Takens' last problem. Whether are there persistent classes of smooth dynamical systems such that the set of initial states which give rise to orbits with
historic behavior has positive Lebesgue measure?

Here we say that a property P is Cr-persistent if the closure YP of the set YP of all diffeomorphisms in Diffr(M) satisfying P has a non-empty interior. Note that any Cr-robust property1 is Cr-persistent but the inverse is not always true. In the recent paper [1], we give an affirmative answer to Takens' last problem. To obtain it, we have to show that the next conjecture about non-trivial wandering domains is true, and moreover prove that the orbit of any point in the wandering domain has historic behavior.

[1] S. Kiriki and T. Soma, Takens' last problem and existence of non-trivial wandering domains, Advances in Math., 306 (2017), 524{588.