Every finite set of natural numbers is realizable as algebraic periods of a Morse–Smale diffeomorphism

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 Möbius 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_ \mathrm g $, of genus $ {\rm g} $, and a Morse–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–Smale diffeomorphisms of $ S_ \mathrm 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–Smale diffeomorphisms, which is exponential in $ \mathrm g $.