Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices

In this paper, we study an analytic curve $\varphi: I=[a,b]\rightarrow \mathrm{M}(m\times n, \mathbb{R})$ in the space of $m$ by $n$ real matrices, and show that if $\varphi$ satisfies certain geometric condition, then for almost every point on the curve, the Diophantine approximation given by Dirichlet's Theorem can not be improved. To do this, we embed the curve into some homogeneous space $G/\Gamma$, and prove that under the action of some expanding diagonal subgroup $A= \{a(t): t \in \mathbb{R}\}$, the translates of the curve tend to be equidistributed in $G/\Gamma$, as $t \rightarrow +\infty$.

1. Introduction 1.1. Diophantine approximation for matrices. In 1842, Dirichlet proved the following result on simultaneous approximation of a matrix of real numbers by integral vectors: Given two positive integers m and n, a matrix Φ ∈ M(m×n, R), and any N > 0, there exist integral vectors p ∈ Z n \{0} and q ∈ Z m such that where · denotes the supremum norm; that is, x := max 1≤i≤k |x i | for any x = (x 1 , x 2 , . . . , x k ) ∈ R k . Now we consider the following finer question: for a particular m by n matrix Φ, could we improve Dirichlet's Theorem? By improving Dirichlet's Theorem, we mean there exists a constant 0 < µ < 1, such that for all large N > 0, there exists nonzero integer vector p ∈ Z n with p ≤ µN m , and integer vector q ∈ Z m such that Φp − q ≤ µN −n . If such constant µ exists, then we say Φ is DT µ -improvable. And if Φ is DT µ -improvable for some 0 < µ < 1, then we say Φ is DT -improvable (here DT stands for Dirichlet's Theorem).
This problem was firstly studied by Davenport and Schmidt in [DS70], in which they proved that almost every matrix Φ ∈ M(m × n, R) is not DT -improvable. In [DS70], they also proved the following result. For m = 1 and n = 2, M(1 × 2, R) = R 2 , one considers the curve φ(s) = (s, s 2 ) in R 2 . Then for almost every s ∈ R with respect to the Lebesgue measure on R, φ(s) is not DT 1/4 improvable. This result was generalized by Baker in [Bak78]: for any smooth curve in R 2 satisfying some curvature condition, almost every point on the curve is not DT µ improvable for some 0 < µ < 1 depending on the curve. Bugeaud [Bug02] generalized the result of Davenport and Schmidt in the following sense: for m = 1, and general n, almost every point on the curve ϕ(s) = (s, s 2 , . . . , s n ) is not DT µ -improvable for some small constant 0 < µ < 1. Their proofs are based on the technique of regular systems introduced in [DS70].
Recently, based on an observation of Dani [Dan84], as well as Kleinbock and Margulis [KM98], Kleinbock and Weiss [KW08] studied this Diophantine approximation problem in the language of homogeneous dynamics, and proved the following result: for m = 1 and arbitrary n, if an analytic curve in M(1 × n, R) ∼ = R n satisfies some non-degeneracy condition, then almost every point on the curve is not DT µ -improvable for some small constant 0 < µ < 1 depending on the curve. Based on the same correspondence, Nimish Shah [Sha09b] proved the following stronger result: for m = 1 and general n, if an analytic curve ϕ : I = [a, b] → R n is not contained in a proper affine subspace, then almost every point on the curve is not DT -improvable. For m = n, Lei Yang [Yan13a] provided a geometric condition and proved that if an analytic curve ϕ : I = [a, b] → M(n × n, R) satisfies the condition, then almost every point on ϕ is not DT -improvable. The geometric condition given there provides some hint on solving the problem for general (m, n), and will be discussed in detail later.
The purpose of this paper is to give a geometric condition for each ( denote an analytic curve. For m = n, we say ϕ is generic at s 0 ∈ I if there exists a subinterval J s 0 ⊂ I such that for any s ∈ J s 0 , ϕ(s) − ϕ(s 0 ) is invertible.
A Lie subgroup L of H = SL(2m, R) is called observable if there exists a finite dimensional linear representation V of H and a nonzero vector v ∈ V such that the subgroup of H stabilizing v is equal to L. A Lie subalgebra l is called an observable Lie subalgebra of sl(2m, R), if it is the Lie algebra of some observable Lie subgroup L ⊂ H.
We say that ϕ is generic (supergeneric) or satisfies generic (supergeneric) condition, if ϕ is generic (supergeneric) at some s 0 ∈ I. Since ϕ is analytic, if it is generic (supergeneric) at one point of I then it will be generic (supergeneric) at all but finitely many points of I.
(1) In [Yan13a], it is proved that for m = n, if there exists s 0 ∈ I and a subinterval J s 0 ⊂ I such that the derivative ϕ (1) (s 0 ) is invertible, ϕ(s) − ϕ(s 0 ) is invertible for any s ∈ J s 0 , and {(ϕ(s) − ϕ(s 0 )) −1 : s ∈ J s 0 } is not contained in any proper affine subspace of M(m × m, R), then almost every point on the curve is not DT -improvable. Here the supergeneric condition is weaker than the condition needed in [Yan13a].
(2) In the case of m and n being co-prime, the generic condition directly implies the supergeneric condition.
(3) For m = 1 and general n, the genericness condition is equivalent to the condition that the curve is not contained in a proper affine subspace of R n .
In this paper we will prove the following result: Theorem 1.3. For any m and n, if an analytic curve is supergeneric, then almost every point on ϕ is not DT -improvable. If (m, n) = 1, then the same result holds for generic analytic curves.
1.2. Equidistribution of expanding curves on homogeneous spaces. Now we describe the correspondence between Diophantine approximation and homogeneous dynamics as follows. Let G = SL(m + n, R), and let Γ = SL(m + n, Z). The homogeneous space G/Γ can be identified with the space of unimodular lattices of R m+n . Every point gΓ corresponds to the unimodular lattice gZ m+n . For r > 0, let B r denote the ball in R m+n centered at the origin and of radius r. For any 0 < µ < 1, the subset contains an open neighborhood of Z m+n in G/Γ. Let us define the diagonal subgroup A = {a(t) : Now we consider the embedding Suppose for some 0 < µ < 1, and any N > 0 large enough, there exist nonzero integer vector p ∈ Z n and integer vector q ∈ Z m such that p ≤ µN m and Φp − q ≤ µN −n . Then the lattice a(log N )u(Φ)Z m+n has a vector a(log N )u(Φ)(−q, p) whose norm is less than µ, i.e., a(log N )u(Φ)Z m+n ∈ K µ for all N > 0 large enough. Thus, to show that Φ ∈ M(m × n, R) is not DT µ -improvable, it suffices to show that the trajectory {a(t)u(Φ)[e] : t > 0} meets K µ infinitely many times. In particular, for an analytic curve if we could show that for almost every point ϕ(s) on the curve the trajectory : t > 0} is dense, then we could conclude that almost every ϕ(s) is not DT -improvable. In particular, if one can prove that the expanding curves a(t)u(ϕ(I))[e] tend to be equidistributed in G/Γ as t → +∞, then (1.6) will follow (see [Sha09b] for detailed proof). It turns out that the equidistribution result described above holds for a much more general setting. In fact, one could prove the following result: Theorem 1.4. Let G be a Lie group containing H = SL(m + n, R), and Γ < G be a lattice of G. Let µ G denote the unique G-invariant probability measure on the homogeneous space G/Γ. Take Let ϕ : I = [a, b] → M(m × n, R) be an analytic curve. We embed the curve into H by For t > 0, let µ t denote the normalized parameteric measure on the curve a(t)u(ϕ(I))x ⊂ G/Γ; that is, for a compactly supported continuous function f ∈ C c (G/Γ), If ϕ is generic, then every weak- * limit measure µ ∞ of {µ t : t > 0} is still a probability measure. If the curve ϕ is supergeneric, then µ t → µ G as t → +∞ in weak- * topology; that is, for any function f ∈ C c (G/Γ), Moreover, if (m, n) = 1, then generic property will imply that µ t → µ G as t → +∞.
(1) To prove Theorem 1.3, we only need the above theorem with G = H = SL(n + m, R), Γ = SL(m + n, Z), and x = [e] = Z m+n ∈ G/Γ. (2) Even in the case G = H = SL(m + n, R), Theorem 1.4 is still much stronger than Theorem 1.3, since it applies for arbitrary lattice Γ < G.
The study of limit distributions of evolution of curves translated by diagonalizable subgroups in homogeneous spaces has its own interest and has a lot of interesting connections to geometry and Diophantine approximation. One could summarize this type of problems as follows: Problem 1.6. Let H be a semisimple Lie group, generated by its unipotent subgroups. Fix a diagonalizable one parameter subgroup A = {a(t) : t ∈ R} ⊂ H, and let U + (A) denote the expanding horospherical subgroup of A in H. Let G be a Lie group containing H, and let Γ be a lattice of G.
Let φ : I = [a, b] → H be a piece of analytic curve in H with nonzero projection on U + (A) (this will make sure that the translates of φ(I) by {a(t) : t > 0} expand). Given a point x = gΓ ∈ G/Γ, Ratner's Theorem tells that the closure of Hx is a finite volume homogeneous subspace F x, where F is a Lie subgroup of G containing H. Let µ F denote the unique probability F -invariant measure supported on F x. One can ask whether the expanding curves {a(t)φ(I)x : t > 0} tend to be equidistributed in F x, i.e., as t → +∞, the normalized parametric measure supported on a(t)φ(I)x approaches µ F in weak- * topology.
Remark 1.7. Without loss of generality, in this paper, we always assume that Hx is dense in G/Γ. If Hx is not dense, suppose its closure is F x, then we may replace G by F , Γ by F ∩ xΓx −1 (which is a lattice of F by the closeness of F x).
Nimish Shah [Sha09c] and [Sha09a] studied the case H = SO(n, 1) and G = SO(m, 1) where m ≥ n. In this case the diagonalizable subgroup A = {a(t) : t ∈ R} is a fixed maximal R-split Cartan subgroup of H. In [Sha09c] it is proved that given an analytic curve and a point x = gΓ ∈ G/Γ, unless the natural visual map Vis : SO(n, 1)/SO(n − 1) ∼ = T 1 (H n ) → ∂H n ∼ = S n−1 sends the curve φ(I) to a proper subsphere of S n−1 , the translates a(t)φ(I)x of φ(I)x will tend to be equidistributed as t → +∞. In [Sha09a], the same result is proved when φ is only C n differentiable. In [Sha09c] and [Sha09a], the obstruction of equidistribution is discussed and possible limit measures are described when the equidistribution fails. This result was generalized by Yang [Yan13b] in the following sense: for H = SO(n, 1) and arbitrary Lie group G containing H, if the same condition on the curve holds, then the expanding curve a(t)φ(I)x tends to be equidistributed as t → +∞. Shah [Sha09b] studied the case m = 1 of the problem we consider in this paper, and proved that if the analytic curve ϕ : I → M(1 × n, R) = R n is not contained in a proper affine subspace of R n , then the equidistribution holds. It turns out that this condition is the same as generic condition for m = 1. Later Yang [Yan13a] studied the case m = n.
When the generic condition holds but supergeneric condition does not, we want to understand the obstruction of equidistribution and describe the limit measures of {µ t : t > 0} to some extent. This requires more subtle argument. In [Sha09c] and [Sha09c], obstruction of equiditribution and description of limit measures are clearly given unconditionally for the case H = SO(n, 1) and G = SO(m, 1) in the set up of Problem 1.6. In our case, the problem becomes much more complicated. In this paper, we only discuss the case n = km, and we conjecture that similar result remains true for general (m, n) such that (m, n) > 1 (for the case (m, n) = 1, generic is the same as supergeneric, so there is nothing in between).
1.3. Relation to extremity of submanifolds in homogeneous spaces. Another direction to study Diophantine properties of a real matrix Φ ∈ M(m × n, R) is to determine whether Φ is very well approximable. We say Φ ∈ M(m × n, R) is very well approximable if there exists some constant δ > 0 such that there exist infinitely many nonzero integer vectors p ∈ Z n and integer vectors q ∈ Z m such that Φp − q ≤ p −n/m−δ . A submanifold U ⊂ M(m × n, R) is called extremal if with respect to the Lebesgue measure on U , almost every point is not very well approximable. Based on the same correspondence due to Dani [Dan84] and due to Kleinbock and Margulis [KM98], this problem can also be studied through homogenous dynamics. Kleinbock and Margulis [KM98] proved that if a submanifold U ⊂ M(1 × n, R) is nondegenerate, then U is extremal. Kleinbock, Margulis and Wang [KMW10] later gave a necessary and sufficient condition of a submanifold of M(m × n, R) being extremal. The condition is stated in terms of a particular representation of H = SL(m + n, R) and could not be translated to a geometric condition. Recently, Aka, Breuillard, Rosenzweig and de Saxcé [Aka+14] gave a family of subvarieties of M(m × n, R) called pencils, and announced a theorem stating that if a submanifold U ⊂ M(m × n, R) is not contained in a pencil, then U is extremal. It turns out that the generic condition implies the condition given in [Aka+14]. We will discuss it in detail in Appendix A.
1.4. Organization of the paper. The paper is organized as follows: In §2, assuming the generic condition on ϕ, we will relate a unipotent invariance to limit measures of {µ t : t > 0}, and show that every limit measure is still a probability measure. This allows us to apply Ratner's theorem. In §3 We will apply Ratner's theorem and the linearization technique to study the limit measure via a particular linear representation of H. Finally we will get a linear algebraic condition on ϕ. In §4, we will recall and prove some basic lemmas on linear representations, which are essential in our proof. In §5, assuming the supergeneric condition, we will give the proof of Theorem 1.4, as we have discussed before, Theorem 1.3 will follow from Theorem 1.4. In §6, assuming the generic condition, we will study the obstruction of equidistribution and limit measures of {µ t : t > 0}. We will only discuss the case n = km, and give a conjecture on general case. In the appendix, we will discuss the condition given in [Aka+14] and its relation to our generic condition.
Notation 1.8. In this paper, we will use the following notation.
For ǫ > 0 small, and quantities A and B, A ǫ ≈ B means that |A − B| ≤ ǫ. Fix a right G-invariant metric d(·, ·) on G, then for x 1 , x 2 ∈ G/Γ, and ǫ > 0, x 1 ǫ ≈ x 2 means x 2 = gx 1 such that d(g, e) < ǫ. For two related variable quantanties A and B, A ≪ B means there exists a constant C > 0 such that A ≤ CB, and A ≫ B means B ≪ A. O(A) denotes some quantity ≪ A or some vector whose norm is ≪ A.
Acknowledgement. The second author thanks The Ohio State University for hospitality during his visit when the project was initiated. Both authors thank MSRI where they collaborated on this work in Spring 2015. The second author thanks Dmitry Kleinbock for helpful discussions on the generic condition and for drawing his attention to the work of Aka, Breuillard, Rosenzweig and de Saxcé [Aka+14].
2. Non-divergence of the limit measures and unipotent invariance 2.1. Preliminaries on Lie group structures. We first recall some basic facts of the group H = SL(m + n, R). Without loss of generality, throughout this paper we always assume m ≤ n.
The centralizer of the diagonal subgroup A, Z H (A), has the following form: : B ∈ GL(m, R), C ∈ GL(n, R), and det B det C = 1 .
The expanding horospherical subgroup of A, U + (A) has the following form: Similarly, the contracting horospherical subgroup U − (A) has the following form: For any z ∈ Z H (A) and u(X) ∈ U + (A), zu(X)z −1 = u(zX) where z · X is defined as follows: Definition 2.1. For any X ∈ GL(m, R), we consider the following three elements in the Lie algebra h of H: Then {n + (X), n − (X −1 ), a} makes a sl(2, R)-triple; that is, they satisfy the following relations Therefore, there is an embedding of SL(2, R) into H that sends 1 1 0 1 to exp(n + (X)), 1 0 1 1 to exp(n − (X −1 )), and e t 0 0 e −t to exp(ta). We denote the image of this SL(2, R) embedding by SL(2, X) ⊂ H. Let us denote It is easy to see that σ(X) corresponds to 0 1 −1 0 ∈ SL(2, R).

Unipotent invariance.
Recall that for t > 0, µ t denotes the normalized parametric measure on the curve a(t)u(ϕ(I))x, and µ G denotes the unique G invariant probability measure on G/Γ. Our aim is to prove that µ t → µ G as t → +∞. We first modify the measures µ t to another measure λ t and show that if λ t → µ G , then µ t → µ G as well. Then we could study {λ t : t > 0} instead. The motivation for this modification is that any accumulation point of {λ t : t > 0} is invariant under a unipotent subgroup. The measure λ t is defined as follows: . Without loss of generality, we may assume that ϕ (1) (s) = 0 for all s ∈ I. Since ϕ is analytic, there exists some integer 1 ≤ b ≤ m, such that the derivative ϕ (1) (s) has rank b for all s ∈ I but finitely many points. Let E b (m) be the m by m matrix defined as follows: Given a closed subinterval J ⊂ I such that ϕ (1) (s) has rank b for all s ∈ J, we define an analytic curve z : Remark 2.3. For any subinterval J ⊂ I, we could similarly define µ J t to be the normalized parameter measure on a(t)u(ϕ(J))x.
Proposition 2.4. Suppose that for any closed subinterval J ⊂ I such that λ J t is defined, we have Proof. Let s 1 , s 2 , . . . , s l ∈ I be all the points where ϕ (1) (s) does not have rank b. For any fixed f ∈ C c (G/Γ) and ǫ > 0, we want to show that for t > 0 large enough, For each i ∈ {1, 2, . . . , l}, one could choose a small open subinterval B i ⊂ I containing s i such that and for any t > 0, Therefore, for t > T r , Then for t > max 1≤r≤p T r , we could sum up the above approximations for r = 1, 2, . . . , p and get Combined with (2.4) and (2.5), the above approximation implies that which is equivalent to Because ǫ > 0 can be arbitrarily small, we complete the proof.
By this proposition, if we could prove the equidistribution of {λ t := λ I t : t > 0} as t → +∞ assuming that ϕ (1) (s) has rank b for all s ∈ I, then the equidistribution of {µ t : t > 0} as t → +∞ will follow. Therefore, later in this paper, we will assume that ϕ (1) (s) has rank b for all s ∈ I and define λ t to be the normalised parametric measure on the curve {z(s)a(t)u(ϕ(s))x : s ∈ I}.
We will show that any limit measure of {λ t : t > 0} is invariant under the unipotent subgroup Proof. Given any f ∈ C c (G/Γ), and r ∈ R, we want to show that Since z(s) and ϕ(s) are analytic and defined on the closed interval I = [a, b], there exists a constant T 1 > 0 such that for t ≥ T 1 , z(s) and ϕ(s) can be extended to analytic curves defined on [a − |r|e −(m+n)t , b + |r|e −(m+n)t ]. Throughout the proof, we always assume that t i ≥ T 1 . Then z(s + re −(m+n)t i ) and ϕ(s + re −(m+n)t i ) are both well defined for all s ∈ I. From the definition of µ ∞ , we have We want to show that In fact, By the definition of z(s), we have the above is equal to When t i is large enough, u(O(e −(m+n)t i )) can be ignored. Therefore, for any δ > 0, there exists . Then from the above argument, we have for t i > T , It is easy to see that when t i > 0 is large enough, Therefore, for t i large enough, Since the above approximation is true for arbitrary ǫ > 0, we have that µ ∞ is W -invariant.
2.3. Non-divergence of limit measures. We also need to show that any limit measure µ ∞ of {λ t : t > 0} is still a probability measure of G/Γ, i.e., no mass escapes to infinity as t → +∞. To do this, it suffices to show the following proposition: Remark 2.7. In this proposition we only assume ϕ is generic.
This proposition will be proved via linearization technique combined with a lemma in linear dynamics as in [Sha09b].
Definition 2.8. Let g denote the Lie algebra of G, and denote d = dim G. We define i Ad(G). This defines a linear representation of G: Remark 2.9. In this paper, we will treat V as a representation of H.
The following theorem due to Kleinbock and Margulis is the basic tool to prove that there is no mass-escape when we pass to a limit measure: Theorem 2.10 (see [Dan84] and [KM98]). Fix a norm · on V . There exist finitely many vectors v 1 , v 2 , . . . , v r ∈ V such that for each i = 1, 2, . . . , r, the orbit Γv i is discrete, and moreover, the following holds: for any ǫ > 0 and R > 0, there exists a compact set K ⊂ G/Γ such that for any t > 0 and any subinterval J ⊂ I, one of the following holds: Remark 2.11. For the case ϕ(s) is polynomial curve, the proof is due to Dani [Dan84], for the case of analytic curve, the proof is due to Kleinbock and Margulis [KM98]. The crucial part of the proofs is to find some constants C > 0 and α > 0 such that in this particular representation, all the coordinate functions of a(t)u(ϕ(·)) are (C, α)-good.
Here a function f : I → R is called (C, α)-good if for any subinterval J ⊂ I and any ǫ > 0, the following holds: Notation 2.12. Let V be a finite dimensional linear representation of a Lie group F . Then for a one-parameter diagonal subgroup D = {d(t) : t ∈ R} of F , we could decompose V as the direct sum of eigenspaces of D; that is, We define and V −0 (D) respectively) with respect to the above direct sums.
The proof of Proposition 2.6 depends on the following property of finite dimensional representations of SL(m + n, R): is generic, then for any nonzero vector v ∈ V , there exists some s ∈ I such that A proof of this statement is one of the most important technical contributions of this paper, and we will postpone its proof to §4.
Proof of Proposition 2.6 assuming Lemma 2.13. Let V be as in Definition 2.8. Since A ⊂ H is a diagonal subgroup, we have the following decomposition: where V λ (A) is defined as in Notation 2.12. Choose the norm · on V to be the maximum norm associated to some choices of norms on V λ (A)'s.
For contradiction we assume that there exists a constant ǫ > 0 such that for any compact subset K ⊂ G/Γ, there exist some t > 0 such that λ t (K) < 1 − ǫ. Now we fix a sequence {R i > 0 : i ∈ N} tending to zero. By Theorem 2.10, for any R i , there exists a compact subset K i ⊂ G/Γ, such that for any t > 0, one of the following holds: S1. There exist γ ∈ Γ and j ∈ {1, . . . , r} such that sup s∈I a(t)u(ϕ(s))gγv j < R i ,
From our hypothesis, for each K i , there exists some t i > 0 such that S2. does not hold. So there exist γ i ∈ Γ and v j(i) such that By passing to a subsequence of {i ∈ N}, we may assume that v j(i) = v j remains the same for all i.
Since Γv j is discrete in V , we have t i → ∞ as i → ∞ and there are the following two cases: Case 1. By passing to a subsequence of {i ∈ N}, γ i v j = γv j remains the same for all i. Case 2. γ i v j → ∞ along some subsequence. For Case 1.: We have a(t i )u(ϕ(s))gγv j → 0 as i → ∞ for all s ∈ I. This implies that which contradicts Lemma 2.13. For Case 2.: After passing to a subsequence, we have By Lemma 2.13, let s ∈ I be such that u(ϕ(s))v ∈ V − (A). Then by (2.9) there exists δ 0 > 0 and Then which contradicts (2.8). Thus Cases 1 and 2 both lead to contradictions.
Remark 2.14. The same proof also shows that any limit measure of {µ t : t > 0} is still a probability measure, which is the non-divergence part of Theorem 1.4.

Ratner's theorem and linearization technique
Take any convergent subsequence λ t i → µ ∞ . By Proposition 2.5 and Proposition 2.6, µ ∞ is a W -invariant probability measure on G/Γ, where W is a unipotent one-parameter subgroup given by (6.2). We will apply Ratner's theorem and the linearization technique to understand the measure µ ∞ .
Notation 3.1. Let L be the collection of proper analytic subgroups L < G such that L ∩ Γ is a lattice of L. Then L is a countable set ( [Rat91]).
Now we want to apply the linearization technique to obtain algebraic consequences of this statement.
Notation 3.3. Let V be the finite dimensional representation of G defined as in Definition 2.8, for L ∈ L, we choose a basis e 1 , e 2 , . . . , e l of the Lie algebra l of L, and define From the action of G on p L , we get a map: Using the fact that ϕ is analytic, we obtain the following consequence of the linearization technique (cf. [Sha09c; Sha09b; Sha10]). SS1. |{s ∈ J : a(t)u(ϕ(s))gΓ ∈ O}| ≤ ǫ|J|. SS2. There exists γ ∈ Γ such that a(t)z(s)u(ϕ(s))gγp L ∈ Φ for all s ∈ J.
The following proposition provides the obstruction to the limiting measure not being G-invariant in terms of linear actions of groups, and it is a key result for further investigations.
Proposition 3.5. There exists a γ ∈ Γ such that Proof (assuming Lemma 2.13). By (3.2), there exists a compact subset C ⊂ N (L, W )) \ S(L, W ) and ǫ > 0 such that µ ∞ (CΓ) > ǫ > 0. Apply Proposition 3.4 to obtain D, and choose any Φ, and obtain a O so that either SS1. or SS2. holds. Since λ t i → µ ∞ , we conclude that SS1. does not hold for t = t i for all i ≥ i 0 . Therefore for every i ≥ i 0 , SS2. holds and there exists γ i ∈ Γ such that Since Γp L is discrete in V , by passing to a subsequence, there are two cases: In Case 1, since Φ is bounded in (3.4), we deduce that z(s)u(ϕ(s))gγp L ⊂ V −0 (A) for all s ∈ I.
In Case 2, by arguing as in the Case 2. of the Proof of Proposition 2.6, using genericness of ϕ and Lemma 2.13, we obtain that a(t i )u(ϕ(s))gγ i p L → ∞. This contradicts (3.4), because z(s) ⊂ Z H (A) and Φ is bounded. Thus Case 2 does not occur.
Our goal is to obtain an explicit geometric condition on ϕ(I) which implies that the linear algebraic condition (3.3) does not hold.
Similarly, for any s ∈ R, we have a(t)u − (s)a(−t) = u − (e −2t s), and hence Using the above relations (4.4), (4.5), (4.6) and (4.7), we get Further if there are all equalities in the above relation, then 4.1. Linear dynamical lemmas for SL(m + n, R) representations. First we give the proof of the basic lemma (Lemma 2.13) that we used more than once in previous sections. The new techniques developed in this section forms the core of this paper, and we expect these techniques to be valuable for other problems.
Proof of Lemma 2.13. We use induction to complete the proof. For the case m = n, the lemma is due to Yang [Yan13a]. We provide a proof here for the sake of self-containedness.
If m > n then by applying a suitable inner automorphism of SL(m + n, R) given by a coordinate permutation σ m,n , we can covert this problem to the case of m < n. Therefore we will assume that m < n.
As inductive hypothesis, we assume that for all (m ′ , n ′ ) such that m ′ ≤ m, n ′ ≤ n and m ′ + n ′ < m + n, the result holds. We want to prove the result holds for (m, n).
Since ϕ is analytic, we have µ 0 (s) = µ 0 for all but finitely many s ∈ I. Also by our assumption we have that (4.8) µ 0 < 0.
For any fixed s ∈ J s 0 , it is straightforward to verify that We express V as the direct sum of common eigenspaces of A 1 and A 2 :

Let us denote
Then because a(t) = a 1 (nt)a 2 (t), we have For any vector v ∈ V , let v δ 1 ,δ 2 denote the projection of v onto the eigenspace V δ 1 ,δ 2 . We also decompose V as the direct sum of irreducible sub-representations of A ⋉ SL(2, ϕ 1 (s)). For any such sub-representation W ⊂ V , let p W : V → W denote the A-equivariant projection. By the theory of finite dimensional representations of SL(2, R), there exists a basis {w 0 , w 1 , . . . , w r } of W such that We claim that each w i is also an eigenvector for A. In fact, and hence b(t) acts on W as a scaler e δt for some δ ∈ R. Therefore, (4.14) a(t)w i = e ((r−2i)(m+n)/2+δ)t w i , for 1 ≤ i ≤ r.
For supergeneric curves, we want to obtain the following stronger conclusion.
for all s ∈ I, then V is a trivial representation.
Proof of Lemma 4.3. The strategy of the proof is similar to that of Lemma 2.13.
We begin with the case m = n. This case is studied in [Yan13a] but the statement proved there is weaker than the statement here.
Taking any s 1 , s 2 ∈ J s 0 , we have This shows that v 0 (A) is fixed by u − (ϕ −1 (s 1 ) − ϕ −1 (s 2 )) for all s 1 , s 2 ∈ J s 0 . By definition, v 0 (A) is also fixed by A. Let L denote the subgroup of H stabilizing v 0 (A), and l denote its Lie algebra. Then from the above argument we have l is observable and contains E ∈ Lie(A) (see (1.2)) and recall that earlier we had replaced ϕ(s) by ϕ(s) − ϕ(s 0 ) and assumed that ϕ(s 0 ) = 0 for notational simplicity. Because ϕ is supergeneric, in view of (1.3) we have that L = H. Since V is an irreducible representation of H, V is trivial. This finishes the proof for m = n.
For the general case we give the proof by an inductive argument. Suppose the statement holds for all (m ′ , n ′ ) such that m ′ ≤ m, n ′ ≤ n and m ′ + n ′ < m + n. We want to prove the statement for (m, n).
We choose a point s 0 and a subinterval J s 0 ⊂ I such that the following statements hold: (1) If we write ϕ(s) = [ϕ 1 (s); ϕ 2 (s)] where ϕ 1 (s) is the first m by m block, and ϕ 2 (s) is the rest m by n − m block, then for any s ∈ J s 0 , ϕ 1 (s) − ϕ 1 (s 0 ) is invertible. (2) The curve ψ(s) = (ϕ 1 (s) − ϕ 1 (s 0 )) −1 (ϕ 2 (s) − ϕ 2 (s 0 )) is supergeneric as a curve from J s 0 to M(m × (n − m), R). Without loss of generality we may assume that ϕ(s 0 ) = 0 and v ∈ V −0 (A). The notations such that u ′ (·), A 2 and v µ 0 (A) have the same meaning as in the proof of Lemma 2.13. Using the same argument as the proof of Lemma 2.13, we could deduce that for all s ∈ J s 0 . By inductive hypothesis, we conclude that v µ 0 (A) is fixed by the whole In particular, v µ 0 (A) is fixed by A 2 . Let the direct sum be as in the proof of Lemma 2.13. From the proof of Lemma 2.13 we know that any nonzero projection (v µ 0 (A)) δ 1 ,δ 2 of v µ 0 (A) with respect to this direct sum satisfies δ 1 (the eigenvalue for A 1 ) is non-negative. Because we have δ 2 = 0 and nδ 1 + δ 2 ≤ 0, we conclude that δ 1 = δ 2 = 0. This implies that µ 0 = 0. By Lemma 4.4, we have v 0 (A) is invariant under {u(hϕ (1) (s 0 )) : h ∈ R}. By our assumption, ϕ (1) (s 0 ) has rank b. By conjugating it with elements in H ′ , we have that u(X) fixes v 0 (A) for any X with rank b. Note that the space spanned by all rank b matrices is the whole space M(m × n, R). This shows that v 0 (A) is invariant under the whole U + (A). Since v 0 (A) is also invariant under A, v 0 (A) is invariant under the whole group H. Since we assume that V is an irreducible representation of H, we conclude that V is trivial. This completes the proof.
Lemma 4.3 is sufficient to prove the equidistribution result under the supergeneric condition. Now we consider the case n = km and the curve is generic but not supergeneric. In this case, we will prove the following result, which can be thought of as a generalization of Corollary 4.2.
then for all s 0 ∈ I satisfying the generic condition, we have This lemma is crucial for describing the obstruction to equidistribution for generic curves as done in §6. In order to prove Lemma 4.5, we will need the following lemma. Proof. Replacing ϕ(s) by ϕ(s) − ϕ(s 0 ), we may assume that ϕ(s 0 ) = 0.
We will prove the statement by induction on k.
As before, let us denote for s ∈ J s 0 . Now we fix a point s 1 ∈ J s 0 and a subinterval J s 1 ⊂ J s 0 such that ψ satisfies the generic condition for s 1 and J s 1 . Replacing ϕ by u ′ (ψ(s 1 )) · ϕ, we get ϕ(s 1 ) = [ϕ 1 (s 1 ); 0; . . . ; 0] and ψ(s 1 ) = 0. Let z 1 denote the following element: By direct calculation, we have that z 1 · ϕ is standard at s 0 with given s 1 , s 2 , . . . , s k . This completes the proof. Now we are ready to prove Lemma 4.5.
Proof of Lemma 4.5. By Lemma 4.7, we may conjugate the curve by some z ′ (s 0 ) ∈ Z H (A), such that the conjugated curve, which we still denote by ϕ, satisfies the following: there exist s 1 , s 2 , . . . , s k ∈ I, such that Replacing v by u(ϕ(s 0 ))v and ϕ(s) by ϕ(s)−ϕ(s 0 ), we may assume that ϕ(s 0 ) = 0 and v ∈ V −0 (A).
For each i = 1, 2, . . . , k, let where e −t I m appears in the (i + 1)-th m × m diagonal block. We denote its Lie algebra by where A i := log a i (1). Let SL(2, ϕ(s i )) denote the SL(2, R) copy in H containing A i as the diagonal subgroup and {u(rϕ(s i )) : r ∈ R} as the upper triangular unipotent subgroup.

Therefore,
This proves (4.17). Let A := log a(1), it is easy to see that A = A 1 + · · · + A k . Therefore, On the other hand, because A 1 , . . . , A k normalize SL(2, ϕ(s i )) for any i = 1, . . . , k, we can decompose V into the direct sum of irreducible representations V p of SL(2, ϕ(s i )) which are invariant under A 1 , . . . , A k : For each V p , we can choose a basis {w 0 , w 1 , . . . , w l }, called a standard basis, of V p such that for each 1 ≤ r ≤ l, w r is contained in some weight space V (δ 1 , δ 2 , · · · , δ k ), and we index the basis elements such that a i (t)w r = e (l−2r)t w r ; that is, if w r ∈ V (δ 1 , δ 2 , · · · , δ k ) then δ i = l − 2r.
Moreover since n(ϕ(s i ))w s is a nonzero multiple of w s−1 for 1 ≤ s ≤ l, by (4.17) we have that (4.20) w r−j ∈ V (δ − je i ), for r − l ≤ j ≤ r.
Let π p : V → V p denote the canonical projection from V to V p with respect to (4.18), and let denote the canonical projection from V to V (δ) with respect to (4.16). Then We call a vector δ ∈ Z k admissible if it can be written as c 1 e 1 + c 2 e 2 + · · · + c k e k , where c 1 , c 2 , . . . , c k are non-negative integers.
Let 1 ≤ i ≤ k be such that δ i = min(δ 1 , . . . , δ k ). Then For this choice of i, consider the decomposition (4.18) of V as V = p V p with respect to the action of SL(2, ϕ(s i )). There exists some V p such that π p (q(δ)v) = 0. If {w 0 , w 1 , . . . , w l } denotes the standard basis of V p , then by (4.19), π p (q(δ)v) is a nonzero multiple of w r for some 0 ≤ r ≤ l such that δ i = l − 2r.
Therefore by (4.1) in Lemma 4.1 applied to V p and the action of SL(2, ϕ i (s i )), we have that π p (u(ϕ i (s i ))v) = u(ϕ i (s i ))π p (v) has a nonzero coefficient on w r−j for some j such that a i (t)w r−j = e (δ i +2j)t w r−j and δ i + 2j ≥ −δ i .
(1) Though our proof works for the special case n = km, we conjecture that the conclusion of Lemma 4.5 should hold for the case of general (m, n).
(2) From the proof we can see, if we assume ϕ(s 0 ) = 0 and v ∈ V −0 (A), then z ′ (s 0 ) · v 0 (A) is fixed by which is the diagonal subgroup generated by A 1 , A 2 , . . . , A k .

Proof of the equidistribution result
In this section we will prove Theorem 1.4. The non-divergence part of the theorem has been proved in §2 (see Remark 2.14). Here we will prove the equidistribution part. The proof is based on Proposition 3.5 and Lemma 4.3.
Proof of Theorem 1.4. Suppose ϕ : I → M(m×n, R) is supergeneric, and the normalized parametric measures {λ t : t > 0} do not tend to the Haar measure µ G along some subsequence t i → +∞. By Proposition 3.5, there exists some L ∈ L and γ ∈ Γ such that u(ϕ(s))gγp L ∈ V −0 (A) for all s ∈ I. Then by Lemma 4.3, we have that v := gγp L is fixed by the whole group H. Hence p L is fixed by the action of γ −1 g −1 Hgγ. Thus This implies G 0 p L = p L where G 0 is the connected component of e. In particular, γ −1 g −1 Hgγ ⊂ G 0 and G 0 ⊂ N 1 G (L). By [Sha09a, Theorem 2.3], there exists a closed subgroup F 1 ⊂ N 1 G (L) containing all Ad-unipotent one-parameter subgroups of G contained in N 1 G (L) such that F 1 ∩ Γ is a lattice in F 1 and π(F 1 ) is closed. If we put F = gγF 1 γ −1 g −1 , then H ⊂ F since H is generated by it unipotent one-parameter subgroups. Moreover, F x = gγπ(F 1 ) is closed and admits a finite F -invariant measure. Then since Hx = G/Γ, we have F = G. This implies F 1 = G and thus L ⊳ G. Therefore N (L, W ) = G. In particular, W ⊂ L, and thus L ∩ H is a normal subgroup of H containing W . Since H is a simple group, we have H ⊂ L. Since L is a normal subgroup of G and π(L) is a closed orbit with finite L-invariant measure, every orbit of L on G/Γ is also closed and admits a finite L-invariant measure, in particular, Lx is closed. But since Hx is dense in G/Γ, Lx is also dense. This shows that L = G, which contradicts our hypothesis that the limit measure is not µ G .
This completes the proof.

Obstruction of equidistribution
We will study the obstruction of equidistribution of the expanding curves {a(t)u(ϕ(I))x : t > 0} as t → +∞ and describe limit measures if equidistribution fails.
At first, if the generic condition fails, then it is possible that the limit measure µ ∞ of {λ t : t > 0} along some subsequence t i → ∞ is not a probability measure. In other words, part of the expanding curves might escape to infinity along this subsequence. In fact, for G = H = SL(m + n, R) and Γ = SL(m + n, Z), it is not hard to construct some special curve ϕ such that a(t)u(ϕ(I))[e] → ∞ as t → ∞. We refer the reader to [KW08] and [KMW10] for more examples. In this paper, we focus on generic curves.
If m and n are co-prime, the generic condition is the same as the supergeneric condition, so there is nothing to discuss in this case.
Therefore we consider the case (m, n) > 1 and the analytic curve is generic but not supergeneric. However, for now, we could only handle the case n = km where k > 1 is some positive integer (to handle general (m, n), we need some general version of Lemma 4.5).
With these assumptions, we want to describe the obstruction of equidistribution of {a(t)u(ϕ(I))x : t > 0} as t → ∞.
In this section, we always assume that n = km and the analytic curve is generic. First note that for L ∈ L, the stabilizer of p L is N 1 G (L) where N 1 G (L) := {g ∈ G : gLg −1 = L and det(Ad(g)| l ) = 1}. Theorem 6.1 (See [Sha09c, Proposition 4.9]). Suppose the expanding curves a(t)u(ϕ(s))x are not tending to be equidistributed along some subsequence t i → +∞. By Proposition 3.5, there exist L ∈ L and γ ∈ Γ such that u(ϕ(s))gγp L ∈ V −0 (A), for all s ∈ I. Then there exist h ∈ P − (A) and some algebraic subgroup F of G containing A such that h −1 F h fixes v := gγp L and u(ϕ(s)) ∈ P − (A)F h for all s ∈ I. Recall that P − (A) = U − (A)Z H (A) denotes the maximal parabolic subgroup of H associated with A.
Proof. Let s 0 ∈ I such that every point in a neighborhood J of s 0 satisfies the generic condition.
Remark 6.2. It may be noted that since by a result of Dani and Margulis, the orbit Γp L is discrete, we have that ΓF 1 is closed in G. Therefore for h = ξ(s 0 ) ∈ P − (A), we have F hgΓ = (hgγ)F 1 Γ is closed. Thus there exist analytic curves ξ − : I → U − (A) and ξ 0 : I → Z H (A) such that u(ϕ(s))x ⊂ ξ − (s)ξ 0 (s)F hx for all s ∈ I. Then as t → ∞, the distance between a(t)u(ϕ(s))x and ξ 0 (s)F hx tends to zero, since F hx = F hgΓ is a proper closed {a(t)}-invariant subset of G/Γ. Thus every limit measure of the sequence {µ t } as t → +∞ is a probability measure whose support is contained in ξ 0 (I)F hx. Replacing F by a smaller subgroup containing A, we can actually ensure that F hx admits a finite F -invariant measure.
We conjecture that in the general case of (m, n) > 1, if ϕ is generic, then Lemma 4.5, Theorem 6.1, and Remark 6.2 should hold.
Appendix A. Relation between the condition given in [Aka+14] and the generic condition It is worth explaining the condition given in [Aka+14] and its relation to our generic condition. We denote M (s) := [I m ; ϕ(s)] ∈ M(m × (m + n), R). Given a subspace W and 0 < r < m dim W m+n , we define the pencil P W,r to be P W,r := {M ∈ M(m × (m + n), R) : dim M W = r}.
In [Aka+14], the following theorem is announced: if a submanifold is not contained in any such pencil, then the submanifold is extremal. In our case, it says that if the curve {M (s) : s ∈ I} is not contained in any pencil P W,r , then the curve is extremal. It is easy to see that if W is a rational subspace, then P W,r is not extremal. So this condition is considered almost optimal. But this contradicts our inductive assumption for case (m, n − m). In fact, W 0 ⊂ R m+(n−m) , N (s) = [I m ; ψ(s)] where the curve ψ(s) ∈ M(m × (n − m), R) is generic. Thus to apply the inductive assumption for (m, n − m), it suffices to check that r ′ < m dim W 0 n . Since r ′ ≤ r, we only need to show that r < m dim W 0 n . The inequality is equivalent to nr < m dim W 0 = m(dim W − r).
It is straightforward to check that it is the same as r < m dim W m + n , which is our assumption. This allows us to apply the inductive assumption and conclude the contradiction. This completes the proof.
Therefore the generic condition implies the condition given in [Aka+14].