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Simultaneous dense and nondense orbits for commuting maps

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Abstract

We show that, for two commuting automorphisms of the torus and for two elements of the Cartan action on compact higher rank homogeneous spaces, many points have drastically different orbit structures for the two maps. Specifically, using measure rigidity, we show that the set of points that have dense orbit under one map and nondense orbit under the second has full Hausdorff dimension.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Ohio State University, Columbus, OH, 43210, USA

    Vitaly Bergelson

  2. Departement Mathematik, ETH Zürich, Rämistrasse 101, 8092, Zürich, Switzerland

    Manfred Einsiedler

  3. School of Mathematics, University of Bristol University Walk, Bristol, BS8 1TW, UK

    Jimmy Tseng

Authors
  1. Vitaly Bergelson
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  2. Manfred Einsiedler
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  3. Jimmy Tseng
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Corresponding author

Correspondence to Vitaly Bergelson.

Additional information

V. B. acknowledges support received from the National Science Foundation via Grant DMS-1162073

M. E. acknowledges support by the SNF (200021-152819).

J. T. acknowledges the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement n. 291147.

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Bergelson, V., Einsiedler, M. & Tseng, J. Simultaneous dense and nondense orbits for commuting maps. Isr. J. Math. 210, 23–45 (2015). https://doi.org/10.1007/s11856-015-1244-y

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  • DOI: https://doi.org/10.1007/s11856-015-1244-y

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