The distribution of periodic torus orbits on homogeneous spaces

2006/07/31 by Manfred Einsiedler, Einsiedler, Manfred, Elon Lindenstrauss +5 · 5 citations
Mathematics · Physics and Astronomy · #Combinatorics #Distribution (mathematics) #Dynamical Systems (math.DS) #FOS: Mathematics #Geography #Geometric and Algebraic Topology #Geometry #Homogeneous #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Number Theory (math.NT) #Quantum chaos and dynamical systems #Torus #math.DS #math.NT

paper · pdf · doi:10.48550/arxiv.math/0607815

published in arXiv (Cornell University) (Cornell University)

arxiv created 2006/07/31 · openalex publication_date 2006/07/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We prove results towards the equidistribution of certain families of periodic torus orbits on homogeneous spaces, with particular focus on the case of the diagonal torus acting on quotients of \PGL_n(\R). After attaching to each periodic orbit an integral invariant (the discriminant) our results have the following flavour: certain standard conjectures about the distribution of such orbits hold up to exceptional sets of at most O(Δ^ε) orbits of discriminant ≤ Δ. The proof relies on the well-separatedness of periodic orbits together with measure rigidity for torus actions. We also give examples of sequences of periodic orbits of this action that fail to become equidistributed, even in higher rank. We give an application of our results to sharpen a theorem of Minkowski on ideal classes in totally real number fields of cubic and higher degrees.

Citations

• Invariant measures for higher-rank hyperbolic abelian actions
• Arithmetic Subgroups of Algebraic Groups

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