KAM rigidity for partially hyperbolic affine Z^k actions on the torus with a rank one factor

2013/08/16 by Danijela Damjanović, Damjanovic, Danijela, Bassam Fayad +1
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1308.3546

openalex publication_date 2013/08/16 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We show that ergodic affine abelian discrete actions on the torus, that have a rank-one factor in their linear part, are locally rigid in a KAM sense if and only if the rank one factor is trivial and the action is higher-rank transversally to this factor. Since it has been proved by Damjanovic and Katok that affine actions with higher-rank linear part are locally rigid, our result completes the local rigidity picture for affine actions on the torus.

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