Logarithm Laws for Unipotent Flows, II

We prove analogs of the logarithm laws of Sullivan and Kleinbock-Margulis in the context of unipotent flows. In particular, we prove results for horospherical actions on homogeneous spaces $G/\Gamma$. We describe some relations with multi-dimensional diophantine approximation.


Introduction
Two important dynamical systems on non-compact manifolds are the geodesic and horocycle flows on the unit tangent bundle of a finite-volume non-compact hyperbolic surface. Both of these flows are known to be ergodic, and thus, generic orbits are dense. A natural question is to understand the behavior of excursions of trajectories into the cusp(s).
For geodesic flows, the statistical properties of these excursions were first studied in [17] by Sullivan (in the context of finite volume hyperbolic manifolds) and later, in the more general context of the actions of oneparameter diagonalizable subgroups on non-compact finite-volume homogeneous spaces, by Kleinbock-Margulis. In [12], they proved the following result: Theorem 1.1. ( [12], Theorem 1.7 and Prop 5.1) Let G be a connected semisimple Lie group without compact factors, g its Lie algebra, Γ ⊂ G an irreducible non-uniform lattice, K a maximal compact subgroup, and d(·, ·) a distance function on G/Γ determined by a right-invariant Riemannian metric on G bi-invariant under K. Let µ denote the measure on G/Γ determined by Haar measure on G. Let a ⊂ g be a Cartan subalgebra, 0 = z ∈ a, and a t = exp(tz). Then there exists a k = k(G/Γ, d) > 0 such that ∀y, • ∃C 1 , C 2 > 0 such that for all t > 0, • For µ-a.e. x, In this paper, we prove results similar to equation (1.2) for several classes of unipotent actions. This paper is a sequel to [4], where we considered the case of unipotent flows on the space of lattices. Subsequently, there has been significant activity in the setting of unipotent logarithm laws, for example the papers [3,5,11].
Our results can broadly be divided into two categories: (1) Horospherical actions. We prove a result (Theorem 2.1) on the excursions of orbits of large subsets of horospherical subgroups. We obtain lower bounds for specific orbits. (2) Almost everywhere results for flows. This result (Theorem 2.5) applies in the most general situation of one-parameter unipotent flows on symmetric spaces, and uses probabilistic techniques (generalized Borel-Cantelli lemmas) and exponential decay of matrix coefficients.
1.1. Organization: The paper is organized as follows: in §2, we state our main results. In §3, we collect technical results on tori and divergent trajectories required for our proofs. In §4, we use these technical results to prove our main theorem on horospherical actions, as well as related corollaries on hyperbolic surfaces. Finally, in §5, we prove our probabilistic results.
be the expanding horospherical subgroup associated to {a t }.
, there is a compact set C ∈ G/Γ and a sequence of times t n → +∞ such that a −tn x 0 ∈ C), that Hx 0 is dense in G/Γ. Our aim is to give a more quantitative version of this result, with regards to visits to neighborhoods of ∞. Let B ⊂ H be a non-empty, bounded, open subset. Set This forms an expanding family of subsets of H.
Let d X denote a right-invariant metric on G arising from the Riemannian metric induced by the Killing form. If G ′ ⊂ G is a sugbroup of G, we let d G ′ denote the induced distance function on G ′ . We let d G/Γ denote the induced distance function on G/Γ. We will drop the subscripts when it is clear on which space we are measuring distances. We will study the behavior of the excursions of B t x 0 away from compact sets by investigating the asymptotic behavior of the quantities for such x 0 . Our main result is about the rate of these excursions.
To formulate our results, we need a little more notation. If the R-rank of G is at least 2, we can assume, by the Arithmeticity Theorem ( [14], Chapter IX), that G = G(R) • and Γ = G(Z), where G is a semisimple algebraic Qgroup. Let S be a maximal Q-split torus. Without loss of generality, we can assume A = S(R) • . Let · denote the norm on A induced by the Killing form. We can write a t = exp tz, with z ∈ a. if the R-rank of G is equal to 1, then there is (up to scaling and conjugation) a unique 1-parameter subgroup, which we again can write as a t = exp tz for z ∈ a, where a is the Lie algebra to {a t }.
We will prove this theorem in §4. Combining this result with Theorem 1.1, which implies that ω − (x 0 ) = 0 for µ-almost every x 0 ∈ G, we have the following corollary: Corollary 2.2. Fix notation as above.

Remarks:
• It is initially somewhat surprising that the typical horospherical excursion should have the same order as the typical excursion for a t . What we will show in the proof is that the behavior of the horosphere is governed by divergent, non-typical, a t -trajectories.
• (2.4) follows relatively easily from the triangle inequality, whereas (2.5) requires a more detailed analysis of divergent {a t }-trajectories.
• We expect that our results will hold for general norm-like pseudometrics, as defined in [1].
we have that Moreover, for allx ∈ M such that Hx is not closed, The following proposition shows that while (2.8) holds for almost every point, the inequality in (2.9) is strict for a (topologically) large set of points: B contains a dense set of second Baire category.

Remarks:
• Note that in the metric on H 2 , d(p(h s ), i) = 2 log s (where by abuse of notation, p : SL(2, R) → H 2 = SO(2)\SL(2, R) is the projection p(g) = SO(2)g), so 2 is the maximum value this lim sup can attain. In fact, for any sequence r n → ∞ in SL(2, R), d(p(r n ), i) ≈ 2 log |r n | (≈ means the ratio goes to 1), where |g| is the supremum of the matrix entries of g.
• By (2.4), the set B must consist of trajectories which diverge at rate 1 under a −t , that is, they must satisfy • In [4], we consider the special case of Γ = SL(2, Z), and obtain several connections to Diophantine approximation. Further results can be found in [3], in which precise conditions for this lim sup to take on certain values for general SL(2, R)/Γ are given, and in [11] where results are obtained for quotients of products of SL(2, R) and SL(2, C).

Upper and lower bounds.
We now return to the case of general semisimple Lie groups G and non-uniform lattices Γ. Now we study the action of one-parameter unipotent subgroups on G/Γ. We have Theorem 2.5. Fix notation as in Theorem 1.1. Let {u t } t∈R ⊂ G denote a one-parameter unipotent subgroup. Then there is a 0 < α ≤ 1 such that for ∀y, µ-a.e. x, We prove this theorem in §5.
Remark: Note that Theorem 1.1 says that if we replace our unipotent subgroup {u t } with a diagonalizable subgroup {a t }, we can always take α = 1. Like Theorem 1.1, Theorem 2.5 is proved using information on decay of matrix coefficients of the regular representation of G on G/Γ, and an appropriately adapted version of the Borel-Cantelli lemma. However, the slower decay of matrix coefficients for unipotent flows as compared to diagonalizable flows does not allow us to conclude that α = 1. It would be very interesting to find examples of unipotent subgroups where α = 1, though we suspect that such subgroups do not exist.

Divergent trajectories
Recall the notation of §2.1: G is a connected semisimple Lie group without compact factors, Γ an irreducible non-uniform lattice, and d the Riemannian metric arising from the Killing form on G. We also use d to denote the metric on G/Γ. A is a maximal Q-diagonalizable subgroup of G, and {a t } a oneparameter subgroup of A. Write a t = exp tz, y ∈ a, and let ν = y . We prove the following result concerning {a t } trajectories in G/Γ.
Moreover, for all x = gΓ with g ∈ G(Q), 3.1. Reduction Theory. We recall some results from reduction theory. Assume the R-rank of G is greater than 1. We can assume, as in §2.1, G = G(R) • and Γ = G(Z), where G is a semisimple algebraic Q-group. Let S be a maximal Q-split torus in G, and set A = S(R) • . Let Φ be a system of Q-roots associated to A and let Φ + and Φ s be the sets of positive and simple roots respectively. We define the positive Weyl chamber a + = {z ∈ d : α(z) ≥ 0 for all α ∈ Φ s }. Using the exponential map, we identify it with A + = exp(a + ) ⊂ A.
Conjugating if necessary, we can assume that z ∈ a + , that is, a t ∈ A + for t > 0. We have the Iwasaswa decomposition G = KAM U (here, K is a maximal compact subgroup, U is unipotent, and M is reductive, with A centralizing M and normalizing U ). Let Q ⊂ M U be relatively compact, and for τ > 0, define a τ = {z ∈ a : α(z) ≥ τ for all α ∈ Φ s } We can define a generalized Siegel set S Q,τ := K exp(a τ )Q.
For appropriate choices of Q and τ , a finite union of translates of S Q,τ form a weak fundamental domain for the Γ-action on G. Precisely, we have Using Theorem 3.2, Leuzinger [13,Theorem 1] proved that there is a b ∈ A + (here, A + denotes the closure of the Weyl chamber A + ) such for any y ∈ a + , with y = 1, a t := exp(ty), any p ∈ M U , any q i , 1 ≤ i ≤ m, and γ ∈ Γ, we have Proof of Proposition 3.1. First note that the upper bound (3.1) follows from the definition of distance on the quotient and the fact that z = ν. To show the limit result (3.2), we first consider the case when R-rank is at least 2, so we can apply Theorem 3.2 and equation (3.3). Since we can write each element g ∈ G(Q) as g = pq i γ 0 for p ∈ P(Q) and γ ∈ Γ, and denoting the bounded error given by the element b by C, we have, for all γ ∈ Γ, 2) follows immidiately.
3.2.1. R-rank 1. Finally, suppose the R-rank of G is 1. Applying standard reduction theory [8] and the density of orbits of parabolic subgroups ( [16], Lemma 8.5) there is a dense set of points diverging under a t at rate ν = z . See also [7,18] for more details on divergent trajectories.

Horospherical actions
We fix notation as in §2.1 and §3. The proof of Theorem 2.1 splits naturally into an upper and lower bound: Proof: The idea is as follows: given the piece of orbit B e T x, we want to show that it has moved depth T into the cusp. We can write B e T x = a t Ba −t x. If a −t x is non-divergent, we can take some T so that a −t x is in a compact set. Using the fact the forward divergent {a t } trajectories are dense, we can find a divergent trajectory (moving at rate ν) in a 'thickening' of the orbit Ba −t x in the directions transverse to H. Since a t does not expand the directions transverse to H, the divergent trajectory (which will be approximately depth νT into the cusp after applying a t ) will be near B e T x, so there is some h ∈ B e T with hx almost depth T into the cusp, as desired. To make this argument precise, we need to use the following Lemma 4.2. Let C ⊂ G/Γ be compact with non-empty interior, and ǫ, φ > 0. Then there is a T C,ǫ,φ such that is ǫ-dense in C.
Proof: Note that by Prop 3.1, is dense in G/Γ. Now let ǫ > 0, C ⊂ G/Γ compact. Let {B(x, ǫ)} x∈C be the cover of C by open ǫ-metric balls. Since C is compact, we can take a finite subcover Let H −0 be the subgroup associated to the neutral/stable directions for a t (t > 0). Let x ∈ G/Γ be such that a −t x is non-divergent. Thus, there is a compact C ′′ ⊂ G/Γ be compact with a non-empty interior and t n → ∞ so that a −tn x ∈ C ′′ for all n.
We fix one more piece of notation: letting G ′ be a subgroup of G, g 0 ∈ G ′ , r > 0, we let B G ′ (g 0 , r) := {g ∈ G ′ : d G ′ (g 0 , g) < r}. Let ǫ 1 be such that for all ǫ < ǫ 1 , there are ǫ + , ǫ − , There is an 0 < ǫ < ǫ 1 , and an ǫ ′ so that (perhaps shrinking ǫ 0 ) we can write Thus, we have as n → ∞,(note that since x ′ n varies in a compact set, it does not matter in the limit whether we measure distance from x or x ′ n ). Thus, which, since φ > 0 was arbitrary yields our result.

Upper bound. Lemma 4.3.
For all x ∈ G/Γ, Proof: Let ǫ > 0. By the definition of ω, and the boundedness of B for all t sufficiently large, for all b ∈ B, By definition d(a log t ba − log t , ba − log t ) ≤ ν log t. Combining these two inequalities, and using the triangle inequality, we have, for all b ∈ B and t sufficiently large, d(a log t ba − log t x, x) < (ω + ν + ǫ) log t.
Since ǫ was arbitrary, we have our result.

Proof of Proposition 2.4:
We need the following lemma, which exploits properties of divergent geodesic trajectories:   (g s z), y) > s − c for all s > 0 (for the rest of this section, we will use the notation a t = e t/2 0 0 e −t/2 ). Fix lifts of y and z to a fundamental domain for Γ in H 2 , call them y 0 and z 0 (y 0 will be a point, z 0 will be a point and a unit tangent vector). There will be a horocycle connecting p(z 0 ) and p(g s z 0 ), and as s → ∞, the inward pointing tangent vector to this will approach the vector z 0 .
More precisely, suppose without loss of generality z 0 is i with the upward pointing tangent vector, i.e., z 0 = e ∈ SL(2, R). Then p(g s z 0 ) = p(g s ) = SO(2)g s , and if v s = r θs ∈ SO(2) is the unit tangent vector (based at i = p(z 0 )) determining the horocycle connecting e s i = p(g s ) and i, we have that v s approaches the upward pointing tangent vector as t → ∞, or equivalently θ s → 0.
In addition, if t = t s is the time it takes for the horocycle to reach e s i, we have SO(2)g s = SO(2)h t r θs , i.e., there is a θ ′ s such that h t = r θ ′ S g s r θs (this is simply the Cartan (or KAK) decomposition). It is an easy calculation that t s ≈ e s/2 , or equivalently, s ≈ 2 log t s . Thus, for s >> 0, r θs z ∈ A, and d(p(h ts r θs z), y) = d(p(g s z), y) > s − c > 2 log t s − C, for some possibly larger C.
f T (x) is increasing in T , and bounded, so we can define f ∞ (x) = lim T →∞ f T (x). The f T 's are continuous for T < ∞, but f ∞ is not. We have Lemma 4.4, and open by the continuity of f n . Thus B is a countable intersection of open dense sets, as desired.

Borel-Cantelli lemmas
In this section we prove Theorem 2.5, using a generalization of the Borel-Cantelli lemma. The classical Borel-Cantelli lemma is as follows: n=0 be a sequence of 0 − 1 random variables, with P (X n = 1) =: p n . Then, if ∞ n=0 p n < ∞, If the X n 's are pairwise independent, we have that if ∞ n=0 p n = ∞. The first statement (non-independent) statement is the 'easy half' of this Lemma, and can be derived by simply doing an expectation calculation.
The first example of a logarithm law can be derived from the lemma as follows. Fix λ > 0. Let {Y n } ∞ n=0 's be independent identically distributed (i.i.d.) exponential random variables with parameter λ. That is, for any Let {r n } ∞ n=0 be a sequence of positive real numbers. Applying Lemma 5.1 to the sequence of random variables implies Y n > r n infinitely often if and only if ∞ n=0 e −λrn = ∞. As a corollary, one obtains that almost surely lim sup n→∞ Y n log n = 1/λ.
To prove Theorem 2.5, we use the following (relatively standard) generalization of Lemma 5.1 to weakly dependent sequences.
Proof: Given measurable X : S → R, we write E(X) := S XdP for the expectation, and V (X) = E(X 2 ) − E(X) 2 for the variance. .
We will show that for any ǫ > 0, P (|Y n − 1| > ǫ) → 0, which will imply that there is a sequence n k such that Y n k → 1 with probability 1, and thus, that (2), so we get that Dividing by ( n i=1 p i ) 2 , we get that the two right hand terms go to zero (by properties (1) and (3) respectively), and thus, we have our result.
We would like to apply this result to the context of group actions on homogeneous spaces. Fixing notation as in §2.1, and given y ∈ G/Γ, we define a sequence of functions Y n : G/Γ → R + by Y n (x) = d(u n x, y). Given a sequence of numbers {r n } n∈N , we set X n (x) := 1 Y n (x) > r n 0 otherwise.
In order to apply Proposition 5.2 to our context, we must estimate two quantities: (1) µ(x : d(x, y) > t) (2) The covariances for the random variables X n . The first estimate follows from equation (1.1), which yields (since u n is measure preserving): C 1 e −krn ≤ p n = µ(x : X n (x) = 1) ≤ C 2 e −krn .
In order to estimate the covariances, we must control the matrix coefficients of the sequence {u k } under the regular representation of G on G/Γ. To do this, we turn once again to [12]. The following result is essentially a combination of Proposition 4.2 and Corollary 3.5 from that paper: Proposition 5.3. [12] There are constants C > 0, 0 < β < 1 such that for all n, m ∈ N |p n,n+m − p n p n+m | ≤ Cp n p n+m m −β , where p i,j = µ(x : X i (x)X j (x) = 1).
Remark: If we were able to obtain β ≥ 2, we would in fact be able to prove α = 1 in the statement of Theorem 2.5 following Prop 4.1 in [12]. However, for reasons beyond the scope of this paper, β < 2.
We will not prove Proposition 5.3 in this paper, instead referring the interested reader to the appropriate sections of [12].
Proof of Theorem 2.5: Let r n > 1 k log n. Then p n is summable, so for almost all x, X n = 1 only finitely often, yielding our upper bound. For our lower bound we apply Proposition 5.2 to our sequence X n , with ψ(m) = m −β . It is a simple calculation that for any γ < β/2, setting r n = γ k log n will yield: Using Proposition 5.2, we have, for µ-a.e. x, β/2k ≤ lim sup t→∞ d(u t x, y) log t ≤ 1/k.
Finally, note that lim sup t→∞ d(u t x, y) log t is a measurable u t -invariant function on G/Γ. Thus, if the u t -action is ergodic, it must be constant almost everywhere. If u t is not ergodic, it must act trivially in some factor of G by the Moore ergodicity theorem [15], and thus we can reduce to the ergodic case.