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Let M be a smooth compact manifold of any dimension. We consider the set of C 1 maps f : M → M which have no absolutely continuous (with respect to Lebesgue) invariant probability measure. We show that this is a residual set in the C 1 topology. Mathematics Subject Classification: 37C40 Statement 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 1 (M, M) be the set of C 1 maps M → M,doi:10.1088/0951-7715/19/11/011 fatcat:vjqumyddpreijetpcrz4mjgh3e