Historic and physical wandering domains for wild blender-horseshoes

We give a contribution to Takens' last problem [Tak08] in the $C^{1}$ topology (note that solutions have been already given in the smooth category of more than or equal to $C^{2}$, see [KS17,BB]). To be specific, we present a one-parameter family $\mathscr{A}$ of $C^1$ diffeomorphisms having wild blender-horseshoes such that every $C^1$ neighbourhood of any element of $\mathscr{A}$ contains two types of diffeomorphisms, one of which has a historic contracting wandering domain, and the other has a non-trivial Dirac physical measure associated with a contracting wandering domain. It is a non-trivial extension of Colli-Vargas' model [CV01] to the higher dimensional dynamics with the use of wild blender-horseshoes.