Classification of orientation reversing periodic homeomorphisms of a two dimensional torus
https://doi.org/10.26907/2949-3919.2025.3.87-109
Abstract
According to J. Nielsen and H. Hang, each class of topological conjugacy of periodic homeomorphisms of orientable compact surfaces is completely described by a finite set of data called the characteristic. For a two-dimensional sphere, exhaustive classification results with the construction of linear representatives in each conjugacy class were obtained by B. Kerekyarto. For a two-dimensional torus, similar results were obtained with the participation of the authors of this article. Here we find all the characteristics of orientation-changing periodic homeomorphisms of a two-dimensional torus. A homeomorphism representing a class of topological conjugacy is constructed for each of them. The classification of periodic homeomorphisms, in addition to being of independent interest, plays a key role in solving the Palis–Pugh problem of constructing stable arcs in the space of discrete dynamical systems, which is included in the list of 50 most important problems of dynamical systems. For all classes of gradient-like diffeomorphisms of surfaces where this problem is solved, the idea of a close connection of such systems with periodic transformations was used. Thus, the obtained result will make it possible to expand the class of systems for which the Palis problem has been solved.
Supporting agencies: This work is an output of a research project implemented as part of the Basic Research Program at the National Research University Higher School of Economics (HSE University).
About the Authors
T. D. Martynov
HSE University
Russian Federation
Russian Federation
Timur Denisovich Martynov
25/12 Bolshaya Pecherskaya str., Nizhny Novgorod, 603155
O. V. Pochinka
HSE University
Russian Federation
Russian Federation
Olga Vitalyevna Pochinka
25/12 Bolshaya Pecherskaya str., Nizhny Novgorod, 603155
E. E. Chilina
HSE University
Russian Federation
Russian Federation
Ekaterina Evgenevna Chilina
25/12 Bolshaya Pecherskaya str., Nizhny Novgorod, 603155
References
1. J. Nielsen, Untersuchungen zur topologie der geschlossenen zweiseitigen Fl¨achen, Acta Math. 50 (1), 189–358 (1927) [In German]. DOI: https://doi.org/10.1007/BF02421324
2. V.Z. Grines, S.Kh. Kapkaeva, O.V. Pochinka, A three-colour graph as a complete topological invariant for gradient-like diffeomorphisms of surfaces, Sb. Math. 205 (10), 1387–1412 (2014). DOI: https://doi.org/10.1070/SM2014v205n10ABEH004423
3. J. Palis, C. Pugh, Fifty problems in dynamical systems, in: Lecture Notes in Math. 468 Springer, Berlin-New York, 345–353 (1975). DOI: https://doi.org/10.1007/BFb0082633
4. E. Nozdrinova, O. Pochinka, Solution of the 33rd Palis–Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere, Discrete Contin. Dyn. Syst. 41 (3), 1101–1131 (2021). DOI: https://doi.org/10.3934/dcds.2020311
5. A.A. Nozdrinov, E.V. Nozdrinova, O.V. Pochinka, Stable isotopy connectivity of gradient-like diffeomorphisms of 2-torus J. Geom. Phys. 207, art. 105352 (2025). DOI: https://doi.org/10.1016/j.geomphys.2024.105352
6. B. von Ker´ekj´art´o, Uber die periodischen Transformationen der Kreisscheibe und der ¨ Kugelfl¨ache, Math. Ann. 80 (1), 36–38 (1919) [in German]. DOI: https://doi.org/10.1007/BF01463232
7. L.E.J. Brouwer, Uber die periodischen Transformationen der Kugel ¨ , Math. Ann. 80 (1), 39–41 (1919) [in German]. DOI: https://doi.org/10.1007/BF01463233
8. J. Nielsen, Die struktur periodischer transformationen von fl¨achen, Math.-fys. Medd. Danske Vid. Selsk. 15 (1), Levin and Munksgaard, 1937.
9. K. Yokoyama, Classification of periodic maps on compact surfaces: I, Tokyo J. Math. 6 (1), 75–94 (1983). DOI: https://doi.org/10.3836/tjm/1270214327
10. K. Yokoyama, Classification of periodic maps on compact surfaces: II, Tokyo J. Math. 7 (1), 249–285 (1984). DOI: https://doi.org/10.3836/tjm/1270153007
11. K. Yokoyama, Complete classification of periodic maps on compact surfaces, Tokyo J. Math. 15 (2), 247–279 (1992). DOI: https://doi.org/10.3836/tjm/1270129455
12. H. Hang, Homology and orientation reversing periodic maps on surfaces, Topology Appl. 229, 1–19 (2017). DOI: https://doi.org/10.1016/j.topol.2017.06.023
13. S. Hirose, Presentations of periodic maps on oriented closed surfaces of genera up to 4, Osaka J. Math. 47 (2), 385–421 (2010).
14. D.A. Baranov, V.Z. Grines, O.V. Pochinka, E.E. Chilina, On a classification of periodic maps on the 2-torus, Russ. J. Nonlinear Dyn. 19 (1) 91–110 (2023). DOI: https://doi.org/10.20537/nd220702
15. D.A. Baranov, O.V. Pochinka, Classification of periodic transformations of an orientable surface of genus two, Zh. SVMO 23 (2), 147–158 (2021) [in Russian]. DOI: https://doi.org/10.15507/2079-6900.23.202102.147-158
16. K.-I. Tahara, On the finite subgroups of GL(3, Z), Nagoya Math. J. 41, 169–209 (1971). DOI: https://doi.org/10.1017/S002776300001415X
17. K.W. Kwun, J.L. Tollefson, PL Involutions of S 1 × S 1 × S 1 , Trans. Amer. Math. Soc. 203, 97–106 (1975). DOI: https://doi.org/10.2307/1997071
18. J. Hempel, Free cyclic actions on S 1 × S 1 × S 1 , Proc. Amer. Math. Soc. 48 (1), 221–227 (1975). DOI: https://doi.org/10.2307/2040721
19. M.A. Natsheh, On cyclic group actions of even order on the three dimensional torus, Bull. Austral. Math. Soc. 37 (2), 189–196 (1988). DOI: https://doi.org/10.1017/S0004972700026721
20. L.E.J. Brouwer, Aufz¨ahlung der periodisken Transformationen des Torus, in: KNAW Proc. 21 II, 1352–1356 (1919).
21. C. Kosniowski, A first course in algebraic topology, Cambridge University Press, Cambridge-New York, 1980. DOI: https://doi.org/10.1017/CBO9780511569296
Review
For citations:
Martynov T.D., Pochinka O.V., Chilina E.E. Classification of orientation reversing periodic homeomorphisms of a two dimensional torus. Mathematics and Theoretical Computer Science. 2025;3(3):87-109. (In Russ.) https://doi.org/10.26907/2949-3919.2025.3.87-109
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