Minimal systems with finitely many ergodic measures

Abstract In this paper it is proved that if a minimal system has the property that its sequence entropy is uniformly bounded for all sequences, then it has only finitely many ergodic measures and is an almost finite to one extension of its maximal equicontinuous factor. This result is obtained as an application of a general criterion which states that if a minimal system under an amenable group action is an almost finite to one extension of its maximal equicontinuous factor and has no infinite independent sets of length k for some k ≥ 2 , then it has only finitely many ergodic measures.