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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
  • 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



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

23518?type=thumb Kadyrov. Notes 243 KB application/pdf Download
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