit of classical closing lemmas, which connect recurrence or pseudo-orbits to actual periodic orbits.

The authors ask for a more directly ergodic formulation of a closing lemma adapted to the $GH^0$ framework, which would strengthen and generalize the periodic approximation principles developed in the paper.