A given self-map $f\colon M\to M$ of a compact manifold determines the sequence $(L(f^n))$ of the Lefschetz numbers of its iterations. We consider its dual sequence $(a_n(f))$ given by the M\"obius inversion formula. The set ${ \mathcal AP}(f)=\{ n\in \mathbb N\ \colon\ a_n(f)\neq 0\}$ is called the set of algebraic periods. We solve an open problem existing in literature by showing that for every finite subset ${ \mathcal A}$ of natural numbers there exist an orientable surface $S_{ \rm g}$, as well as a non-orientable surface $N_{ \rm g}$, of genus ${ \rm g}$, and a Morse$\unicode{ x2013}$Smale diffeomorphism $f$ of this surface such that $\mathcal{ AP}(f)=\mathcal{ A}$. For such a map it implies the existence of points of a minimal period $n$ for each odd $n \in \mathcal{ A}$. For the orientation-reversing Morse$\unicode{ x2013}$Smale diffeomorphisms of $S_{ \rm g}$, we identify strong restrictions on ${ \mathcal AP}(f)$. Our method also provides an estimate of the number of conjugacy classes of mapping classes containing Morse$\unicode{ x2013}$Smale diffeomorphisms, which is exponential in ${ \rm g}$.
Categories: math.DS