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Ergodic theory and the duality principle on homogeneous spaces

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Abstract

We prove mean and pointwise ergodic theorems for the action of a lattice subgroup in a connected algebraic Lie group on infinite volume homogeneous algebraic varieties. Under suitable necessary conditions, our results are quantitative, namely we establish rates of convergence in the mean and pointwise ergodic theorems, which can be estimated explicitly. Our results give a precise and in most cases optimal quantitative form to the duality principle governing dynamics on homogeneous spaces. We illustrate their scope in a variety of equidistribution problems.

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Authors and Affiliations

  1. School of Mathematics, University of Bristol, Bristol, UK

    Alexander Gorodnik

  2. Department of Mathematics, Technion, Haifa, Israel

    Amos Nevo

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  1. Alexander Gorodnik
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  2. Amos Nevo
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Correspondence to Amos Nevo.

Additional information

Alexander Gorodnik was supported in part by EPSRC, ERC, and RCUK. Amos Nevo was supported by ISF grant.

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Gorodnik, A., Nevo, A. Ergodic theory and the duality principle on homogeneous spaces. Geom. Funct. Anal. 24, 159–244 (2014). https://doi.org/10.1007/s00039-014-0257-8

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