Let X = G/Γ, where G is a Lie group and Γ is a lattice in G, let U be an open subset of X, and let {gt} be a one-parameter subgroup of G. Consider the set of points in X whose gt-orbit misses U ; 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 has been proved when X is compact or when G is a simple Lie group of real rank 1. In this paper we prove this conjecture for the case G = SLm+n(R), Γ = SLm+n(Z) and gt = diag(e nt , . . . , e nt , e −mt , . . . , e −mt ), in fact providing an effective estimate for the codimension. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on SLm+n(R)/ SLm+n(Z). We also discuss an application to the problem of improving Dirichlet's theorem in simultaneous Diophantine approximation.
Let X = G/Γ, where G is a Lie group and Γ is a lattice in G, and let U be a subset of X whose complement is compact. We use the exponential mixing results for diagonalizable flows on X to give upper estimates for the Hausdorff dimension of the set of points whose trajectories miss U . This extends a recent result of Kadyrov [10] and produces new applications to Diophantine approximation, such as an upper bound for the Hausdorff dimension of the set of weighted uniformly badly approximable systems of linear forms, generalizing an estimate due to Broderick and Kleinbock [2].
We prove that for every flat surface
\omega
, the Hausdorff dimension of the set of directions in which Teichmüller geodesics starting from
\omega
exhibit a definite amount of deviation from the correct limit in Birkhoff’s and Oseledets’ theorems is strictly less than 1. This theorem extends a result by Chaika and Eskin who proved that such sets have measure 0. We also prove that the Hausdorff dimension of the directions in which Teichmüller geodesics diverge on average in a stratum is bounded above by 1/2, strengthening a classical result due to Masur. Moreover, we show that the Hausdorff codimension of the set of non-weakly-mixing IETs with permutation
(d, d-1,\ldots,1)
, where
d \geq 5
is an odd number, is at least 1/2, thus strengthening a result by Avila and Leguil. Combined with a recent result of Chaika and Masur, this shows that the Hausdorff dimension of this set is exactly 1/2.
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