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Homoclinic groups, IE groups, and expansive algebraic actions

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We give algebraic characterizations for expansiveness of algebraic actions of countable groups. The notion of \(p\)-expansiveness is introduced for algebraic actions, and we show that for countable amenable groups, a finitely presented algebraic action is \(1\)-expansive exactly when it has finite entropy. We also study the local entropy theory for actions of countable amenable groups on compact groups by automorphisms, and show that the IE group determines the Pinsker factor for such actions. For an expansive algebraic action of a polycyclic-by-finite group on \(X\), it is shown that the entropy of the action is equal to the entropy of the induced action on the Pontryagin dual of the homoclinic group, the homoclinic group is a dense subgroup of the IE group, the homoclinic group is nontrivial exactly when the action has positive entropy, and the homoclinic group is dense in \(X\) exactly when the action has completely positive entropy.

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Acknowledgments

The second named author was partially supported by NSF grants DMS-0701414 and DMS-1001625. He is grateful to Doug Lind and Klaus Schmidt for very interesting discussions. We thank David Kerr for helpful comments.

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  1. Nhan-Phu Chung

    Present address: Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 , Leipzig, Germany

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  1. Department of Mathematics, SUNY at Buffalo, Buffalo, NY, 14260-2900, USA

    Nhan-Phu Chung & Hanfeng Li

  2. Center of Mathematics, Chongqing Institute of Mathematics, Chongqing University, Chongqing, 401331, China

    Hanfeng Li

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Chung, NP., Li, H. Homoclinic groups, IE groups, and expansive algebraic actions. Invent. math. 199, 805–858 (2015). https://doi.org/10.1007/s00222-014-0524-1

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