Takens’ the last problem and Stein-Ulam spiral type maps. (English) Zbl 07852386
Summary: 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 F. Takens’ last problem [Nonlinearity 21, No. 3, T33–T36 (2008; Zbl 1147.37013)] is resolved within the class of SUS-t maps.
MSC:
| 37H12 | Random iteration |
| 60H25 | Random operators and equations (aspects of stochastic analysis) |
| 47A35 | Ergodic theory of linear operators |
| 47H25 | Nonlinear ergodic theorems |
Citations:
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