Develop an analogous criterion for the non-periodic case

Develop an analogue of the periodic-flow criterion that ensures non-local exponentiality for C^∞(M) ⋊_α R in the case of non-periodic flows σ: R × M → M on compact manifolds M by identifying verifiable conditions under which the exponential map fails to be a local diffeomorphism at the identity.

Background

Within Section 3, the authors present criteria establishing non-local exponentiality for several situations, including periodic flows on compact manifolds, and they derive concrete non-surjectivity of the associated β_s operators. They note that these results yield non-local exponentiality in various cases.

However, they point out a gap for non-periodic flows, stating that an analogous criterion has not been proved, and they express the belief that such groups are never locally exponential—echoing the broader Conjecture A.