A brief remark on orbits of SL(2,Z) in the euclidean plane
AA BRIEF REMARK ON ORBITS OF SL (2 , Z ) IN THEEUCLIDEAN PLANE
ANTONIN GUILLOUX
F. Ledrappier [5] proved the following theorem as an application of Ratner theo-rem on unipotent (cid:29)ows (A. Nogueira [7] proved it for SL (2 , Z ) with di(cid:27)erent meth-ods):Theorem 1 (Ledrappier, Nogueira). Let Γ be a lattice of SL (2 , R ) of covolume c (Γ) , (cid:107) . (cid:107) the euclidean norm on the algebra of × -matrices M (2 , R ) , and v ∈ R with non-discrete orbit under Γ .Then we have the following limit, for all ϕ ∈ C c ( R \ { } ) : T (cid:88) γ ∈ Γ , (cid:107) γ (cid:107)≤ T ϕ ( γv ) T →∞ −−−−→ | v | c (Γ) (cid:90) R \{ } ϕ ( w ) dw | w | . We may draw a picture of this equidistribution theorem, for example with
Γ = SL (2 , Z ) . Here is shown the orbit of the point (cid:18) π (cid:19) under the ball of radius .We draw only the points which are falling in some compact to avoid the rescalingof the picture (in the theorem, ϕ has to have compact support):Figure 1. Orbit under the ball of radius 1000A striking phenomenon is the gaps around lines of simple rational slopes. Thisappears for any initial point. We will describe here these gaps in a fully elementaryway for the lattice Γ = SL (2 , Z ) (which is enough to describe it for all arithmeticlattices). Let us mention that our analysis is carried on in the arithmetic case forsake of elementariness but a similar analysis can be done for non-arithmetic lattices.Another experimentation with a cocompact lattice does not show these gaps. Itwill be clear from the analysis below that this comes from the unique ergodicity ofthe unipotent (cid:29)ow in SL (2 , Z ) \ SL (2 , R ) . a r X i v : . [ m a t h . D S ] F e b ANTONIN GUILLOUX
1. The plane and the horocyclesThe key point in the theorem of Ledrappier is the identi(cid:28)cation of R \ { } andthe space of horocycles SL (2 , R ) /U , where U = { u ( t ) = (cid:18) t (cid:19) for t ∈ R } is theupper triangular unipotent subgroup of SL (2 , R ) . The projection from SL (2 , R ) tothe plane is given by the (cid:28)rst column of the matrix. We will use the followingsection from R \ ( { } × R ) to SL (2 , R ) : σ (cid:18) ab (cid:19) (cid:55)→ (cid:18) a b a − (cid:19) . Then we have: σ (cid:18) ab (cid:19) u ( t ) = (cid:18) a tab a − + tb (cid:19) , which in turn projects to the samepoint (cid:18) ab (cid:19) .The theorem of Ledrappier is proven using the fact that a large portion of adense orbit of U in SL (2 , Z ) \ SL (2 , R ) becomes equidistributed in this space. With-out any more detail on this, we may just remark that if ab is rational, the orbit σ (cid:18) ab (cid:19) U projects in a periodic horocycle in SL (2 , Z ) \ SL (2 , R ) . This means thatthe application R → SL (2 , Z ) \ SL (2 , R ) given by t (cid:55)→ SL (2 , Z ) σ (cid:18) ab (cid:19) u ( t ) is peri-odic. Another way to state it: there exists t ∈ R ∗ and γ ∈ SL (2 , Z ) such that wehave γσ (cid:18) ab (cid:19) = σ (cid:18) ab (cid:19) u ( t ) . The period of this application is called the period ofthe orbit σ (cid:18) ab (cid:19) U .2. Periods and heights of points with rational slopeConsider a point v = (cid:18) ab (cid:19) in R \ { } with ba ∈ Q or a = 0 . Then we mayde(cid:28)ne the following number:De(cid:28)nition 2.1.The period ρ ( v ) of v is the period of the orbit σ ( v ) U in the space SL (2 , Z ) \ SL (2 , R ) It is not hard to e(cid:27)ectively compute this period:Proposition 2. Write v = t (cid:18) pq (cid:19) with p and q two coprime integers.Then the period of v is given by ρ ( v ) = t .Proof. We assume here that p (cid:54) = 0 (if not you can change the section σ ). Thepoint t (cid:18) pq (cid:19) correspond via σ to the matrix (cid:18) tp tq ( tp ) − (cid:19) . So we have to solve theequation: γ (cid:18) tp tq ( tp ) − (cid:19) = (cid:18) tp tq ( tp ) − (cid:19) u ( s ) for γ in SL (2 , Z ) and s real. Thatis: (cid:18) t ( ap + bq ) b ( tp ) − t ( cp + dq ) d ( tp ) − (cid:19) = (cid:18) tp stptq stq + ( tp ) − (cid:19) , for a , b , c , d integers verifying ad − bc = 1 and s real. We check that b and d − have to be divisible by p hence s has to belong to t Z . Now we easily check thefollowing equality, thus proving the proposition : (cid:18) pq p q − pq (cid:19) (cid:18) tp tq ( tp ) − (cid:19) = (cid:18) tp tq ( tp ) − (cid:19) (cid:18) t (cid:19) . (cid:3) This computation is an elementary way to check that the period of a pointwith rational slope is invariant under the action of SL (2 , Z ) : the image under anelement of SL (2 , Z ) of a point (cid:18) pq (cid:19) with coprime p and q is still a point of thisform. Of course, a more intrinsic way to see this is to look at the de(cid:28)nition of theperiod which is clearly invariant under SL (2 , Z ) . Anyway this simple fact is the keyremark. Indeed the set of points of (cid:28)xed period is a discrete subset of the plane.Call P ( ρ ) := { v ∈ R of rational slope with period ρ } . The previous propositiondescribe these sets as P ( ρ ) = √ ρ Z ∧ Z where Z ∧ Z stands for the set of points withcoprime integer coordinates.Moreover we may de(cid:28)ne the height of a point of rational slope (using the heightfunction on the space P ( Q ) ) by this simple formula: h ( t (cid:18) pq (cid:19) ) = (cid:112) p + q = | (cid:18) pq (cid:19) | (as usual p and q are coprime integers). We have the following tautologicalformula for any point v of rational slope in the plane : ρ ( v ) | v | = h ( v ) .
3. Spectrum of periodsConsider v a point in the plane (not ). Then for each ρ > , the distance of v to the set P ( ρ ) is a nonnegative real number. Moreover if v has irrational slope,this number is positive for each ρ . We then de(cid:28)ne a function, called spectrum ofperiods, for v :De(cid:28)nition 3.1. Let v be a point in the plane of irrational slope. Then its spectrumof periods D v is the function : D v : R ∗ + → R ∗ + ρ (cid:55)→ d ( v, P ( ρ )) The description of the sets P ( ρ ) made above allows the following rewriting of D v : D v ( ρ ) = √ ρ d ( √ ρv, Z ∧ Z ) . This last expression shows that for ρ big enough thisfunction encodes the diophantine property of the slope of v , and may be interestingto study precisely. But a (cid:28)rst remark is that D v ( ρ ) is always smaller than √ ρ ;moreover for ρ ≤ | v | ) , D v ( ρ ) is bigger than √ ρ :Lemma 3. For ρ ≤ | v | ) , we have √ ρ ≤ D v ( ρ ) ≤ √ ρ . Moreover, as ρ → , D v ( ρ ) is equivalent to √ ρ .Proof. If ρ is less than | v | ) , the modulus of √ ρv is less than . So its distance to Z ∧ Z is more than , proving the inequality. The equivalence is straightforward. (cid:3) We are now able to state the desired property: the orbit of v under the set Γ T = { γ ∈ SL (2 , Z ) such that (cid:107) γ (cid:107) ≤ T } cannot come too close of the points ofrational slopes.Proposition 4. Let w be a point of rational slope in the plane. Then the distanceof Γ T v to w is bounded from below by D v ( ρ ( w )) T = D v ( h ( w )2 | w | ) T .Let us prove the proposition before giving a more geometric description. ANTONIN GUILLOUX
Proof. Consider an element γ of SL (2 , Z ) of euclidean norm less than T . Then itmultiplies length by at most T Let us suppose that the point γv is very close tosome w with rational slope: | γv − w | = (cid:15)T for some (cid:15) ; we immediately get that | v − γ − w | ≤ (cid:15) . But the point γ − w has same period as w by invariance and thusbelongs to P ( ρ ( w )) . So by de(cid:28)nition of D v and the tautological formula on theperiod, we get that γv cannot be too close to w : | γv − w | ≥ D v ( ρ ( w )) T ≥ D v ( h ( w ) | w | ) T . (cid:3)
Now if we are interested at how the orbit of v comes close some half-line ofrational slopes R ∗ + (cid:18) pq (cid:19) , we (cid:28)x the height h ( w ) . If we furthermore add the condition | w | ≥ | v | h ( w ) we may use the easy bound on D v to get: | γv − w | ≥ | w | h ( w ) T , for all γ of norm less than T. We see on this last formula that the simpler is the slope (as a rational number) theharder it is to come close. The linear behavior suggests a picture in coordinates(radius, slope) to see clearly the gaps. Here we draw the whole orbit (check thatthe radius of points goes up to 1900) for T = 1000 in a small neighborhood of thehorizontal axis. The gap is fairly evident. The graphs of the functions D v and − D v are drawn in blue. The previous proposition states that no point of this orbitmay fall between this two graphs. Once again we are in coordinates (radius,slope):Figure 2. The gap around the horizontal axis4. Two kinds of optimalityLet us mention that the optimality of the described gap seen on the previouspicture is easy to understand. Indeed the next lemma states that some points ofthe orbit Γ T v are almost as close as possible to points of rational slope.Lemma 5. There exist a γ ∈ SL (2 , Z ) with (cid:107) γ (cid:107) ≤ T and some point w of rationalslope such that we have for all T ≥ : | γv − w | − D v ( ρ ( w )) T ≤ D v ( ρ ( w )) T . Proof. Consider the matrix γ = (cid:18) T −
10 1 (cid:19) of Γ T . Let us note v = (cid:18) ab (cid:19) . Let usassume (cid:28)rst that | a | ≤ b . Then we have γv = (cid:18) a + ( T − bb (cid:19) . Now consider thepoint w = (cid:18) a + ( T − b (cid:19) of slope . First we get that the distance | γv − w | isequal to | b | . Second we check that ( T − | b | ≤ D v ( ρ ( w ) ≤ T | b | using the formula for the function D v . That means that we have: D v ( ρ ( w )) T ≤ | b | − | b | T ≤ | γv − w | − TT − D v ( ρ ( w )) T ≤ | γv − w | − D v ( ρ ( w )) T So the lemma is proven in this case. If we had | b | < | a | , we may then considerthe matrix γ = (cid:18) T − (cid:19) and the point w = (cid:18) T − a + b (cid:19) which lead to thesame estimate via the same computation ! (cid:3) But of course this consideration is somehow deceptive, as it describes a generalfact veri(cid:28)ed for any initial point and do not re(cid:29)ects the diophantine properties ofthis point. So let us show that the diophantine information about the beginningpoint e(cid:27)ectively lies in the evolution of the orbit. Consider a point v = (cid:18) ab (cid:19) withirrational slope s = ba . Recall that the best approximation of s by a rational number pq gives us the point (cid:18) aa pq (cid:19) which realizes the distance D v ( q a ) . Hence we are onlyinterested in the periods ρ of the form q a .According to the following lemma, we do always get points in the orbit undera ball of big enough size T which almost realizes the minimal predicted distance D v ( ρ ) T to the set P ( ρ ) .Lemma 6. Let v = (cid:18) ab (cid:19) be a point with irrational slope, and (cid:28)x ε > . Then thereexists a real T such that for all T > T , and every integer q > , there is a point γ.v in Γ T .v and a point w in P ( ρ (cid:48) ) such that: | ρ (cid:48) q a − | ≤ ε ) D v ( q a ) aT and the distance between γ.v and w is at most (1 + ε ) D v ( q | a | ) T .
Proof. Once again the proof is elementary. We just have to (cid:28)nd in Γ T a contractingelement γ and apply it to a well-chosen vector. I let the reader verify that thefollowing construction veri(cid:28)es the above estimates. Take N the biggest integersuch that N + 2 ≤ T , and consider the matrix γ = (cid:18) N −
11 0 (cid:19) of Γ T . This matrixcontracts the vector (cid:18) N (cid:19) to the vector (cid:18) (cid:19) . ANTONIN GUILLOUX
Hence, let w = (cid:18) ab (cid:48) (cid:19) be the point of P ( q a ) realizing the in(cid:28)mum distance D v ( q a ) . Eventually, consider α and λ the solutions of αw − v = λ (cid:18) N (cid:19) (which has solutions for all but possibly one integer N ).We have α = Na − bNa − b (cid:48) and λ = a ( b − b (cid:48) ) Na − b (cid:48) .Now consider w = γ ( αw ) . We have: w − γ.v = γ ( αw − v ) = λ (cid:18) (cid:19) Hence the distance between w and γ.v is λ which is as near as wanted of D v ( q | a | ) T (recall that D v ( q a ) = b − b (cid:48) ).Moreover the period ρ (cid:48) of w is the one of αw , i.e. ρα . Hence we get the desiredcontrol on ρ (cid:48) by checking that, for N big enough (but independent of q ): | α − | = ( N a − b (cid:48) ) − ( N a − b ) ( N a − b (cid:48) ) ≤ (1 + ε )2 D v ( q a ) T a (cid:3)
This previous result allows us to get the best rationnal approximation of theslope by the following limit:Proposition 7. Let q be a positive integer and v = (cid:18) ab (cid:19) be a point with irrationalslope s = ba .Then we have the following equality: inf (cid:26) | s − pq | for p ∈ Z (cid:27) = lim T →∞ Ta inf (cid:26) d (Γ T .v, P ( ρ (cid:48) )) for (cid:12)(cid:12)(cid:12)(cid:12) a ρ (cid:48) q (cid:12)(cid:12)(cid:12)(cid:12) ≤ qT (cid:27) Proof. The previous lemma ensure that the limsup of the right side is correct.So we just have to prove that the liminf is bigger than the left-hand side: let ρ (cid:48) belong to the segment [ q a − qT a ; q a + qT a ] , w be a point in P ( ρ (cid:48) ) and γ ∈ Γ T besuch as d ( γ.v, w ) ≤ aAT .Then, as usual, we get D v ( ρ (cid:48) ) ≤ d ( v, γ − w ) ≤ aA . And, as the formulas givenfor D v show, D v ( ρ (cid:48) ) − D v ( ρ ) = O ( | ρ (cid:48) ρ − | ) . We conclude by seeing that | ρ (cid:48) ρ − | isa big O of T . Hence, we have aA ≥ D v ( ρ ) + O ( T ) , which proves that the liminf isgreater than D v ( ρ ) a = inf (cid:110) | s − pq | for p ∈ Z (cid:111) . (cid:3) Remark. Of course this is not a valid way to compute the left-hand side of theequality ! It only shows that we may (cid:28)nd the dipophantine information in theorbit, hence gives us the hope that one may (cid:28)nd a direct proof of some results ondiophantine approximation from this viewpoint and generalize it to other situations(see below).Eventually let’s restrict our attention to some compact, for example an annulus A . Ledrappier’s theorem describe the asymptotic distribution of the sets Γ T v ∩ A ,i.e. the points of the orbit of v under Γ T which are inside A . Around every line L of rational slope and for every positive T , the proposition 4 gives us a domain of area (in fact the cone over a Cantor set) c L T - where c ( L ) only depends on L - inwhich no point of Γ T v ∩ A lies. So, globally speaking, we have found a set of areaat least cT , for some constant c , such that no point of the orbit of Γ T v ∩ A falls inthis set.As Ledrappier’s theorem implies that the number of points in Γ T v ∩ A is equiv-alent to a constant times T , the information given by proposition 4 seems to be avaluable one. 5. GeneralizationsThis concluding section is a mostly speculative one and far less elementary thanthe previous description. The point is that the method and the result concerning therepartition of the orbits of SL (2 , Z ) in the plane has been generalized, for exampleby Gorodnik [2], Gorodnik-Weiss [3] Ledrappier-Pollicott [6] and the author [4] toa wide variety of situations, which may be described with some simpli(cid:28)cations asfollows.Let G be a closed simple subgroup of GL ( n, R ) or GL ( n, Q p ) or a (cid:28)nite productof them. Let H be a closed subgroup of G that is either unipotent or simple (orsemidirect product of them, but with additional assumptions [4]), and Γ a latticein G . As G is included in a matrix algebra, we may choose a norm to compute thesize of an element of Γ thus de(cid:28)ning the ball Γ T . Remark that in all these knowncases, any lattice of H is (cid:28)nitely generated. Let x be a point of H \ G with denseorbit under Γ . Then the repartition of the orbit Γ T .x in H \ G may be described inthe same way as in theorem 1. For example, orbits of SL ( n, Z ) in R n belong to the known situations. Andthe same analysis as before leads to exactly the same conclusions, including thediophantine part. Moreover, we may give a description of the gaps in a moregeneral situation. Suppose that H \ G is embedded in a vector space, on which G acts linearily and the G -actions are compatible. Then H \ G may be equipped witha distance coming from a norm on the vector space. This situation is not so rareand may be found under some hypotheses using Chevalley’s theorem [1]. Moreoversuppose H has closed orbit in G/ Γ .We check below that the set of points in H \ G corresponding to closed orbit of H in G/ Γ of a given covolume ρ is a closed set. If this holds, the distance from a givenpoint x of dense orbit to this set is de(cid:28)ned and strictly positive, and the ball Γ T , asa (cid:28)nite set of invertible linear transformations, has a bounded contraction. Hencewe follow the description of the gaps made before for SL (2 , Z ) without di(cid:30)culties.So we conclude this paper on the following (may be well-known) lemma:Lemma 8. Let G be a locally compact group, H a closed subgroup of G with all itslattices (cid:28)nitely generated and Γ a lattice in G such that H ∩ Γ is a lattice in H ofcovolume one (to normalize the Haar measure on H ). Suppose that, if g n belongsto H for some g ∈ G and n integer, then g belongs to H .Then, for all ρ > , the subset P ( ρ ) of H \ G consisting of classes Hg such that g Γ g − ∩ H is a lattice in H of covolume ρ is a closed set.Remark. I tried to state it in a general enough setting, so there is in the statementthe two ad-hoc hypotheses I need below. It is easy to check that in the abovedescribed cases they are ful(cid:28)lled.Proof. Let x n = Hg n be a sequence of points in P ( ρ ) converging to x = Hg in G/ Γ . Suppose we made the choices such that g n converges to g in G . I do not want to state it precisely, nor will I be very precise in the following, as the settingsrequire some technical hypotheses useless to discuss here.
ANTONIN GUILLOUX
Let A be a compact subset in H of volume strictly greater than ρ . Then, byde(cid:28)nition, for every n , there is an element γ n in Γ such that A ∩ g n γ n g − n A is notempty. As A is compact and g n tends to g , the choices for the γ n ’s stay inside acompact subset, hence are in (cid:28)nite number. So there is a (cid:28)xed γ ∈ Γ such thatfor in(cid:28)nitely many n , the intersection A ∩ g n γg − n A is not empty. Conclusion: A ∩ gγg − A is not empty and g Γ g − ∩ H is a lattice in H of covolume at most ρ .We now prove that g Γ g − ∩ H e(cid:27)ectively has covolume ρ in H . We even provethe stronger fact: the sequence of subgroups Γ ∩ g − n Hg n is a stationnary sequence.Hence for n and m big enough, g − n g m normalizes H and let its Haar measureinvariant. The subgroup of the normalizer of H letting its Haar measure invariantis closed, so g − n g belongs to it, thus proving that g Γ g − ∩ H is of covolume ρ .As g Γ g − ∩ H is (cid:28)nitely generated, we just have to show that for any γ ∈ Γ , if g − γg is in H , then g − n γg n is in H for n big enough. So let A be a compact subsetin H of positive measure α such its images under g − Γ g ∩ H are disjoints. And let A (cid:48) be of the form A (cid:48) = ∪ ki =0 g − γgA , where k is bigger than ρα . Then for all n , thereexist a γ n ∈ Γ ∩ g − n Hg n such that, g n γ n g − n A (cid:48) ∩ A (cid:48) is not empty. As before, thereis only a (cid:28)nite number of possibilities for γ n , hence it takes some value γ (cid:48) in(cid:28)nitelymany times. Therefore gγ (cid:48) g − A (cid:48) ∩ A (cid:48) is not empty. By construction, γ (cid:48) is a power γ k of γ , and for in(cid:28)nitely many n , g n γ k g − n belongs to H . Now the hypothesis on H shows that g n γg − n also belongs to H .At this point we showed that for any γ in Γ ∩ g − Hg , there is an in(cid:28)nite numberof n such that γ belongs to Γ ∩ g − n Hg n . Using this fact along any subsequence,it shows that for n big enough, γ belongs to Γ ∩ g − n Hg n . And for n big enough, Γ ∩ g − n Hg n contains all the generators of Γ ∩ g − Hg . Hence for n big enough, thesubgroups Γ ∩ g − n Hg n and Γ ∩ g − Hg are the same one. This concludes the proofof this lemma. (cid:3) References
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