The invariant connection of a \({1\over 2}\)-pinched Anosov diffeomorphism and rigidity
type: Item
instance of: scholarly article
title: The invariant connection of a \({1\over 2}\)-pinched Anosov diffeomorphism and rigidity
zbMATH Open document ID: 0855.58045
author: Renato Feres
published in: Pacific Journal of Mathematics
publication date: 1997-02-11T00:00:00Z
review text: Let \(f\) be a \(C^\infty\) Anosov diffeomorphism of a compact manifold \(M\), preserving a smooth measure. If \(f\) satisfies the \({1\over 2}\)-pinching assumption defined below, it must preserve a continuous affine connection for which the leaves of the Anosov foliations are totally geodesic, geodesically complete, and flat (its tangential curvature is defined along individual leaves). If this connection, which is the unique \(f\)-invariant affine connection on \(M\), is \(C^r\)-differentiable, \(r\geq 2\), then \(f\) is conjugate via a \(C^{r+2}\)-affine diffeomorphism to a hyperbolic automorphism of a geodesically complete flat manifold. If \(f\) preserves a smooth symplectic form, has \(C^3\) Anosov foliations, and satisfies the \(2:1\)-nonresonance condition (an assumption that is weaker than pinching), then \(f\) is \(C^\infty\) conjugate to a hyperbolic automorphism of a complete flat manifold. (In the symplectic case, the invariant connection is the one previously defined by Kanai in the context of geodesic flows.) If the foliations are \(C^2\) and the holonomy pseudo-groups satisfy a certain growth condition, the same conclusion holds.
Mathematics Subject Classification ID: 37D99
Mathematics Subject Classification ID: 37C85
Mathematics Subject Classification ID: 53C05
zbMATH DE Number: 912138
zbMATH Keywords: Anosov diffeomorphisms
zbMATH Keywords: invariant connections
zbMATH Keywords: rigidity
MaRDI profile type: Publication
full work available at URL: pjm.1995.171.139
OpenAlex ID: W1530975305
DOI: 10.2140/PJM.1995.171.139
description: scientific article; zbMATH DE number 912138