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New limiting notions of the IP type in the enveloping semigroup

Abstract An IP set in ℕ is a subset of ℕ which coincides with the set of finite sums taken from an infinite sequence in ℕ with not necessarily distinct terms. It has been established by H. Furstenberg and others that there is a rich connection between IP sets and idempotents in the enveloping semigroup E(X) of a compact metric dynamical system. Our aim in this paper is to further develop this program by considering special elements of E(X) with various IP properties.