Let $\mathcal M_T(X)$ denote the simplex of invariant measures of the dynamical system $(X,T)$. For any point $x\in X$ and any positive integer $n$ denote by $m(x,n)$ the measure $1/n(\delta(x)+\ldots+\delta(T^{n-1}(x)))$ and by $\hat\omega(x)$ the set of all accumulation points of the sequence $\{m(x,n)\}_{n\in\mathbb N}$ with respect to the weak-* topology. During the talk we will show that if a dynamical system has the asymptotic average shadowing property, then for every compact connected and non-empty set $V\subset \mathcal M_T(X)$ there exists $x\in X$ such that $\hat\omega(x)$ is equal to $V$. In particular it means that every invariant measure has a generic point. In the proof we will use the Besicovitch pseudometric.

The talk is based on the joined work with Dominik Kwietniak and Piotr Oprocha.