| Abstract: |
| A point without time averages for a (nonautonomous) dynamical system is said to have historic behaviour.
It is known:
(i) There is a persistent class of dynamical systems such that the set of points with historic behaviour is of positive Lebesgue measure (Kiriki and Soma 2017);
(ii) For any expanding maps, the set of points with historic behaviour is residual (Takens 2008);
(iii) For any (non-trivial) nonautonomous dynamical system under independent and identically distributed (i.i.d.) noise, the set of points with historic behaviour is of zero Lebesgue measure (Araujo 2000).
Based on the theorems, in this talk, we present the following two results about historic behaviour for nonautonomous dynamical systems.
(1) For any nonautonomous expanding maps under ergodic noise (including i.i.d. noise), the set of points with historic behaviour is residual (arXiv:1510.00905).
(2) There is a (nontrivial) nonautonomous dynamical system under a non-i.i.d. noise such that the set of points with historic behaviour is of positive Lebesgue measure (arXiv:1703.03163).
The second result is a joint work with Kiriki and Soma (although it is independent of their theorem in 2017). |
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