Dold Coefficients of Quasi-unipotent Homeomorphisms of Orientable Surfaces

Grzegorz Graff; Wacław Marzantowicz; Łukasz Patryk Michalak

doi:10.1007/s12346-025-01275-1

Dold Coefficients of Quasi-unipotent Homeomorphisms of Orientable Surfaces

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{\"o}bius inversion formula. The set $\cAP(f) = \{ n \colon a_n(f) \neq 0 \}$ 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 system 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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Articles

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artykuły w czasopismach

Published in:

Qualitative Theory of Dynamical Systems no. 24,
ISSN: 1575-5460

Language:

English

Publication year:

2025

Bibliographic description:

Graff G., Marzantowicz W., Michalak Ł. P.: Dold Coefficients of Quasi-unipotent Homeomorphisms of Orientable Surfaces// Qualitative Theory of Dynamical Systems -Vol. 24 art 116,iss. 3 (2025), s.1-22

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