Hielsen numbers of self-naps of surfaces

The Hielsen number $N(f)$ of a self-map $f$ of a compact polyhedronis a classical invariant of $f$ – defined in terms of fixed points of $f$. The Nielsen number $N(f)$ is a lower bound for the number of fixed points for all maps homotopic to $f$. There is the following classical question about exactness of this bound: given a map $f$, whether there is a map homotopic to $f$ with precisely $N(f)$ fixed points? It is known that this bound is exact for self-maps of every compact polyhedron without local separating points which is not a surface. The main result of the paper asserts that this bound is exact for homotору autoequivalences of compact surfaces. The proof of this theorem is based on Thurston's theory of diffeomorphisms of surfaces. Besides that some examples of self-maps of compact surfaces are discussed. It seems that the above bound is not exact in these examples.