A Generic C 1 Map Has No Absolutely Continuous Invariant Probability Measure
نویسندگان
چکیده
Let M be a smooth compact manifold (maybe with boundary, maybe disconnected) of any dimension d ≥ 1. Let m be some (smooth) volume probability measure in M. Let C(M,M) be the set of C maps M → M, endowed with the C topology. Given f ∈ C(M,M), we say that μ is an acim for f if μ is an f -invariant probability measure which is absolutely continuous with respect to m. Theorem 1. The set R of C maps f : M → M which have no acim is a residual (dense Gδ) subset of C (M,M).
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