Bounded and divergent trajectories and expanding curves on homogeneous spaces

Abstract

Suppose g t is a 1-parameter Ad-diagonalizable subgroup of a Lie group G and Γ < G is a lattice. We study differentiable curves of the form ϕ : [0, 1] → U + satisfying certain non-degeneracy conditions, where U + is the expanding horospherical subgroup of g t . For a class of examples that includes products of real rank one Lie groups, we obtain sharp upper bounds on the Hausdorff dimension of the set of points s for which the forward orbit (g t ϕ(s)x 0 ) t 0 is divergent on average in G/Γ, for any basepoint x 0 ∈ G/Γ. Moreover, we prove that the set of points s for which (g t ϕ(s)x 0 ) t 0 remains bounded is winning in the sense of Schmidt. We describe applications of our results to problems in diophantine approximation. Our methods also yield the following result for square systems of linear forms: suppose ϕ(s) = sY + Z where Y ∈ GL(n, R) and Z ∈ M n,n (R). Then, the dimension of the set of points s such that ϕ(s) is singular is at most 1/2 while badly approximable points have Hausdorff dimension equal to 1.Lemma 4.5. There exists a constant C 1 > 1, such that for all x ∈ X, natural numbers n with α(n) ≥ 1/γ, t > 0 and all subintervals J ⊂ [−1, 1] of radius at least e −α(nt) , one hasProof. First, we note that for all r ∈ [−1, 1], we haveUsing positivity of f , (4.5) and a change of variable, we getThen, Fubini's theorem and the fact that ϕ is C 1+γ imply the following. J f (g (n+1)t u(ϕ(s))x) ds J 1 −1 f (g (n+1)t u(ϕ(s) + re −α(nt)φ (s) + O(e −(1+γ)α(nt) ))x) dr dsMoreover, by definition of g t and u(Y ), we haveThus, by our assumption that α(n) 1/γ, we getIn particular, by (4.7), we getSince e α(t) ∈ N, for each 1 ≤ j ≤ k, the partition P k is a refinement of P j . This implies the following inclusion.Hence, the following inequality follows from (4.8), (4.9), and the fact that f is non-negative.(4.10) Iterating (4.10), by induction, we obtain the following exponential decay integral estimate. B(M,n−1)∩J 0 f (g (m+n)t u(ϕ(s))x) ds (2C 1c ) n e −βα(nt) J∈Pm J∩B(M,n−1)∩J 0 =∅ J f (g mt u(ϕ(s))x) ds = (2C 1c ) n e −βα(nt) J 0 f (g mt u(ϕ(s))x) ds (4.11)where on the second line, we used the following consequence of P m being a partition.

Singular Vectors on Fractals and Projections of Self-similar Measures

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