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Entropy in the cusp and singular systems of linear forms

Advances in Homogeneous Dynamics May 11, 2015 - May 15, 2015

May 11, 2015 (02:55 PM PDT - 03:20 PM PDT)
Speaker(s): Shirali Kadyrov (Nazarbayev University)
Location: MSRI: Simons Auditorium
Tags/Keywords
  • Dirichlet's theorem for linear forms

  • singular points

  • Hausdorff dimension

  • Dani's correspondence

  • unstable horospherical subgroup

  • measures with entropy bounds

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

14234

Abstract

Singular systems of linear forms were introduced by Khintchine in the 1920s, and it was shown by Dani in the 1980s that they are in one-to-one correspondence with certain divergent orbits of one-parameter diagonal groups on the space of lattices. We give a (conjecturally sharp) upper bound on the Hausdorff dimension of singular systems of linear forms (equivalently the set of points with divergent trajectories). This extends work by Cheung, as well as by Chevallier and Cheung, on the vector case. For a diagonal action on the space of lattices, we also consider the relation of the entropy of an invariant measure to its mass in a fixed compact set. Our technique is based on the method of integral inequalities developed by Eskin, Margulis, and Mozes. This is a joint work with D. Kleinbock, E. Lindenstrauss, and G. A. Margulis

Supplements
23518?type=thumb Kadyrov. Notes 243 KB application/pdf Download
Video/Audio Files

14234

H.264 Video 14234.mp4 106 MB video/mp4 rtsp://videos.msri.org/data/000/023/425/original/14234.mp4 Download
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