Fixed points of measure preserving torus homeomorphisms
type: Item
instance of: scholarly article
title: Fixed points of measure preserving torus homeomorphisms
zbMATH Open document ID: 0722.58027
DOI: 10.1007/BF02568764
author: Martin Flucher
published in: Manuscripta Mathematica
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155527
review text: Theorem: Assume a homeomorphism of the 2-torus is homotopic to the identity and preserves a measure \(\mu\) which is positive on open sets. If it has a lift f: \({\mathbb{R}}^ 2\to {\mathbb{R}}^ 2\) with vanishing mean translation, i.e. \(\int (f(x)-x)d\mu (x)=0,\) then it has at least two fixed points. In particular 2 of the 3 fixed points conjectured by Arnold for diffeomorphisms preserving Lebesgue measure also exist for homeomorphisms. As a corollary we derive an analogous fixed point theorem for the annulus which is a variant of Birkhoff's last geometric theorem. The proof is based on a generalization of Brouwer's lemma on translation arcs and the ``shift decomposition formula'' presented in this paper. This formula also applies to the n-torus showing that f is chain transitive on all of \({\mathbb{R}}^ n\) if it has vanishing mean translation.
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 4190763
zbMATH Keywords: periodic disk chains
zbMATH Keywords: chain transitivity
zbMATH Keywords: Arnold's conjecture
zbMATH Keywords: measure preserving torus homeomorphisms
zbMATH Keywords: 2-torus
zbMATH Keywords: vanishing mean translation
zbMATH Keywords: fixed points
zbMATH Keywords: shift decomposition
zbMATH Keywords: n-torus
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4190763