EQUIDISTRIBUTION OF DILATED CURVES ON NILMANIFOLDS

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1 EQUIDISTRIBUTION OF DILATED CURVES ON NILMANIFOLDS BRYNA KRA, NIMISH A. SHAH, AND WENBO SUN Abstract. Generalizing classic results for a family of measures in the torus, for a family (µ t ) t of measures defined on a nilmanifold, we study conditions under which the family equidistributes, meaning conditions under which the measures µ t converge as t in the weak topology to the Haar measure on. We give general conditions on a family of measures defined by a dilation process, showing necessary and sufficient conditions for equidistribution as the family dilates, along with conditions such that this holds for all dilates outside some set of density zero. Furthermore, we show that these two types of equidistribution are different.. Introduction.. Limiting distributions of measures. A classic problem for billiards is of illumination: in a polygonal room, a light source is located at some point and the question is if there is some point not illuminated by this source. Chaika and Hubert [5] recently studied a related problem for circles of light, rather than points, showing that dilated circles around a fixed point weakly equidistribute outside a set of density zero (in their terminology, this phenomenon is called weak illumination). Motivated by their work, we prove weak and strong equidistribution results for a dilated family of measures on a nilmanifold. The study of equidistribution results on a nilmanifold originates in the work of Green [], where he showed that a flow is either equidistributed or there is a nontrivial obstruction to this flow arising from horizontal character on the nilmanifold. More precise distributional results were obtained by Shah [2], who described the iting behavior for a polynomial series of iterates in a flow, and by Leibman [6], who proved convergence results for polynomial sequences of iterates. The asymptotic behavior of dilates of a measure supported on a curve in a nilmanifold was studied by Björklund and Fish [4], and among other The first author was partially supported by NSF grant DMS-567 and the second author was partially supported by NSF grant DMS-7394.

2 2 BRYNA KRA, NIMISH A. SHAH, AND WENBO SUN results we answer a conjecture of theirs about the general behavior of such dilates (this result is given in Theorem.2). For example, we consider a family of measures that are linear expansions on the Lie algebra g associated to some nilpotent Lie group G. A sample result is necessary and sufficient conditions that this family equidistributes as it dilates, and whether or not these equidistribution results for all sufficiently large dilates or only for all dilates outside a set of density zero depends on the derivative of the dilation. The simplest case of our results is for a torus, where such strong equidistribution results for a curve are implicit in the literature (though we are unaware of an explicit result like this). For a nilmanifold, if we consider a continuous curve with the induced measure on this curve, it is easy to check that the dilates equidistribute weakly so long as the curve contains no linear segment of positive length (this corresponds to a nontrivial horizontal character picking up positive measure). More general equidistribution results on a nilmanifold, covering a broader class of dilations, require significantly more work and these characterizations are our main focus. To give the precise formulations of our results in Sections.6,.7, and.8, we start by defining the objects that give us sufficient (and in some cases necessary) conditions for equidistribution..2. Equidistribution and weak equidistribution. Let = G/Γ be a compact nilmanifold, meaning that G is a simply connected nilpotent Lie group and Γ G is a cocompact closed subgroup of G. The group G acts on = G/Γ by left translation, and there is a unique G-invariant Borel probability measure µ on (the Haar measure). A family of probability measures (µ t ) t on is equidistributed on if µ t converges in the weak topology as t to the Haar measure µ of. Letting C() denote the space of continuous functions on, this means that the family of measures (µ t ) t is equidistributed if for all f C(), f dµ t = f dµ. (.) t Throughout, we let λ denote the Lebesgue measure. The family of measures (µ t ) t is said to be weakly equidistributed on if the convergence in (.) holds for t in a set of asymptotic density in [, ), meaning there exists A [, ) such that λ(a [, T ]) T T =

3 EQUIDISTRIBUTION OF DILATED CURVES 3 and for all f C(), f dµ t = f dµ. t t A If we consider discrete time, then the family (µ n ) n N of measures is weakly equidistributed on if the convergence in (.) holds for a set of parameters n with asymptotic density, meaning along a set A N A {,...,N} satisfying N =, where denotes the cardinality of N the set. Equivalently, (µ t ) t is weakly equidistributed if and only if for every f C() with f dµ =, T 2 f dµ t dt =, (.2) T T and for discrete iterates (µ n ) n N, this becomes N N n= f dµ n 2 =. (.3) While it is clear that strong equidistribution implies weak equidistribution, for a family of measures in a nilmanifold these two notions are not equivalent. Various examples illustrating the difference are given in Section Dilation associated to the Lie algebra. We note that Ad(Γ) is a Zariski dense subgroup of Ad(G), see [, Chapter 2]. Since Γ normalizes its connected component of the identity, say Γ, we have that Γ is a normal Lie subgroup of G. We note that Γ acts trivially on G/Γ, so G acts on = G/Γ via G/Γ, or we can say that G/Γ acts on = G/Γ as follows: (gγ ) x = gx, for all g G and x. Let g denote the Lie algebra associated to the nilpotent Lie group G. Then g/ Lie(Γ ) is the Lie algebra associated to G/Γ. We define a family of dilations (ρ t ) t R such that ρ t : g g/ Lie(Γ ) is a linear transformation, and each matrix entry of ρ t with respect to any bases of g and g/ Lie(Γ ) is a polynomial in t. In other words, for some m and for some linear maps B j : g g/ Lie(Γ ), j m, we have ρ t = m j= tj B j for all t R.

4 4 BRYNA KRA, NIMISH A. SHAH, AND WENBO SUN.4. Dynamics of measures under dilations. Let ν be a probability measure on g and let x G/Γ. For t, let µ ν,x,ρ t denote the measure on defined as follows: for any f C(), f dµ ν,x,ρ t = f(exp ρ t (y) x ) dν(y), (.4) where exp : g/ Lie(Γ ) G/Γ is the exponential map from Lie algebra to Lie group. An interesting example of ν is as follows: Given a measurable map φ: (, ) g, we can take ν to be the pushforward of the Lebesgue measure λ restricted to (, ) under φ. Then for t, taking µ φ,x,ρ t to be the measure on defined by f dµ φ,x,ρ t = f(exp ρ t φ(u) x ) dλ(u) (.5) for all f C(), then µ φ,x,ρ t = µ ν,x,ρ t..5. Abelianization of a nilsystem. Let [G, G] denote the Lie subgroup of G corresponding to the commutator algebra [g, g]. For a cocompact, closed subgroup Γ of G, we have that [G, G]Γ is a closed subgroup of G, and := G/[G, G]Γ can be identified with a compact torus R m /Z m for some m (see for example [, Corollary of Theorem 2.3]). We call the abelianization of and use t = R m to denote the associated Lie algebra of and q : = G/Γ = G/[G, G]Γ to denote the natural quotient map. Let dq : g/ Lie(Γ ) t denote the differential of the map q at the identity coset. Then q(exp(y) x ) = exp dq(y) + q(x ) (.6) for all y g/ Lie(Γ ) and all x. Then dq ρ t : g t can be expressed as d dq ρ t = t i A i, (.7) where d N and each A i : g t is a linear map. When Γ is discrete; that is, Γ = {e}, a natural choice for a family of dilations is to consider (ρ t ) t R, where ρ t v = tv for all v g. In this case, d =, A = dq, and A =. Let denote the space of (continuous) unitary characters on the torus. Then elements of are in one-to-one correspondence with unitary characters on G such that the kernel of the map contains Γ. i=

5 EQUIDISTRIBUTION OF DILATED CURVES 5 For any χ, let dχ: t R denote the differential of χ, meaning that for y t, χ(y + Z m ) = e 2πidχ(y). (.8).6. Weak equidistribution of dilated measures. Our first result provides necessary and sufficient conditions for weak equidistribution of the family (µ t := µ ν,x,ρ t ) t. Theorem.. Let ν be a probability measure on g, x, and (µ t := µ ν,x,ρ t ) t be the family of measures defined in (.4). If for all χ \ {} and (z,..., z d ) R d we have ν({v g: dχ(a i v) = z i for all i d }) =, (.9) then the families of measure (µ t ) t and (µ n ) n N are weakly equidistributed on. Furthermore, if A =, then condition (.9) is necessary for weak equidistribution of (µ n ) n N, and also for (µ t ) t..7. Equidistribution of dilated analytic curves. We maintain the same notation in this section. Carrying out the constructions of the family of measures in (.5) with an analytic map φ: (, ) g, we resolve a conjecture stated in Björklund and Fish [4] (see the discussion following Theorem 7 in their paper): Theorem.2. Assume φ: (, ) g is analytic, let x, and let (µ t = µ φ,x,ρ t ) t R be as defined in (.5). If for every χ \{}, there exists i d such that the map u dχ(a i φ(u)) (.) is not constant, then (µ t ) t is equidistributed on. Furthermore, if A =, then the conditions on the maps given in (.) are also necessary for the equidistribution of (µ t ) t on..8. Equidistribution of dilated differentiable curves. Our main result of this article provides a condition on tangents to the curve for (strong) equidistribution, when φ is a sufficiently differentiable curve. Still maintaining the assumptions and notation stated at the beginning of this section, we have: Theorem.3. Assume that = G/Γ is a compact nilmanifold. There exists a natural number D, which can be expressed in terms of the degrees of polynomials in t defining ρ t such that the following holds: suppose that φ (D) (u) exists for (Lebesgue) almost all u (, ) and that for every χ \ {} and (Lebesgue) almost every u (, ), A i φ () (u) / ker dχ (.)

6 6 BRYNA KRA, NIMISH A. SHAH, AND WENBO SUN for some i d. Then the family of measures (µ φ,x,ρ t ) t is equidistributed on G/Γ. Our proof is based on a stronger equidistribution result for dilates of curves that shrink (the precise statement is given in Theorem 3.). The following special case is of interest: Corollary.4. Let = R n /Z n and φ: (, ) R n be an almost everywhere differentiable map. Suppose that for each v Z n \ {}, φ () (u) is not orthogonal to v for (Lebesgue) almost every u (, ). Then for any f C(R n /Z n ) and x R n /Z n, t f(tφ(s) + x ) dλ(s) = R m /Z m f dµ, where λ denotes the Lebesgue measure on [, ] and µ the Haar measure on R n /Z n. For analytic φ, the conclusion of Corollary.4 was obtained in Randol [9]. In Section 5 we provide examples where condition in (.9) holds and so (µ t ) t is weakly equidistributed on, but (µ t ) t is not equidistributed on. 2. Weak equidistribution for measures This section is devoted to proving Theorem.. We give a detailed proof for the discrete case, and then indicate the modifications needed for the continuous setting. The classic equidistribution theorem in the discrete setting is due to Weyl: Theorem 2. (Weyl [3]). Let p(t) = a d t d + + a t + a be a real polynomial. Then e 2πip(n) = N N if and only if at least one of a,..., a d is not rational. If G is a nilpotent Lie group, a,..., a m G, and p,..., p m : N N are polynomials taking integer values on the integers, then a sequence (g(n)) n N in G of the form a p (n) a p 2(n) 2... am pm(n) is a polynomial. If g denotes the Lie algebra of G, we say that ζ : N g is a polynomial map if exp(ζ(n)) = g(n) for some polynomial (g(n)) n N. Generalizing Weyl s equidistribution result, Leibman [6] showed (we write his result in terms of the Lie algebra):

7 EQUIDISTRIBUTION OF DILATED CURVES 7 Theorem 2.2 (Leibman [6]). Let ζ : N g be a polynomial map such that for every nontrivial character χ on, dχ dq ζ(n) dχ dq ζ() mod Q for some n N. (2.) Then for any f C() and any x, N N f(exp ζ(n) x) = f dµ. We make use of these results in the proof of the weak equidistribution result: Proof of Theorem. (in the discrete setting). First we prove the necessity of condition (.9) for weak equidistribution. Assume that A = and suppose that (.9) fails to hold, and we want to prove that the families (µ t ) t and (µ n ) n N are both not weakly equidistributed. Thus there exist a nontrivial character χ on and z,..., z d R such that ν({v g: dχ(a j v) = z j for all j d }) (2.2) is positive. Since χ is a nontrivial character on the torus and its pushforward q µ is the translation invariant (Haar) probability measure on the compact torus, we have χ q dµ = χ d(q µ) =. (2.3) Therefore in view of (.3), in order prove that (µ n ) n N is not weakly equidistributed, we need to show that sup N N Therefore it suffices to prove that M N N χ q dµ n 2 >. χ q dµ M!n 2 >, (2.4) because for any sequence {a n } of non-negative reals, and M N, M!N M!N a n M! N a M!n.

8 8 BRYNA KRA, NIMISH A. SHAH, AND WENBO SUN For all t R, χ q dµ t = χ(q(exp ρ t (y) x )) dν(y) g = χ(q(x )) χ(dq ρ t (y) + Z m ) dν(y) g = χ(q(x )) e 2πidχ(dq ρt(y)) dν(y) g = χ(q(x )) e 2πi d j= tj dχ(a j y) dν(y), g where we have used equations (.6), (.8), and (.7). Writing y j := dχ(a j y) and using the assumption that A =, this last quantity is the same as χ(q(x )) e 2πi d j= tj y j dν(y). g Combining this expression for χ q dµ t with the fact that χ(q(x )) =, we have that M N = = N M N g g [ N M N N χ q dµ M!n 2 g g e 2πi(nM!)j d j= (y j y j ) d(ν ν)(y, y ) e 2πi d j= nj ((M!) j (y j y j )) ] d(ν ν)(y, y ). We claim that this integral over g g is the same as integrating the indicator function of the set W = {(y, y ) g g : y j y j mod Q for all j d }. To see this, fix y, y g. If y j y j mod Q for some j d, by Weyl Equidistribution (Theorem 2.) the it equals to. If y j = y j mod Q for all j d, then (M!) j y j = (M!) j y j mod Z for all j d and all M sufficiently large, and so the it as N equals to.

9 Thus N g N EQUIDISTRIBUTION OF DILATED CURVES 9 χ q dµ n 2 = W d(ν ν)(y, y ) g g = ν({y g: dχ(a j y) y j mod Q for all j d }) dν(y ) = w,...,w d R ν({y g: dχ(a j y) w j mod Q for all j d }) ν({y g: dχ(a j y ) = w j for all j d }) ν({v g: dχ(a j v) = z j for all j d }) 2. By (2.2), this is strictly positive and so (2.4) holds, which completes the proof that (µ n ) n N is not weakly equidistributed on. The converse implication. Assume that condition (.9) of Theorem. holds. Our goal is to show that (.3) holds for the family (µ n := µ ν,x,ρ n ) n N and any given f C() with f dµ =. We have: N N = N N = N N [ = g g [ N f dµ n 2 ] f f d(µ n µ n ) [ ] f(exp ρ n (y) x )f(exp ρ n (y ) x ) d(ν ν)(y, y ) g g ] f N f(exp ζ y,y (n) (x, x )) d(ν ν)(y, y ), where ζ y,y : N g/ Lie(Γ ) g/ Lie(Γ ) is the map given by (2.5) ζ y,y (t) = (ρ t (y), ρ t (y )) for all t N. (2.6) From the definition of (ρ t ) t R, it follows that (exp ζ y,y (n)) n N polynomial on (G/Γ ) (G/Γ ). is a

10 BRYNA KRA, NIMISH A. SHAH, AND WENBO SUN For every (χ, χ ) \ {(, )}, we have d(χ χ ) = dχ d χ, and by (.7), (dχ d χ ) ζ y,y (n) = dχ(dq ρ n (y)) dχ (dq ρ n (y )) d = [dχ(a j y) dχ (A j y )]n j + dχ(a y) dχ (A y ). j= Therefore in view of condition (2.) of Theorem 2.2, (dχ d χ ) ζ y,y (n) dχ(a y) dχ (A y ) for all n Z is equivalent to the condition that dχ(a j y) dχ (A j y ) mod Q mod Q for all j d. In order to apply Theorem 2.2 to ζ y,y and, define B to be the set of (y, y ) g g such that for all (χ, χ ) \ {(, )}, there exists j {,..., d } such that dχ(a j y) dχ(a j y ) mod Q. Define C to be the set of y g such that for all χ \ {}, there exists j {,..., d } such that dχ(a j y) Q and define E y = {y g: (y, y ) B}. Since Q is countable, by the hypothesis given in (.9), we have that ν(c) =. Also B y C{y} (g \ E y ). (2.7) For any y C, χ, χ \ {}, define E y,χ,χ to be the set {y g: dχ (A j y ) dχ(a j y) mod Q for all j d }. Then since Q is countable, by the hypothesis (.9), we have that ν(e y,χ,χ ) =. (2.8) Suppose that y C, y E y, and (χ, χ ) \ {(, )} are such that dχ(a j y) dχ (A j y ) mod Q for all j d. (2.9) By definition of the set C, we can choose j with j d such that dχ(a j y) / Q. Then by (2.9), χ (A j y ) Q. Therefore χ. By definition of the set E y, E y = E y,χ,χ. (2.) χ \{} χ

11 EQUIDISTRIBUTION OF DILATED CURVES Since is countable, by (2.8), we have ν(e y ) =. Hence by (2.7), (ν ν)(b) =. By Theorem 2.2 applied to, for any f C() with f dµ =, for all (y, y ) B, we have N N f f(exp ζ y,y (n) (x, x )) =. Now since (ν ν)((g g) \ B) =, it follows from (2.5) that N = N g g [ f(exp ρ n (y) x ) dν(y) N g N Thus (µ n ) n N is weakly equidistributed. 2 f f(exp ] ζ y,y (n) (x, x )) d(ν ν)(y, y ) =. The case of continuous parameter. The proof in this case is similar. Instead of using the discrete version of Weyl s Theorem to show necessity, we use: Theorem 2.3 (Weyl [3]). Let p(t) = a d t d + + a t + a be a polynomial with real coefficients. Then T T T e 2πip(t) dt = if and only if at at least one of a,..., a d is nonzero. We also replace the use of Leibman s Theorem by the following result of Shah (which is generalizes Theorem 2.3): Theorem 2.4 (Shah [2]). Let ζ : R g be a polynomial map such that for every nontrivial character χ on, dχ dq ζ(t) dχ dq ζ() for some t R. (2.) Then for any f C() and any x, T f(exp ζ(t) x) dt = T T f dµ. Up to obvious changes in notation, the proof is then the same as in the discrete case; in fact it gets simplified as one can replace equivalence mod Q by just equality, and we omit further details.

12 2 BRYNA KRA, NIMISH A. SHAH, AND WENBO SUN 3. Strong equidistribution 3.. A stronger equidistribution result for dilations of curves. The key ingredient for proving Theorem.3 is a stronger equidistribution of dilation of shrinking curves as stated below. First we need to recall the notation. We continue to use λ to denote the Lebesgue measure on (, ) or R, depending on the context. We consider the dynamics on = G/Γ, where G is a simply connected nilpotent Lie group and Γ a closed subgroup of G. Let g denote the Lie algebra of G. We write ḡ = g/ Lie(Γ ). For t R, let ρ t : g ḡ denote the linear map such that ρ t is a polynomial in t with coefficients which are linear maps from g to ḡ. Let t denote the Lie algebra associated to the compact torus = G/[G, G]Γ. Let dq : ḡ t = ḡ/[ḡ, ḡ] denote the natural quotient map. Then dq ρ t : g t can be expressed as d dq ρ t = t i A i, where each A i : g t is a linear map. Consider the lower central series ḡ (k) = [ḡ, ḡ (k ) ] for k. Put ḡ () = ḡ. Let κ be an integer such that ḡ (κ) =. For k κ, let Q k : ḡ ḡ/ḡ (k) denote the natural quotient map and choose an integer D k such that the map Q k ρ t : g ḡ/ḡ (k) is a polynomial of degree at most D k in t and D D 2 D κ. Set { n } n D = max D km : k m κ, k m κ, n κ. (3.) m= m= The purpose of such a definition of D becomes apparent later. Theorem 3.. Suppose that φ: (, ) g is a function such that φ (D) (u) exists for (Lebesgue) almost all u (, ). Assume further that for every nontrivial character χ on and (Lebesgue) almost every u (, ), A i φ () (u) / ker dχ for some i d. (3.2) Then there exists W (, ) with λ(w ) = such that for all s W the following holds: Let (l t ) t be a family such that l t >, and l t and l t t as t. Then s +l tt f(exp ρ t l t t t φ(ξ) x ) dξ = f dµ. (3.3) s i=

13 EQUIDISTRIBUTION OF DILATED CURVES 3 We defer the proof of this result to the next two sections. Before this, we explain its use in completing the proof Theorem.3 and obtain some of its immediate consequences. Proof of Theorem.3. Let f C(G/Γ) be such that f dµ =, G/Γ and without loss of generality we can assume that f. Given ε >, choose W (, ) with λ(w ) = as in Theorem 3.. The conclusion of Theorem 3. can be interpreted as follows: for every s W, there exists l s such that for any l l s, there exists t s,l > l( s) such that for all t t s,l, s+lt f(exp ρ t φ(ξ) x ) dξ lt ε. (3.4) s For L >, let W (L) = {s W : l s L}. By our choice of l s, there exists L > such that λ(w \ W (L)) ε. For t, let W (L, t) = {s W (L): t s,l t}. By our choice of t s,l, there exists T L such that λ(w (L)\W (L, T L )) < ε. Pick a compact set W 2 W (L, T L ) such that λ(w (L, T L )\W 2 ) ε. Let t T L. We cover W 2 by disjoint intervals [s i, s i + Lt ] for i n, where s = min W 2, and if s i W 2 is chosen for some i, then choose s i+ = min ( W 2 \ (, s i + Lt ) ) W 2, if s i + Lt max(w 2 ), otherwise put n = i and stop the induction. Since s n W (L, T L ), and t T L, we have s n + Lt <. Therefore the intervals [s i, s i + Lt ) for i n are disjoint and contained in (, ). Hence (n + )(Lt ). Let W 2 = n i= [s i, s i + Lt ). Then W 2 W 2 and by (3.4), n si +Lt f(exp ρ t φ(s) x ) ds f(exp ρ t φ(s) x ) ds W 2 i= s i (n + )(Lt )ε ε. Since λ(w 2) λ(w 2 ) 3ε, for all t T L, f(exp ρ t φ(s) x ) ds ε + 3ε f 4ε. The following special case of Theorem.3 is of interest:

14 4 BRYNA KRA, NIMISH A. SHAH, AND WENBO SUN Corollary 3.2. Let κ be such that ḡ (κ) =. Suppose that that for k κ, Q k ρ t : g ḡ/ḡ (k) is a polynomial of degree at most k in t. (3.5) Let φ: (, ) g be such that φ (κ) (u) exists for almost all u (, ). Suppose that the condition on χ \{} given in (.) holds. Then the family of measures (µ φ,x,ρ t ) t is equidistributed on as t. Note that the example of ρ t being multiplication by t on g is an example satisfying (3.5) and generalizing Corollary.4 to nilpotent groups. Other natural examples of family of dilations (ρ t ) t R satisfying the condition given in (3.5) are given in [2, 3, 8]. In particular, when ρ t is a Lie algebra automorphism of g, then ρ t satisfies (3.5). Proof of Corollary 3.2. Given the assumption in (3.5), the bound D k on the degree of the quotient map Q k ρ t satisfies D k = k for all k κ. Therefore in (3.), we have n n D km = k m κ, m= m= and hence D κ. Thus the result is a special case of Theorem.3. We also complete the proof of Theorem.2: Proof of Theorem.2. Since φ is an analytic curve, the condition on the χ given in (.) implies the condition given in (.). Therefore the equidistribution follows from Theorem.3. For the converse, suppose that the condition on the map given in (.) fails to hold and A =. Then there exists a nontrivial character χ on such that dχ(a i φ () (u)) = for all u (, ) and all i d. But then by analyticity of φ, for every i d, there exists z i R, such that dχ(a i φ(u)) = z i for all u (, ). Hence, since A =, d d dχ(dq ρ t φ(u)) = t i dχ(a i φ(u)) = t i z i. (3.6) i= Let f = χ q C(). Then f dµ = χ d(q µ) =, because χ is a nontrivial character on the torus and q µ is the invariant probability measure on. For any x, by (.8), f(exp ρ t φ(u) x ) = e 2πidχ(dq ρt φ(u)) χ(q(x )). (3.7) i=

15 EQUIDISTRIBUTION OF DILATED CURVES 5 If t f dµt exists, then by (3.6) and (3.7) t e 2πi d i= ti z i exists. We conclude that z i = for all i. It follows that the it is equal to χ(q(x )). Therefore the sequence µ t does not converge to µ as t with respect to the weak topology Equidistribution of expanding translates of shrinking curves. First we describe a quantitative condition that we want for s (, ) to hold, so that (3.3) holds. Assume that φ: (, ) g is a function satisfying the hypotheses of Theorem 3.. That is, φ (D) (u) exists for almost all u (, ), where D is defined as in (3.), and for every χ \ {}, for almost all u (, ), there exists i d, such that A i φ () (u) ker dχ. Define a map ψ t : {(u, ξ): u + ξ, u (, )} g/ Lie(Γ ) by ψ t (u, ξ) = log[exp(ρ t φ(u + ξ)) exp( ρ t φ(u))], (3.8) where exp : g/ Lie(Γ ) G/Γ is the exponential map, and log is its inverse. Definition 3.3 (Property W(Γ )). We say that s (, ) has property W(Γ ) if the following holds: there exists α such that for any sequences t n and l n with l n t n and for any ɛ >, there exists a compact set I ɛ [, ] with λ([, ] \ I ɛ ) < ɛ such that the following holds: for any s I ɛ and ζ R, for each n N, if we put u n = s + sl n t n and ξ n = ζt α n, then ψ t n (u n, ξ n ) = η(ζ), (3.9) n where η : R g/ Lie(Γ ) is a non-constant polynomial map. If Λ is any closed subgroup of G containing Γ, then let dp : ḡ = g/ Lie(Γ ) g/ Lie(Λ ) be the natural quotient map. Then we replace the dilations ρ t with dp ρ t : g g/ Lie(Λ ). With these modification, property W(Λ ) is also defined. Proposition 3.4. Suppose that s (, ) has property W(Λ ) for every closed subgroup Λ of G containing Γ. For all t, let l t > be such that as t, l t and l t t. For all t such that s + l t t <, let ν t be the probability measure on such that all f C(), f dν t = s +l tt f(exp(ρ l t t t φ(ξ) x ) dξ = s f(exp ρ t φ(s + sl t t ) x ) ds. Then ν t µ with respect to the weak topology as t.

16 6 BRYNA KRA, NIMISH A. SHAH, AND WENBO SUN Proof. The result is trivial if dim(g/γ ) =. We intend to prove the result by induction on dim(g/γ ). In particular, we can assume that the result is valid for all closed subgroups Λ of G containing Γ such that dim(λ ) > dim(γ ). In other words, if p: G/Γ G/Λ is the natural quotient map, and p denotes the corresponding pushforward of measures then as t, p (ν t ) converges to the G-invariant probability measure on G/Λ with respect to the weak topology. Let t n be any given sequence. It suffices to show that after passing to a subsequence of {t n }, we have that ν tn µ in the weak topology as n, where µ denotes the G-invariant probability measure on G/Γ. Since G/Γ is compact, after passing to a subsequence, we may assume that ν tn ν in the space of probability measures on G/Γ with respect to the weak topology as n. Claim The measure ν is invariant under a connected subgroup U of G properly containing Γ. Proof. Let f C(G/Γ). Since s has property W(Γ ), let α be such that (3.9) holds. For the remainder of the proof, we make use of some shorthand to simplify formulas. For all n N, we write t = t n, for all s [, ], we write u = s + sl t t, and for all ζ R, we write ξ = ζt α. Then, since ν tn ν as n, by (3.3) we have G/Γ f(x) dν(x) = n f(exp ρ tn φ(u) x ) ds. (3.) Now ξl t t = ζl t t (α ), because α and l t as n. Therefore (3.) can be rewritten as n = I ε+ξl t t n I ε+ξl t = n f(exp ρ t φ(u) x ) ds + O( f ε) t f(exp ρ t φ(s + sl t t ) x ) ds + O( f ε) I ε f(exp ρ t φ(s + (s + ξl t t)l t t ) x ) ds + O( f ε) = n I ε f(exp ρ t φ(u + ξ) x ) ds + O( f ε).

17 Applying (3.8), this becomes n EQUIDISTRIBUTION OF DILATED CURVES 7 f(exp(ψ t (u, ξ)) exp(ρ t φ(u)) x ) ds + O( f ε) I ε = f(exp(η(ζ)) exp(ρ t φ(u)) x ) ds + O( f ε) n I ε = f(exp(η(ζ)) x) dν tn (x) + O(2 f ε), n G/Γ where we have used (3.9) in the second to last step. Taking the it, in view of (3.), f(x) dν(x) f(exp(η(ζ))x) dν(x) O(2 f ε). G/Γ G/Γ Thus ν is invariant under the action of U, where U is the closure of the subgroup of G/Γ generated by {exp(η(ζ)): ζ R}. But ζ η(ζ) is a non-constant polynomial map and so U is a nontrivial connected subgroup of G/Γ. Let U = π (U ). Then ν is U-invariant, U is connected, and properly contains Γ. Claim The measure ν is invariant under a closed connected normal subgroup F of G containing U such that F Γ/Γ is compact. Proof. Let H be the collection of all closed connected subgroups H of G containing Γ with HΓ/Γ is compact. Then H is countable. By Lesigne [7, Theorem 2], or more generally Ratner s Theorem [], any U-ergodic invariant probability measure on G/Γ is of the form gµ H for some H H, where µ H denotes the H-invariant probability measure on HΓ/Γ, and g G is such that UgΓ ghγ, or equivalently, g (U, H), where (U, H) = {g G: g Ug H}. (3.) Note that (U, H) is an algebraic subvariety of G with respect to any algebraic group structure on G. Let F H be a subgroup of smallest possible dimension such that We note that ν((u, F )Γ/Γ) >. (3.2) (U, H)Z = (U, H), (3.3) where Z G is the inverse image of the center of G/Γ. Now Λ := ZΓ is closed and G/Λ is compact, see [, Chapter II]. Consider the natural quotient map p: G/Γ G/Λ. Let p denote corresponding the pushforward map of measures. Then p (ν tn ) p (ν) as t n.

18 8 BRYNA KRA, NIMISH A. SHAH, AND WENBO SUN If G = Γ, the lemma is trivial. So we assume that G/Γ is a nontrivial connected nilpotent group. Therefore its center is of strictly positive dimension. Therefore dim(g/z) < dim(g/γ ). So by our induction hypothesis stated on the beginning of the proof of Proposition 3.4, we conclude that p (ν) is the G-invariant probability measure on G/ZΓ. By (3.2) and (3.3), p (ν)((u, F )ZΓ/ZΓ) = ν((u, F )Γ/Γ) >. Since (U, F )Z = (U, F ) and p (ν) is the G-invariant measure on G/ZΓ, the Haar measure of (U, F ) is strictly positive. As we noted before, (U, F ) is an algebraic subvariety of G. Therefore (U, F ) = G; that is, Ug gf for all g G. Now we show that that F is normal in G, and ν is F -invariant. By minimality of F, where ν(s(u, F )Γ/Γ) =, S(U, F ) := {(U, H): H H, dim H < dim F }. Therefore ν((u, F )Γ/Γ \ S(U, F )Γ/Γ) =. As a consequence, almost every U-ergodic component of ν is of the form gµ F for some g G. Now G \ S(U, F ). Let g G \ S(U, F ). We claim that UgΓ = gf Γ. To see this note that Ug gf and F Γ is closed, so UgΓ gf Γ, and by the orbit closure theorem (cf. Lesigne [7, Theorem 2]), UgΓ = LgΓ and g Lg H. Therefore g Lg F. Since g (U, g Lg) and g S(U, F ), we conclude that dim(g Lg) dim F, and hence g Lg = F, and so UgΓ = gf Γ. Now let γ Γ. Since g, gγ G = (F, U), we have that Ugγ gγf, and hence gf Γ = UgΓ gγf Γ. Therefore gf g gγf γ g, and hence F = γf γ. This proves that F is normalized by Γ. But Ad Γ is Zariski dense in Ad G; see [, Chapter 2]. Hence F is normalized by G. As we noted above, almost every U-ergodic component of ν is of the form gµ F for some g G, so it is gf g = F -invartiant. Therefore ν is F -invartiant. This completes the proof of the claim. Now it remains to prove that ν is G-invariant. Let p: G/Γ G/F Γ be the natural quotient map, and let p denote the corresponding pushforward map of measures. Then p (ν tn ) p (ν) as n. Since U F, we have that dim((f Γ) ) > dim(γ ). Therefore by the induction hypothesis stated in the begining of the proof of Proposition 3.4,

19 EQUIDISTRIBUTION OF DILATED CURVES 9 p (ν) is the G-invariant probability measure on G/F Γ. Since ν is F - invariant, it follows that ν is G-invariant. Since the Haar measure is the unique G-invariant probability measure on, we have that ν = µ, completing the proof of Proposition Points with property W(Γ ) Now we determine which points s (, ) satisfy the property W(Γ ). For this purpose, we study the following function using Taylor s expansion and Baker-Campbell-Hausdorff formula: for u (, ) and ξ (, ) such that u + ξ (, ), we have ψ t (u, ξ) = log[exp(ρ t φ(u + ξ)) exp( ρ t φ(u))] ḡ. (4.) Let u (, ) be such that φ (D) (u) exists. By Taylor s formula, for any ξ R such that u + ξ (, ), we have D φ(u + ξ) = l! φ(l) (u)ξ l + ε(u, ξ)ξ D, (4.2) l= where ε(u, ξ) as ξ. Since ḡ (k) ḡ (k ) for all k κ, we can choose a subspace V k ḡ (k ) such that ḡ (k ) = V k ḡ (k). Then ḡ = V V κ and ḡ (k) = V k+ V κ. Let P k : ḡ V k denote the corresponding projection. By the choice of the integers D k (the polynomial Q k ρ t is of degree at most D k ), for all k κ and for each i D k, there exists a linear map A i,k : g V k, such that Then ρ t φ(u + ξ) = = D k P k ρ t = t i A i,k. (4.3) i= κ (P k ρ t ) φ(u + ξ) k= ( κ D k D ) t i ξ l l! A i,kφ (l) (u) + t i ξ D A i,k ε(u, ξ). (4.4) k= i= l= By the Baker-Campbell-Hausdorff formula, since ḡ (κ) =, using the definition of ψ t in (4.), we have ψ t (u, ξ) = κd i= κd l= t i ξ l Z (i,l) (u, ξ), (4.5)

20 2 BRYNA KRA, NIMISH A. SHAH, AND WENBO SUN where the functions Z (i,l) can be expressed as Z (i,l) (u, ξ) = Y (i,l) (u) + ε (i,l) (u, ξ), (4.6) where Y (i,l) (u) is a fixed linear combination of nested commutators of the form [, [ 2, [, [ n, n ]]]], where n κ. (4.7) and m = A im,km φ (lm) (u) V km, for m n, (4.8) and ε (i,l) (u, ξ) is a fixed linear combination of similar nested commutators where one or more of the m s with l m = D are replaced by m = A im,km ε(u, ξ) V km. (4.9) Claim 4.. We have the following: () ξ ε (i,l) (u, ξ) =. (2) For all i, Z (i,) =. (3) If l < D, then ε (i,l) =. (4) If i > D, then Z (i,l) =. In particular, where ψ t (u, ξ) = (i,l) P t i ξ l Y (i,l) (u) + D κd i= l=d t i ξ l ε (i,l) (u, ξ), (4.) P = {(i, l): i D, l κd}. (4.) Proof. From the description of ε (i,l) (u, ξ) and (4.9), due to (4.2) we conclude that () holds. Next (2) holds, because ψ t (u, ) =. In view of (4.4), the following relations hold between various indices in (4.8) and (4.9): k m κ, i m D km, and l m D. (4.2) In view of equations (4.5)-(4.9), we get n n i m = i and l m = l. (4.3) i= m= For every commutator appearing in the expression for ε (i,l) (u, ξ), we have l m = D for at least one m, and hence l D by (4.3). Therefore if l < D then ε (i,l) =, which proves (3). By (4.8) m V km, and by (4.9) m V km. Also V km ḡ (km ) and [ḡ (k ), ḡ (k ) ] ḡ (k+k ) for all k, k.

21 EQUIDISTRIBUTION OF DILATED CURVES 2 Therefore the nested commutator as in (4.7) and its analogue involving m s belong to ḡ (( n m= km) ). Since ḡ (κ) =, if the nested commutator is nonzero then n k m κ. (4.4) m= By (4.2) and (4.3), we have that if the nested commutator as in (4.7) or its analogue involving m s is nonzero then n n i = i m D km. (4.5) m= m= Now in view of (4.7), (4.4) and (4.5), we recall (3.): { n } n D = max D km : k m κ, k m κ, n κ. m= m= Therefore by (4.5), if i > D then Z (i,l) =. This proves (4) Claim 4.2. For i d, dq(y (i,) (u)) = A i φ () (u). Proof. First note that dq = dq P. Now P Y (i,) (u) = A i, φ () (u), (4.6) because l = and by (4.7) and (4.8) we have n =, and hence by (4.7) l = l =. So by (4.3), dq ρ t = dq (P ρ t ) = dq ( D ) D t i A i, = t i dq A i,. (4.7) i= i= Therefore by (.7), d D and { A i if i d dq A i, = if d < i D. Hence the claim follows from (4.6). (4.8) Proposition 4.3. Given ε >, there exists a Borel measurable set J (, ) such that λ(j ) ε and the following conditions hold: () The derivative φ (κ) (u) exists and is uniformly continuous for u J and it is bounded on J. In particular, for each (i, l) P, Y (i,l) (u) is uniformly continuous and bounded for u J.

22 22 BRYNA KRA, NIMISH A. SHAH, AND WENBO SUN (2) For every u J and every nontrivial unitary character χ on the torus G/[G, G]Γ =, we have A i φ () (u) ker dχ for some i d. (3) For each (i, l) P, ε (i,l) (u, ξ) as ξ, and this convergence is uniform for u J. Proof. By our assumption, φ (κ) (u) exists for almost all u (, ), hence by Lusin s Theorem there exists a compact set J 2 (, ) such that φ (κ) is uniformly continuous on J 2 and λ(j 2 ) ε /2. By the condition given in (.), since \ {} is countable, there exists a Borel set J 3 J 2 such that λ(j 2 \ J 3 ) = and for every χ \ {} and every u J 3, there exists i d such that A i φ () (u) ker dχ. Given p, q, define the set J p,q to be {u J 3 : for all (i, l) P, ε (i,l) (u, ξ) /p for all ξ /q}. Then by Claim 4.(()), the sets J p,q form a nested sequence of sets growing to J 3 as q. Choose, q p such that λ(j 3 \ J p,qp ) < 2 p (ε /2) and set J = p= J p,q p. Then λ(j 3 \J ) ε /2, and for every (i, l) P, ε i,l (u, ξ) as ξ uniformly for u J. Also λ(j ) ε. Corollary 4.4. If J satisfies conditions (),(2) and (3) of Proposition 4.3 for Γ, then J also satisfies the same three conditions when we replace Γ by any closed subgroup Λ of G containing Γ and ρ t by dp ρ t, where dp : g/ Lie(Γ ) g/ Lie(Λ ) it the natural quotient map. Proof. When Λ replaces Γ, each Y (i,l) gets replaced by dp(y (i,l) ) and ε (i,l) gets replaced by dp(ε (i,l) ). Therefore it directly follows that Conditions () and (3) hold. Now = G/[G, G]Λ is the abelianization of G/Λ with t = Lie(G/[G, G]Λ ) being its Lie algebra. Let dr : t t the natural quotient map. Then dq gets replaced by dq : Lie(G/Λ ) t such that dq dp = dr dq. Then in view of (.7), d dq (dp ρ t ) = dr (dq ρ t ) = t i (dr A i ). Thus in case of Λ replacing Γ, we have that A i gets replaced by dr A i in Condition (2). To verify this condition, let χ \ {} be given. Put χ = χ r \ {}. Then i= dχ (dr A i ) = dχ A i.

23 EQUIDISTRIBUTION OF DILATED CURVES 23 For any u J, by Condition (2) for Γ, pick i d such that dχ(a i φ () (u)). Then dχ ((dr A i )φ () (u)). Let F Γ denote the collection of all normal connected normal subgroups F of G such that F Γ and F Γ is closed. In particular, F/F Γ = F Γ/Γ is compact. By [, Chapter II], there exists a Q- structure on G such that Γ is a Q-subgroup of G and the image of Γ on G/Γ consists of integral points with respect to the Q-structure on the quotient algebraic group G/Γ. Moreover for any F F Γ, we have that F/Γ must be an algebraic Q-subgroup of G/Γ. Therefore F Γ is countable. For any F F Γ, let π F : G G/F be the natural quotient map and let dπ F : g g/f denote its differential, where f denotes the Lie algebra of F. For all (i, l) P and F F Γ, define K (i,l),f = {u (, ): φ (D) (u) exists and dπ F (Y (i,l) (u)) = }. (4.9) Let J be a Borel measurable set which satisfies the conditions (),(2) and (3) of Proposition 4.3. For any (i, l) P and F F Γ, let S (i,l),f = {s J K (i,l),f : s is not a Lebesgue Density point of J K (i,l),f }. (4.2) Then by the Lebesgue Density theorem, λ(s (i,l),f ) =. Let S Γ = S (i,l),f. (4.2) (i,l) P, F F Γ Since P is finite and F Γ is countable, λ(s Γ ) =. (4.22) If Λ is a closed subgroup of G containing Γ, then Λ F Γ. Hence F Λ F Γ. Therefore using Corollary 4.4 and (4.2), we have that S Λ S Γ. (4.23) Proposition 4.5. Let J be a Borel set which satisfies all the three conditions of Proposition 4.3. Let s J \ S Γ. Then s has property W(Γ ). Proof.. Let In view of (4.9), P = {(i, l) P : Y (i,l) (s ) }. (4.24) s J K, where K := (i,l) P\P K (i,l),γ. (4.25)

24 24 BRYNA KRA, NIMISH A. SHAH, AND WENBO SUN Set α = max{i/l : (i, l) P }. (4.26) We first show that α. It suffices to show that Y (i,) (s ) for some i d. (4.27) Let χ be a nontrivial character on G/[G, G]Γ =. Since s J, by condition (2) of Proposition 4.3 pick i d such that dχ(a i φ () (s )). Therefore A i φ () (s ). Therefore by Claim 4.2, dq(y (i,) (s )) = A i φ () (s ). Therefore by (4.24) and (4.27), there exists i d such that (i, ) P. Hence by (4.26) α i/. (4.28) Fix ε >. For every (i, l) P \ P, s J K (i,l),γ by (4.25), and since s S Γ, s is a Lebesgue density point of J K (i,l),λ by (4.2). For every n, we can choose k n n and a compact set I n [, ] such that if we put t = t kn, then and s + sl t t J K (i,l),γ for all s I n and (i, l) P \ P, λ([, ] \ I n ) 2 n ε. Let I ε = i= I n. Then λ((, ) \ I ε ) ε. For any s I ε, ζ R, and n, put t = t kn, u = s + sl t t J K, and ξ = ζt α. (4.29) Then the following statements hold: () For l > i, since α, by () of Proposition 4.3, t i ξ l Y (i,l) (u) as n. (4.3) (2) For l D i, since α and u J, by (3) of Proposition 4.3, t i ξ l ε (i,l) (u, ξ) = ζ l t (αl i) ε (i,l) (u, ξ) as n. (4.3) (3) For all (i, l) P \ P, u K K (i,l),γ, so by (4.9), Y (i,l) (u) =. (4.32) (4) Since s, u J, for each (i, l) P we have that Y (i,l) (u) Y (i,l) (s ) as n, (4.33) by continuity of Y (i,l) on J.

25 (5) For each (i, l) P, EQUIDISTRIBUTION OF DILATED CURVES 25 n ti ξ l = ζ l t i αl = n { ζ l, if i/l = α, if i/l < α. In view of (4.), by the above list of observations (4.34) ψ t(u, ξ) = ζ l Y (αl,l) (s ) := η(ζ), (4.35) n l L where L = {l : l κd, (αl, l) P }, which is nonempty by the definition of α. By (4.24), Y (αl,l) for all l L. Therefore s has property W(Γ ). Proof of Theorem 3.. Let ɛ >. Obtain J as in Proposition 4.3 such that λ(j ) ɛ. Let S Γ be as defined in (4.2). Then by (4.22) we have λ(s Γ ) =. Let W ɛ = J \ S Γ. Then λ(w ɛ ) ɛ. Let s W ɛ. By Corollary 4.4 and (4.23), and Proposition 4.5, s has property W(Λ ) for every closed subgroup Λ of G containing Γ. Therefore (3.3) holds by Proposition 3.4. Now let W = n N W /n. Then λ(w ) = and (3.3) is satisfied for every s W. 5. Weak equidistribution does not imply equidistribution In this section we provide different instances where weak equidistribution of sequence of measures hold, but (strong) equidistribution does not. As before let λ denote the Lebesgue measure on R restricted to (, ). Proposition 5.. Let = T = G/Γ, where G = R and Γ = Z. Let ρ t (v) = tv for all v R = Lie(G) and t R, and let x. There exists a measure ν on R such that the family of measures (µ t := µ ν,x,ρ t ) t is weakly equidistributed but not equidistributed on (, µ), where µ is the Haar measure on. Proof. We first construct a non-atomic probability measure ν on R such µ 3 m = µ for all m N. To construct such a measure, let ψ : R R be the function defined by ψ(u) = (5.) 3 n for u = {n : a n(u)=} a n (u) 3 n mod, where a n (u) {,, 2}, and then take ν be the pushforward of λ under the map ψ to R = Lie(G). We now check that it has the stated properties.

26 26 BRYNA KRA, NIMISH A. SHAH, AND WENBO SUN Note that if < u < u 2 < and ψ(u ) = ψ(u 2 ) then x 2 x = a n/3 n, where a n {, 2}. So u 2 u belongs to the standard Cantor middle third set, which has zero Lebesgue measure. Therefore λ(ψ ({y})) = for all y R. Hence ν is non-atomic. Let f C(T). Then for all t R, f dµ t = f(tψ(u) + x ) du. For any u [, /3) and b {,, 2}, 3ψ(b/3 + u) ψ(3u) mod Z. (5.2) 3n {n : a n(u)=} Thus it follows that for any x T, f(3ψ(u) + x ) du = = = = 2 b/3+/3 b= b/3 /3 2 b= 2 (/3) b= f(3ψ(u) + x ) du f(ψ(3u) + x ) du From this we conclude that for any m N, f(3 m ψ(u) + x ) du = f(ψ(u) + x ) du. f(ψ(u) + x ) du f(3 m ψ(u) + x ) du. It follows that µ 3 m = µ for all m. Thus the measures ν and µ t satisfy the stated conditions. Since µ differs from the Haar measure µ, the family of measures (µ t ) t is not equidistributed with respect to µ. On the other hand, since ker(dχ) is countable set for all nontrivial unitary characters χ, we have that ν(ker(dχ) + v) = for all v Lie(G) = R, since ν is non-atomic. By Theorem., (µ t ) t is weakly equidistributed. This example can be generalized to higher dimensional tori: Corollary 5.2. For every torus = T d = G/Γ, where G = R d, Γ = Z d, let ρ t (v) = tv for all v Lie(G) = R d and t R n. There exists a measure σ on R d such that for any x G/Γ, the family of measures

27 EQUIDISTRIBUTION OF DILATED CURVES 27 (µ t := µ σ,x,ρ t ) t is weakly equidistributed but not equidistributed on (, µ), where µ is the Haar measure on. Proof. Let ν be the measure of R as defined in Proposition 5. and define σ = ν ν. We claim that σ satisfies these conditions. Suppose that x = (y,..., y d ). Then µ 3 n = µ ν,y,ρ 3 n µ ν,yd,ρ 3 n = µ ν,y,ρ 3 µ ν,yd,ρ 3, and so (µ t ) t is not equidistributed. On the other hand, for every nontrivial unitary character χ on T d and every v Lie(G) = R d, there exist finitely many linear functionals g,..., g k : R d R and some i d such that k v + ker(dχ) = {(y,..., y d ) R d : g m (y,..., ŷ i,..., y d ) = y i }. m= Assuming without loss of generality that i = d, for all m k, σ({(y,..., y d ): g m (y,..., y d ) = y d }) = ν({y d g: y d = g m (y,..., y d )}) dν(y )... dν(y d ) R d =, because ν is non-atomic. Thus σ(v +ker(dχ)) = and by Theorem., (µ t ) t is weakly equidistributed. Theorem 5.3. Let = T 2 = G/Γ, where G = R 2 and Γ = Z 2 and let ρ t (v) = tv for all v Lie(G) = R 2 and t R. There is a function φ: (, ) g such that for any x T 2, the family of measures (µ t := µ φ,x,ρ t ) t is weakly equidistributed but not equidistributed on. Proof. Let φ: (, ) Lie(G) = R 2 be defined by φ(u) = (u, ψ(u)), u (, ), where ψ : (, ) R is defined by (5.). Let µ t = µ φ,x,ρ t. We first show that (µ t ) t is weakly equidistributed. Let ν be the probability measure on R 2 which is the pushforward of the Lebesgue measure on (, ) under the map φ. Let C = {(u, ψ(u)) R 2 : u (, )}. Let P : R 2 R denote the projection on the first factor. Then for any E R 2 we have ν(e) = λ(p (C E)). Then µ φ,x,ρ t = µ ν,x,ρ t for all t R. Therefore by Theorem., to prove the weak equidistribution of the corresponding family, it suffices to show that for all (p, q) Z 2 \ {(, )} and z R, if we put L(p, q, z) = {(x, y) : px + qy = z}

28 28 BRYNA KRA, NIMISH A. SHAH, AND WENBO SUN then ν(l(p, q, z)) = λ(p (L(p, q, z) C) =. First suppose that p =. Then q and P (L(p, q, z) C) = ψ (z/q), which is contained in a translate of a Cantor middle set. Therefore λ(p (L(p, q, z) C)) =. Now assume that p. For N N and a,..., a N {,, 2}, set B a,...,a N = [ N n=n+ a n 3, N a n n 3 + n 3 N ) [ a Note that for b n {,, 2} for all n N +, a n u = 3 + b n n 3 = N a Also Therefore (, ) a,...,a N {,,2} C B N = 3 n ψ(u) a [ N Therefore it suffices to prove that a,...,a N {,,2} 3, n 3 + n 2 3 N a N ) a n 3, a n n 3 +. n 3 N B a,...,a N. λ(p (L(p, q, z) B N )) =. N 3 + n n=n+ ]. 3 n. It is easy to see that there exist s,..., s 2 N (, ) such that for all a,..., a N {,, 2}, a = s 3 n i for some i 2 N. Let C j,n = B a,...,an. a,...,a N {,,2}, a 3 n =s j Then Now B N = 2 N j= C j,n. P (L(p, q, z) C j,n ) {(z/p (q/p)y : y [s j, s j + /(2 3 N )]}. Therefore Hence λ(p (L(p, q, z) C j,n )) q/p /(2 3 N ). λ(p ((L(p, q, z) B N )) 2 N q/2p /3 N as N.

29 EQUIDISTRIBUTION OF DILATED CURVES 29 Thus we can now conclude that the family of measures (µ t ) t is weakly equidistributed. We now prove that (µ t ) t is not equidistributed. By definition, for any continuous function f on T 2, we have f dµ t = f((tu, tψ(u)) + x ) du. T 2 So Recall that by (5.2), For any u [, /3) and b {,, 2}, 3ψ(u + b/3) ψ(3u) mod Z. (5.3) 3n {n : a n(u)=} Therefore, for any f C(T 2 ), and b {,, 2} 3 + b 3 b 3 f((3u, 3ψ(u)) + x ) du = /3 3 f((3u, 3ψ(u)) + x ) du = 3 From this we can deduce that for any m, f((3 m u, 3 m ψ(u)) + x ) du = = f((3u, ψ(3u)) + x ) du. f((3u, ψ(3u)) + x ) du f((u, ψ(u)) + x ) du. f((3 m u, 3 m ψ(u)) + x ) du. Therefore, µ 3 m = µ 3 m = µ. Since µ is not a Haar measure on T 2, (µ t ) t is not equidistributed. References [] L. Auslander, L. Green, & F. Hahn. Flows on homogeneous spaces. With the assistance of L. Markus and W. Massey, and an appendix by L. Greenberg. Annals of Mathematics Studies, 53 Princeton University Press, Princeton, N.J [2] E. Breuillard, & E. Le Donne. On the rate of convergence to the asymptotic cone for nilpotent groups and subfinsler geometry. Proc. Natl. Acad. Sci. USA (23), no. 48, [3] E. Breuillard, & E. Le Donne. Nilpotent groups, asymptotic cones, and subfinsler geometry. [4] M. Björklund & A. Fish. Equidistribution of dilations of polynomial curves in nilmanifolds. Proc. Amer. Math. Soc. 37 (29), no. 6, [5] J. Chaika & P. Hubert. Circle averages and disjointness in typical flat surfaces on every Teichmüller disc. Bull. Lond. Math. Soc., to appear.

30 3 BRYNA KRA, NIMISH A. SHAH, AND WENBO SUN [6] A. Leibman. Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold. Ergodic Theory Dynam. Systems 25 (25), no., [7] E. Lesigne. Sur une nil-variété, les parties minimales associés á une translation sont uniquement ergodiques. Ergodic Theory Dynam. Systems (99), no. 2, [8] P. Pansu. Croissance des boules et des géodésiques fermées dans les nilvariétés. Ergodic Theory Dynam. Systems 3 (983), no. 3, [9] B. Randol. The behavior under projection of dilating sets in a covering space. Trans. Amer. Math. Soc. 285 (984), no. 2, [] M.S. Raghunathan. Discrete Subgroups of Lie groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68. Springer-Verlag, New York- Heidelberg, 972. ix+227 pp. [] M. Ratner. Strict measure rigidity for unipotent subgroups of solvable groups. Invent. Math. (99), no. 2, [2] N. A. Shah. Limit distributions of polynomial trajectories on homogeneous spaces. Duke Math. J. 75 (994), no. 3, [3] H. Weyl. Über die Gleichverteilung von Zahlen mod Eins. Math. Ann. 77 (96), Northwestern University, Evanston, IL 628 USA address: kra@math.northwestern.edu The Ohio State University, Columbus, Ohio 432 USA address: shah@math.osu.edu The Ohio State University, Columbus, Ohio 432 USA address: sun.99@osu.edu