It is for many years that structural stability of dynamical systems has been taken into consideration by dynamicists and there are various studies, results, and open problems on this topic. This kind of stability cares about the change of the topological behavior of `all' points for the maps nearby the initial map. However, if your aim is to study a dynamical system only from a statistical point of view, this is too restrictive. You can ignore the change of behavior of orbits on a set of zero measure. Moreover, from a statistical point of view, it is not important that for which iterations the orbit of a point meets a subset of the phase space, the only thing which is important is the proportion of times that an orbit meets a given subset. In other words, orbits with different topological behavior may have the same statistical behavior. So it is natural to think about another version of stability while working with statistical properties of your maps. In this talk, using some examples, I would like to present a version of the notion of `statistical stability', and state a few theorems about it.

References:
A. Talebi, Non-statistical rational maps. arXiv:2003.02185, (2020).
A. Talebi, Statistical (in)stability and non-statistical dynamics. arXiv:2012.14462, (2020).

[IPM Youth Seminars on Topology and Dynamics]
Venue: meet.google.com/qix-iqiy-txt