We consider the problem of when the ascending HNN extension of a free group is residually finite. This is equivalent to the problem of when an endomorphism of a free group can be extended to an automorphism of a pro-finite closure of the free group group. We show that this problem can be reduced to a problem about counting periodic points of polynomial maps on algebraic varieties over fields of positive characteristics. Using some results from algebraic geometry (the intersection theory, resolution of singularities, etc.) we proved (together with Alexander Borisov) that, in particular, every ascending HNN extension of the free group of rank 2 is residually finite.