Historic wandering domains near cycles

Abstract

Abstract We explain how to obtain non-trivial historic contractive wandering domains for a dense set of diffeomorphisms in certain type of C r -Newhouse domains of homoclinic tangencies in dimension d ⩾ 3 and r ⩾ 1. In particular, this gives for the first time a contribution to Takens’ last problem in the C 1 topology and in dimension d > 2. We show how these Newhouse domains can be obtained arbitrarily close to diffeomorphisms exhibiting heterodimensional cycles (in dimension d = 3) or non-transverse equidimensional cycles (in any dimension d ⩾ 3) associated with periodic points with non-real complex leading multipliers.