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Inverse limit stability for semiflows

On semiflows on Banach manifolds, Quandt (1988 and 1989) stated that some special semiflows were inverse limit stable. However, the concepts and proofs used to show inverse limit stability of semiflows are rough and ambiguous. We consider semiflows on finite-dimensional compact manifolds or on finite-dimensional compact non-singular branched manifolds. In this paper we show that more special semiflows are inverse limit stable. That is, Anosov semiflows are inverse limit stable.