Cartan Actions of Higher Rank Abelian Groups and their Classification

2019/01/19 by Ralf Spatzier, Kurt Vinhage, Spatzier, Ralf +1
Computer Science · Mathematics · #Cellular Automata and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1901.06559

openalex publication_date 2019/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study ℝ^k × ℤ^ℓ actions on arbitrary compact manifolds with a projectively dense set of Anosov elements and 1-dimensional coarse Lyapunov foliations. Such actions are called totally Cartan actions. We completely classify such actions as built from low-dimensional Anosov flows and diffeomorphisms and affine actions, verifying the Katok-Spatzier conjecture for this class. This is achieved by introducing a new tool, the action of a dynamically defined topological group describing paths in coarse Lyapunov foliations, and understanding its generators and relations. We obtain applications to the Zimmer program.

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