Finite orbits for nilpotent actions on the torus
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- by S. Firmo and J. Ribón PDF
- Proc. Amer. Math. Soc. 146 (2018), 195-208 Request permission
Abstract:
A homeomorphism of the $2$-torus with Lefschetz number different from zero has a fixed point. We give a version of this result for nilpotent groups of diffeomorphisms. We prove that a nilpotent group of $2$-torus diffeomorphims has finite orbits when the group has some element with Lefschetz number different from zero.References
- F. Béguin, P. Le Calvez, S. Firmo, and T. Miernowski, Des points fixes communs pour des difféomorphismes de $\Bbb S^2$ qui commutent et préservent une mesure de probabilité, J. Inst. Math. Jussieu 12 (2013), no. 4, 821–851 (French, with English and French summaries). MR 3103133, DOI 10.1017/S1474748012000898
- Christian Bonatti, Un point fixe commun pour des difféomorphismes commutants de $S^2$, Ann. of Math. (2) 129 (1989), no. 1, 61–69 (French, with English summary). MR 979600, DOI 10.2307/1971485
- Christian Bonatti, Difféomorphismes commutants des surfaces et stabilité des fibrations en tores, Topology 29 (1990), no. 1, 101–126 (French). MR 1046627, DOI 10.1016/0040-9383(90)90027-H
- Robin B. S. Brooks, Robert F. Brown, Jingyal Pak, and Douglas H. Taylor, Nielsen numbers of maps of tori, Proc. Amer. Math. Soc. 52 (1975), 398–400. MR 375287, DOI 10.1090/S0002-9939-1975-0375287-X
- S. Druck, F. Fang, and S. Firmo, Fixed points of discrete nilpotent group actions on $S^2$, Ann. Inst. Fourier (Grenoble) 52 (2002), no. 4, 1075–1091 (English, with English and French summaries). MR 1926674
- S. Firmo, A note on commuting diffeomorphisms on surfaces, Nonlinearity 18 (2005), no. 4, 1511–1526. MR 2150340, DOI 10.1088/0951-7715/18/4/005
- S. Firmo and J. Ribón, Global fixed points for nilpotent actions on the torus, to appear.
- S. Firmo, J. Ribón, and J. Velasco, Fixed points for nilpotent actions on the plane and the Cartwright-Littlewood theorem, Math. Z. 279 (2015), no. 3-4, 849–877. MR 3318253, DOI 10.1007/s00209-014-1396-1
- John Franks, Michael Handel, and Kamlesh Parwani, Fixed points of abelian actions on $S^2$, Ergodic Theory Dynam. Systems 27 (2007), no. 5, 1557–1581. MR 2358978, DOI 10.1017/S0143385706001088
- John Franks, Michael Handel, and Kamlesh Parwani, Fixed points of abelian actions, J. Mod. Dyn. 1 (2007), no. 3, 443–464. MR 2318498, DOI 10.3934/jmd.2007.1.443
- Étienne Ghys, Sur les groupes engendrés par des difféomorphismes proches de l’identité, Bol. Soc. Brasil. Mat. (N.S.) 24 (1993), no. 2, 137–178 (French, with English summary). MR 1254981, DOI 10.1007/BF01237675
- Michael Handel, Commuting homeomorphisms of $S^2$, Topology 31 (1992), no. 2, 293–303. MR 1167171, DOI 10.1016/0040-9383(92)90022-A
- M. I. Kargapolov and Ju. I. Merzljakov, Fundamentals of the theory of groups, Graduate Texts in Mathematics, vol. 62, Springer-Verlag, New York-Berlin, 1979. Translated from the second Russian edition by Robert G. Burns. MR 551207
- Svetlana Katok, Fuchsian groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992. MR 1177168
- Kiran Parkhe, Smooth gluing of group actions and applications, Proc. Amer. Math. Soc. 143 (2015), no. 1, 203–212. MR 3272745, DOI 10.1090/S0002-9939-2014-12231-3
- Elon L. Lima, Commuting vector fields on $S^{2}$, Proc. Amer. Math. Soc. 15 (1964), 138–141. MR 159342, DOI 10.1090/S0002-9939-1964-0159342-8
- MichałMisiurewicz and Krystyna Ziemian, Rotation sets for maps of tori, J. London Math. Soc. (2) 40 (1989), no. 3, 490–506. MR 1053617, DOI 10.1112/jlms/s2-40.3.490
- J. F. Plante, Fixed points of Lie group actions on surfaces, Ergodic Theory Dynam. Systems 6 (1986), no. 1, 149–161. MR 837981, DOI 10.1017/S0143385700003345
- Javier Ribón, Fixed points of nilpotent actions on $\Bbb {S}^2$, Ergodic Theory Dynam. Systems 36 (2016), no. 1, 173–197. MR 3436759, DOI 10.1017/etds.2014.58
- Jerry Shurman, Geometry of the quintic, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1997. MR 1427489
Additional Information
- S. Firmo
- Affiliation: Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rua Mário Santos Braga s/n - Valonguinho, 24020 - 140 Niterói, Rio de Janeiro, Brasil
- MR Author ID: 303468
- Email: firmo@mat.uff.br
- J. Ribón
- Affiliation: Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rua Mário Santos Braga s/n - Valonguinho, 24020 - 140 Niterói, Rio de Janeiro, Brasil
- Email: javier@mat.uff.br
- Received by editor(s): October 4, 2016
- Received by editor(s) in revised form: January 26, 2017, and February 3, 2017
- Published electronically: August 1, 2017
- Additional Notes: This work was supported in part by CAPES
- Communicated by: Yingfei Yi
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 195-208
- MSC (2010): Primary 37E30, 37E45, 37A15, 37A05, 54H20; Secondary 55M20, 37C25
- DOI: https://doi.org/10.1090/proc/13686
- MathSciNet review: 3723133