2021/03/01 by Jonas Deré, Deré, Jonas, Thomas Witdouck +1
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2103.00914
openalex publication_date 2021/03/01 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
A few years ago, the first example of a closed manifold admitting an Anosov diffeomorphism but no expanding map was given. Unfortunately, this example is not explicit and is high-dimensional, although its exact dimension is unknown due to the type of construction. In this paper, we present a family of concrete 12-dimensional nilmanifolds with an Anosov diffeomorphism but no expanding map, where nilmanifolds are defined as the quotient of a 1-connected nilpotent Lie group by a cocompact lattice. We show that this family has the smallest possible dimension in the class of infra-nilmanifolds, which is conjectured to be the only type of manifolds admitting Anosov diffeomorphisms up to homeomorphism. The proof shows how to construct positive gradings from the eigenvalues of the Anosov diffeomorphism under some additional assumptions related to the rank, using the action of the Galois group on these algebraic units.