Simultaneous Dense and Nondense Orbits and the Space of Lattices
Shi, R; Tseng, J
Date: 8 February 2015
Journal
International Mathematics Research Notices
Publisher
Oxford University Press (OUP)
Publisher DOI
Abstract
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 ...
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.
Mathematics and Statistics
Faculty of Environment, Science and Economy
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