Dimension drop for diagonalizable flows on homogeneous spaces

2022/07/26 by Kleinbock, Dmitry, Mirzadeh, Shahriar
#37A25 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT) #Primary: 37A17 #Secondary: 11J13

paper · doi:10.48550/arxiv.2207.13155

Abstract

Let X = G/Γ, where G is a Lie group and Γ is a lattice in G, let O be an open subset of X, and let F = \g_t: t≥ 0\ be a one-parameter subsemigroup of G. Consider the set of points in X whose F-orbit misses O; it has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture is proved when X is compact or when G is a simple Lie group of real rank 1, or, most recently, for certain special flows on the space of lattices. In this paper we prove this conjecture for arbitrary Ad-diagonalizable flows on irreducible quotients of semisimple Lie groups. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on G/Γ. We also derive an application to jointly Dirichlet-Improvable systems of linear forms.

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