Let $M$ be a two-sided surface of genus $g>1$. In 1936 A. Weil singled out the problem of generalization of the Poincarè rotation numbers to the minimal flows $\phi$ on $M$. In this paper we suggest a solution to this problem by constructing the rotation numbers which we call Artin's. These numbers are invariants of the $K_0$-group of a crossed product $C^*$-algebra $C(X)\rtimes_{\phi}\Z$. It is shown that the dynamics of $\phi$ is completely subordinate to diophantine properties of the Artin numbers.