Dold coefficients of quasi-unipotent homeomorphisms of orientable surfaces

Abstract

The sequence of Dold coefficients $(a_n(f))$ of a self-map $f:X\to X$ forms a dual sequence to the sequence of Lefschetz numbers $(L(f^n))$ of iterations of $f$ under the Möbius inversion formula. The set $\mathcal{A}P(f)=\{n:a_n(f)\ne 0\}$ 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.

References in corpus (2)

• Dold sequences, periodic points, and dynamics
• On the periodic structure of $C^1$ self-maps on the product of spheres of different dimensions

Cited by in corpus (1)

• Homological data on the periodic structure of self-maps on wedge sums