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Let f : X → X be continuous and onto, where X is a compact metric space. Let Y := lim ←− (X, f ) be the inverse limit and F : Y → Y the induced homeomorphism. Suppose that µ is an f -invariant measure, and let m be the measure induced on Y by (µ, µ, . . .). We show that B is a basin of µ if and only if π −1 1 (B) is a basin of m. From this it follows that if µ is an SRB measure for f on X , then the induced measure m on Y is an inverselimit SRB measure for F. Conversely, if m is andoi:10.1017/s0143385709000388 fatcat:ttudoyesizb4jd4p6owrrbnl2a