Takens' the last problem and Stein-Ulam Spiral Type Maps

The main issue of Takens' problem is solved by introducing a new class of stochastic operators called Stein-Ulam Spiral type (SUS-t) maps on a finite-dimensional simplex. Each SUS-t map is established as non-ergodic, i.e., it possesses historical behaviour. Through the usage of the new introduced class, it propels the work forward with focus on the power of the SUS-t map. Hence this paper establishes that any power of SUS-t map also has historical behaviour. The obvious corollary of the main result is that Takens' last problem [33] is resolved within the class of SUS-t maps.