Hyperplane absolute winning property of bounded orbits under diagonalizable flows on SL₃(ℂ)/SL₃(O_\mathbbK)

2023/10/25 by Gaurav Sawant, Sawant, Gaurav
Mathematics · #Mathematical Dynamics and Fractals #Advanced Differential Equations and Dynamical Systems #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2310.16671

Abstract

We extend the work of An, Guan and Kleinbock on bounded orbits of diagonalizable flows on SL₃(ℝ)/SL₃(ℤ) to SL₃(ℂ)/SL₃(O_\mathbbK), where \mathbbK is an imaginary quadratic field. To achieve this, we first prove a complex analogue of Minkowski's Linear Forms Theorem. We then set up an appropriate Schmidt game in ℂ³ such that bounded orbits correspond to a hyperplane-absolute-winning set consisting of certain vectors in ℂ³ relative to an approximation by imaginary quadratic rationals in \mathbbK.

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