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Global fixed points for nilpotent actions on the torus

Part of: Smooth dynamical systems: general theory Classical Topics in Algebraic Topology Low-dimensional dynamical systems Ergodic 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 [email protected], [email protected]
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 [email protected], [email protected]
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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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