Simultaneous dense and nondense orbits for commuting maps
Israel Journal of Mathematics
Springer Verlag for Hebrew University Magnes Press
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.
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.
This is the author accepted manuscript. The final version is available from the publisher via the DOI in this record.
Vol. 210, pp. 23 - 45