Flavio ABDENUR and Martin ANDERSSON: Ergodic theory of generic continuous maps (seminar of F.A. at Parix-XIII).
Both generic endomorphisms of manifolds of any dimension and generic homeomorphisms of manifolds of dimension greater than one exhibit highly pathological ergodic properties with respect to Lebesgue measure: they are weird in the sense that they support neither physical measures nor absolutely continuous measures but at the same time the orbit of Lebesgue almost-every point of M does converge in the Birkhoff sense.
They also study the ergodic properties of homeomorphisms f which are generic within the conjugacy class of expanding maps of the circle. It turns out that the Birkhoff-averaged iterated push-forwards of Lebesgue measure by f accumulate on every Borel invariant probability on M; this implies in particular that the dynamics is wicked: the orbit of Lebesgue-a.e. point of M does NOT converge in the Birkhoff sense.
(adapted from the announcement of the seminar).