Nontrivial semigroups in based monodromy sets

Determine whether a based monodromy set $H(i_0)$ associated with the group extension of an expanding dynamical system necessarily contains a nontrivial subsemigroup, and characterize when such a subsemigroup can recover the whole group or a sufficiently large finite-index subgroup.

Background

The based monodromy set $H(i_0)$ collects group elements obtained from orbit segments sharing a prescribed initial cylinder. The paper explains that the strongest possible situation would be for this set itself to be a semigroup equal to the ambient group, while geometric examples may only yield a finite-index subgroup.

The existence of such semigroup structure is important because the transfer-operator argument requires sufficiently many monodromy elements to interact with the spectral gap supplied by expansion. The authors impose embedding assumptions instead, leaving the general structural question unresolved.