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.