Slow entropy of higher rank abelian unipotent actions

Adam Kanigowski; Philipp Kunde; Kurt Vinhage; Daren Wei

doi:10.3934/jmd.2022018

Article Contents

2022, Volume 18: 575-607. Doi: 10.3934/jmd.2022018

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Slow entropy of higher rank abelian unipotent actions

1.
Department of Mathematics, University of Maryland, College Park, MD 20742, USA
2.
Department of Mathematics, University of Hamburg, Bundesstrasse 55, 20146, Hamburg, Germany
3.
Department of Mathematics, The University of Utah, Salt Lake City, UT 84112, USA
4.
Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 9190401, Israel

Received: June 30, 2020

Revised: July 13, 2022

AK: Partially supported by the NSF grant DMS-1956310.
PK: Acknowledges support from a DFG Forschungsstipendium under Grant No. 405305501.
DW: Partially supported by the NSF grant DMS-16-02409 and ERC 2020 grant HomDyn (grant no. 833423).

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Abstract

We study slow entropy invariants for abelian unipotent actions $ U $ on any finite volume homogeneous space $ G/\Gamma $. For every such action we show that the topological slow entropy can be computed directly from the dimension of a special decomposition of $ {{\rm{Lie}}}(G) $ induced by $ {{\rm{Lie}}}(U) $. Moreover, we are able to show that the metric slow entropy of the action coincides with its topological slow entropy. As a corollary, we obtain that the complexity of any abelian horocyclic action is only related to the dimension of $ G $. This generalizes the rank one results from [14] to higher rank abelian actions.

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Mathematics Subject Classification: Primary: 37A35; Secondary: 37A17, 37A15.

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References

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