Dold coefficients of quasi-unipotent homeomorphisms of orientable surfaces
Grzegorz Graff, Wacław Marzantowicz, Łukasz Patryk Michalak
TL;DR
The paper develops a complete framework linking Dold coefficients $a_n(f)$ and algebraic periods ${\mathcal A}{\mathcal P}(f)$ for orientation-preserving self-maps of closed orientable surfaces, focusing on quasi-unipotent maps whose Lefschetz numbers are bounded. It establishes a precise correspondence between the spectrum of $H_1(f)\in Sp(2g,\mathbb{Z})$ and the Dold sequence via a finite multiplicity vector $\{r_k\}$ of primitive roots of unity, yielding $2g=\sum_k r_k\varphi(k)$ and explicit formulas $a_1=2-\sum_k\mu(k)r_k$, $a_n=-\sum_{k: n|k} \mu(k/n) r_k$ for $n\ge2$; it proves realizability of any finitely supported $a_n$ under parity constraints ($r_1,r_2$ even) and provides an algorithm to enumerate all possible Dold coefficients for a given genus, as well as a method to minimize the genus realizing a given algebraic-period set ${\mathcal A}$. The work also connects these algebraic invariants to dynamical properties of transversal maps, giving lower bounds on the number of minimal-period points, and offers an explicit constructive Appendix ensuring the existence of the required symplectic matrices with prescribed cyclotomic characteristic polynomials. Together, these results enable explicit, algorithmic construction of surface maps with prescribed periodic data and provide practical tools for studying periodic points in surface dynamics.
Abstract
The sequence of Dold coefficients $(a_n(f))$ of a self-map $f\colon X \to X$ forms a dual sequence to the sequence of Lefschetz numbers $(L(f^n))$ of iterations of $f$ under the Möbius inversion formula. The set ${\mathcal AP}(f) = \{ n \,\colon\, a_n(f) \neq 0 \}$ 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.