Existence of a lower mmd-expansive but non-mmd-expansive map

Determine whether there exists a map that is lower mmd-expansive but not mmd-expansive.

Background

The paper introduces mmd-expansiveness as a generalization based on metric mean dimension. A map is called upper mmd-expansive when the upper metric mean dimension of every sufficiently small dynamical Bowen ball is zero, and lower mmd-expansive when the corresponding lower metric mean dimension is zero. Since upper mmd-expansiveness implies lower mmd-expansiveness, the unresolved issue is whether the converse can fail.

The authors prove that the shift map on the Hilbert cube is not lower mmd-expansive by exhibiting invariant subsets inside arbitrarily small dynamical Bowen balls with positive lower metric mean dimension. This result does not resolve whether any map can satisfy the weaker lower condition without satisfying the stronger upper condition.