Sensitive Group Actions on Regular Curves of Almost ≤ n Order

Let X be a regular curve and n be a positive integer such that for every nonempty open set U ⊂ X , there is a nonempty connected open set V ⊂ U with the cardinality ∣∂ x ( V )∣≤ n . We show that if X admits a sensitive action of a group G , then G contains a free subsemigroup and the action has positive geometric entropy. As a corollary, X admits no sensitive nilpotent group action.