A generic C1 map has no absolutely continuous invariant probability measure

Artur Avila, Jairo Bochi
2006 Nonlinearity  
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,
more » ... dowed with the C 1 topology. Given f ∈ C 1 (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 1 maps f : M → M which have no acim is a residual (dense G δ ) subset of C 1 (M, M). Since the set of all expanding maps and the set of all diffeomorphisms are open subsets of C 1 (M, M), we have the following immediate consequences. (i) The C 1 -generic expanding map has no acim. (ii) The C 1 -generic diffeomorphism has no acim. Result (i) was previously obtained in the case where M is a circle by Quas [Q]. Of course, (i) does not hold in the C 1+Hölder topology. It seems possible that result (ii) holds in higher topologies. An old result by Livsic and Sinai implies that the C ∞ -generic Anosov map has no acim, see [LS] and also [C]. (In fact,
doi:10.1088/0951-7715/19/11/011 fatcat:vjqumyddpreijetpcrz4mjgh3e