The sequence of Dold coefficients \((a_n(f))\) of a self-map \(f:X \rightarrow X\) forms a dual sequence to the sequence of Lefschetz numbers \((L(f^n))\) of iterations of f under the Möbius inversion formula. The set \({\mathcal {A}}P(f) = \{ n :a_n(f) \ne 0 \}\) is called the set of algebraic periods of f. Both the set of algebraic periods and sequence of Dold coefficients play an important role in dynamical systems and periodic point theory. In this work we provide a description of surface homeomorphisms with bounded \((L(f^n))\) (quasi-unipotent maps) in terms of Dold coefficients. We also discuss the problem of minimization of the genus of a surface for which one can realize a given set of natural numbers as the set of algebraic periods. Finally, we compute and list all possible Dold coefficients and algebraic periods for a given orientable surface with small genus and give some geometrical applications of the obtained results.
