Abstract
We provide various counter examples for quantitative multiple recurrence problems for systems with more than one transformation. We show that:
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There exists an ergodic system (X,X, μ,T1, T2) with two commuting transformations such that, for every 0 < ℓ < 4, there exists A ∈ X such that
$$\mu \left( {A \cap T_1^{ - n}A \cap T_2^{ - n}A} \right) < \mu {\left( A \right)^\ell }foreveryn \ne 0$$ -
There exists an ergodic system (X,X, μ,T2, T3) with three commuting transformations such that, for every ℓ > 0, there exists A ∈ X such that
$$\mu \left( {A \cap T_1^{ - n}A \cap T_2^{ - n}A \cap T_3^{ - n}A} \right) < \mu {\left( A \right)^\ell }foreveryn \ne 0$$ -
There exists an ergodic system (X,X, μ,T1, T2) with two transformations generating a 2-step nilpotent group such that, for every ℓ > 0, there exists A ∈ X such that
$$\mu \left( {A \cap T_1^{ - n}A \cap T_2^{ - n}A} \right) < \mu {\left( A \right)^\ell }foreveryn \ne 0$$
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The first author is supported by Fondecyt Iniciación en Investigación grant 11160061 and CMM-Basal grant PFB-03.
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Donoso, S., Sun, W. Quantitative multiple recurrence for two and three transformations. Isr. J. Math. 226, 71–85 (2018). https://doi.org/10.1007/s11856-018-1690-4
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DOI: https://doi.org/10.1007/s11856-018-1690-4