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Global fixed points for nilpotent actions on the torus

Published online by Cambridge University Press:  11 July 2019

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]

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.

Type
Original Article
Copyright
© Cambridge University Press, 2019

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