Simultaneous Dense and Nondense Orbits and the Space of Lattices
International Mathematics Research Notices
Oxford University Press (OUP)
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We show that the set of points nondense under the ×n -map on the circle and dense for the geodesic flow after we identify the circle with a periodic horospherical orbit of the modular surface has full Haudorff dimension. We also show the analogous result for toral automorphisms on the 2 -torus and a diagonal flow. Our results can be interpreted in number-theoretic terms: the set of well-approximable numbers that are nondense under the ×n -map has full Hausdorff dimension. Similarly, the set of well-approximable 2 -vectors that are nondense under a hyperbolic toral automorphism has full Hausdorff dimension. Our result for numbers is the counterpart to a classical result of Kaufmann.
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. R.S. is supported by NSFC (11201388), NSFC (11271278), ERC starter grant DLGAPS 279893.
This is the author accepted manuscript. The final version is available from Oxford University Press via the DOI in this record.
Vol. 2015, Iss. 21, pp. 11276 - 11288