Dold coefficients of quasi-unipotent homeomorphisms of orientable surfaces

TL;DR

This paper investigates the Dold coefficients of quasi-unipotent homeomorphisms on orientable surfaces, linking algebraic periods to surface dynamics, and explores minimal genus surfaces realizing specific periodic behaviors.

Contribution

It characterizes surface homeomorphisms with bounded Lefschetz numbers using Dold coefficients and analyzes the minimal genus surfaces for given algebraic periods.

Findings

Classification of Dold coefficients for small genus surfaces

Identification of algebraic periods for specific surface types

Geometrical applications 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 M\"obius 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.