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Global fixed points for nilpotent actions on the torus

Part of: Ergodic theory Classical Topics in Algebraic Topology Low-dimensional dynamical systems Smooth dynamical systems: general theory

Published online by Cambridge University Press:  11 July 2019

S. FIRMO  and
J. RIBÓN
S. FIRMO
Affiliation:
Instituto de Matemática e Estatística, Universidade Federal Fluminense, Campus do Gragoatá, Rua Marcos Valdemar de Freitas Reis s/n, 24210-201 Niterói, Rio de Janeiro, Brasil email firmo@mat.uff.br, jribon@id.uff.br
J. RIBÓN
Affiliation:
Instituto de Matemática e Estatística, Universidade Federal Fluminense, Campus do Gragoatá, Rua Marcos Valdemar de Freitas Reis s/n, 24210-201 Niterói, Rio de Janeiro, Brasil email firmo@mat.uff.br, jribon@id.uff.br
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Abstract

An isotopic to the identity map of the 2-torus, that has zero rotation vector with respect to an invariant ergodic probability measure, has a fixed point by a theorem of Franks. We give a version of this result for nilpotent subgroups of isotopic to the identity diffeomorphisms of the 2-torus. In such a context we guarantee the existence of global fixed points for nilpotent groups of irrotational diffeomorphisms. In particular, we show that the derived group of a nilpotent group of isotopic to the identity diffeomorphisms of the 2-torus has a global fixed point.

Keywords

rotation vectorglobal fixed pointirrotational homeomorphismergodic probability measurenilpotent group

MSC classification

Primary: 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces 37E45: Rotation numbers and vectors 37A15: General groups of measure-preserving transformations 37A05: Measure-preserving transformations
Secondary: 55M20: Fixed points and coincidences 37C25: Fixed points, periodic points, fixed-point index theory
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 12 , December 2020 , pp. 3339 - 3367
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Copyright
© Cambridge University Press, 2019

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References

Atkinson, G.. Recurrence of co-cycles and random walks. J. Lond. Math. Soc. (2) 13(3) (1976), 486–488.Google Scholar
Austin, T.. Integrable measure equivalence for groups of polynomial growth. Groups Geom. Dyn. 10(1) (2016), 117–154.Google Scholar
Béguin, F., Le Calvez, P., Firmo, S. and Miernowski, T.. Des points fixes pour des difféomorphismes de 𝕊2 qui commutent et préservent une mesure de probabilité. J. Inst. Math. Jussieu 12(4) (2013), 821–851.Google Scholar
Bonatti, C.. Un point fixe commun pour des difféomorphismes commutants de 𝕊2. Ann. of Math. (2) 129 (1989), 61–79.Google Scholar
Bonatti, C.. Difféomorphismes commutants des surfaces et stabilité des fibrations en tores. Topology 29(2) (1990), 205–209.Google Scholar
Brooks, R. B. S., Brown, R. F., Pak, J. and Taylor, D. H.. Nielsen numbers of maps of tori. Proc. Amer. Math. Soc. 52 (1975), 398–400.Google Scholar
Brown, M. and Kister, J. M.. Invariance of complementary domains of a fixed point set. Proc. Amer. Math. Soc. 91(3) (1984), 503–504.Google Scholar
Choquet, G.. Lectures on analysis. Representation Theory (Mathematics Lecture Note Series, 25). Third printing. Vol. II. Eds. Marsden, J., Lance, T. and Gelbart, S.. W.A. Benjamin, Advanced Book Program, Reading, MA, 1976.Google Scholar
Druck, S., Fang, F. and Firmo, S.. Fixed points of discrete nilpotent group actions on 𝕊2. Ann. Inst. Fourier (Grenoble) 52(4) (2002), 1075–1091.Google Scholar
Firmo, S.. A note on commuting diffeomorphisms of surfaces. Nonlinearity 18(4) (2005), 1511–1526.Google Scholar
Firmo, S. and Ribón, J.. Finite orbits for nilpotent actions on the torus. Proc. Amer. Math. Soc. 146(1) (2018), 195–208.Google Scholar
Firmo, S., Ribón, J. and Velasco, J.. Fixed points for nilpotent actions on the plane and the Cartwright–Littlewood theorem. Math. Z. 279 (2015), 849–877.Google Scholar
Franks, J.. Recurrence and fixed points of surface homeomorphisms. Ergod. Th. & Dynam. Sys. 8(8*) (1988), 99–107.Google Scholar
Franks, J., Handel, M. and Parwani, K.. Fixed points of abelian actions on 𝕊2. Ergod. Th. & Dynam. Sys. 27(5) (2007), 1557–1581.Google Scholar
Franks, J., Handel, M. and Parwani, K.. Fixed points of abelian actions. J. Mod. Dyn. 1(3) (2007), 443–464.Google Scholar
Gambaudo, J.-M.. Periodic orbits and fixed points of a C 1 orientation-preserving embedding of D 2. Math. Proc. Cambridge Philos. Soc. 108 (1990), 307–310.Google Scholar
Hamstrom, M. E.. Homotopy groups of the space of homeomorphisms on a 2-manifold. Illinois J. Math. 10(4) (1966), 563–573.Google Scholar
Handel, M.. Commuting homeomorphisms of 𝕊2. Topology 31 (1992), 293–303.Google Scholar
Kargapolov, M. I. and Merzljakov, Ju. I.. Fundamentals of the Theory of Groups (Graduate Texts in Mathematics, 62). Springer, New York–Heidelberg–Berlin, 1979, Translated from the second Russian edition by Robert G. Burns.Google Scholar
Koropecki, A. and Tal, F.. Bounded and unbounded behavior for area-preserving rational pseudo-rotations. Proc. Lond. Math. Soc. (Print) 109 (2014), 785–822.Google Scholar
Lima, E.. Commuting vector fields on S 2. Proc. Amer. Math. Soc. 15(1) (1964), 138–141.Google Scholar
Misiurewicz, M. and Ziemian, K.. Rotation sets for maps of tori. J. Lond. Math. Soc. (2) 40(3) (1989), 490–506.Google Scholar
Yu Olshanskii, A.. Distortion functions for subgroups. Geometric Group Theory Down Under, Proc. Special Year in Geometric Group Theory (Canberra, Australia, 14–19 July 1996). Ed. Cossey, J. et al. . Walter de Gruyter, Berlin, 1996, pp. 281–291.Google Scholar
Parkhe, K.. Smooth gluing of group actions and applications. Proc. Amer. Math. Soc. 143(1) (2015), 203–212.Google Scholar
Plante, J. F.. Fixed points of Lie group actions on surfaces. Ergod. Th. & Dynam. Sys. 6(1) (1986), 149–161.Google Scholar
Ribón, J.. Fixed points of nilpotent actions on 𝕊2. Ergod. Th. & Dynam. Sys. 36(1) (2016), 173–197.Google Scholar
Schmitt, B.. Sur les plongements, admettant zero ou un point fixe, du disque dans le plan. Topology 14 (1975), 357–365.Google Scholar