Extend zero-entropy GH^0 approximation beyond full-support measures

Determine whether the $C^0$–Gromov–Hausdorff approximation-by-zero-entropy theorem holds for homeomorphisms that admit an invariant probability measure whose support is not the entire space; specifically, ascertain whether every homeomorphism of a compact metric space with an invariant measure lacking full support can still be approximated in the $C^0$–Gromov–Hausdorff topology by homeomorphisms with zero topological entropy.

Background

The paper’s main theorem proves that if a homeomorphism on a compact metric space admits an invariant probability measure with full support, then it can be approximated in the $C^0$–Gromov–Hausdorff topology by homeomorphisms with zero topological entropy. The proof exploits the ergodic decomposition together with points having dense positive orbits in the supports of selected ergodic components that collectively cover the whole space.

If the invariant measure does not have full support, this covering argument may fail because the supports of ergodic components need not globally cover the ambient space. The authors therefore ask whether an analogous approximation result still holds without the full-support hypothesis.