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arXiv · 2411.19313

Dold coefficients of quasi-unipotent homeomorphisms of orientable surfaces

Grzegorz Graff·Wacław Marzantowicz·Łukasz Patryk Michalak
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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.

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Grzegorz Graff, Wacław Marzantowicz, Łukasz Patryk Michalak. 2024-11-28. Dold coefficients of quasi-unipotent homeomorphisms of orientable surfaces. https://doi.org/10.1007/s12346-025-01275-1

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