Every finite set of natural numbers is realizable as algebraic periods of a Morse$\text{unicode}x2013$Smale diffeomorphism

A given self-map $f\text{\hspace{0.17em}⁣}:M\to M$ 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 $\mathcal{A}P(f)=\{n\in \mathbb{N}\text{\hspace{0.17em}⁣}:a_n(f)\mathrm{\ne }0\}$ 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 $S_{\mathrm{g}}$, as well as a non-orientable surface $N_{\mathrm{g}}$, of genus $\mathrm{g}$, and a Morse$\text{unicode}x2013$Smale diffeomorphism $f$ of this surface such that $\mathcal{A}\mathcal{P}(f)=\mathcal{A}$. 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$\text{unicode}x2013$Smale diffeomorphisms of $S_{\mathrm{g}}$, we identify strong restrictions on $\mathcal{A}P(f)$. Our method also provides an estimate of the number of conjugacy classes of mapping classes containing Morse$\text{unicode}x2013$Smale diffeomorphisms, which is exponential in $\mathrm{g}$.