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Database of algebraic periods of quasi-unipotent orientation-preserving homeomorphisms of orientable surfaces
Description
The set of algebraic periods of a map contains important information about periodic points and, in addition, is a homotopy invariant of the map. It is determined by indices of nonzero Dold coefficients which are computed purely algebraically from maps induced on homology groups of a considered space. In this dataset, we include for a given g=1,2,...,30, all possible algebraic periods of quasi-unipotent orientation-preserving homeomorphisms of a closed orientable surface of genus g.
In fact, the database consists of three parts. In Algebraic_periods_orientable_surface.zip there are three folders, and each of them contains 30 text files, whose names differ only in the genus number #g=1,2,...,30.
- The folder Algebraic_periods consists of 30 files genus#g_AP.txt containing the list of all algebraic periods of quasi-unipotent orientation-preserving homeomorphisms of a closed orientable surface of genus #g.
- The folder Odd_algebraic_periods consists of 30 files genus#g_APodd.txt containing the list of all odd algebraic periods, known also as the minimal sets of Lefschetz periods, of quasi-unipotent orientation-preserving homeomorphisms of a closed orientable surface of genus #g.
- The folder Minimal_algebraic_periods consists of 30 files min_genus#g_AP.txt containing the list of all algebraic periods of quasi-unipotent orientation-preserving homeomorphisms of a closed orientable surface of genus #g, for which the genus #g is minimal, i.e. they do not occur on the lists for smaller genera.
Odd algebraic periods, used by other authors under the name minimal set of Lefschetz periods, were intensively studied in the context of Morse-Smale diffeomorphisms, which are also quasi-unipotent. Note that if a map is not quasi-unipotent, then it has infinitely many algebraic periods.
The data was produced in python using SageMath. The algorithm is based on the description of spectra of finite order integral symplectic matrices of dimension 2g, which represent maps on the first homology group of the surface, and connection with Dold coefficients using the Moebius inversion formula. The details can be found in the paper G. Graff, W. Marzantowicz and Ł. P. Michalak, Dold coefficients of quasi-unipotent homeomorphisms of orientable surfaces, preprint (2024).
Dataset file
hexmd5(md5(part1)+md5(part2)+...)-{parts_count} where a single part of the file is 512 MB in size.Example script for calculation:
https://github.com/antespi/s3md5
File details
- License:
-
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CC BYAttribution
Details
- Year of publication:
- 2024
- Verification date:
- 2024-11-06
- Dataset language:
- English
- Fields of science:
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- mathematics (Natural sciences)
- DOI:
- DOI ID 10.34808/pfes-fh25 open in new tab
- Funding:
- Verified by:
- Gdańsk University of Technology
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