Chapter 2. Periodic points of toral. automorphisms. 2.1 General introduction

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1 Chapter 2 Periodic poits of toral automorphisms 2.1 Geeral itroductio The automorphisms of the two-dimesioal torus are rich mathematical objects possessig iterestig geometric, algebraic, topological ad measure theoretic properties. The torus T 2 = R2 Z 2 is here viewed as the topological group [0, 1) [0, 1) with coordiate-wise additio modulo 1. Let GL(2, Z) be the set of all 2 2 matrices A = ad Det(A) = ad bc = ±1. a c b d, where a, b, c, d Z Each such matrix Agives a liear map o R 2 by x 1 x 2 A x 1 x 2 We defie a automorphism o the torus T A : T 2 T 2 by T A (x 1, x 2 ) = (ax 1 + bx 2, cx 1 + dx 2 )(mod1). Now we have, 22

2 CHAPTER 2. PERIODIC POINTS OF TORAL AUTOMORPHISMS 23 Propositio [29] Every automorphism T A (as defied above) o the torus is a homeomorphism. Proof. The map T is clearly cotiuous, sice if x 1 y 1, x 2 y 2 < ɛ the (T A (x 1, x 2 )) 1 (T A (y 1, y 2 )) 1 < ( a + b )ɛ ad (T A (x 1, x 2 )) 2 (T A (y 1, y 2 )) 2 < ( c + d )ɛ.(here suffix 1 refer to the first coordiate ad 2 refers to the secod coordiate.) To see that T A is ivertible we ote that if we write the iverse matrix A 1 = d ad bc c ad bc b ad bc a ad bc the sice ad bc = ±1 we see that A 1 GL(2, Z). The iverse to T A is the the toral automorphism associated to A 1, i.e. T A 1. O the other had, i the followig propositio we prove that every cotiuous automorphism o the torus is iduced by a matrix from GL(2, Z). Let Aut(T 2 ) to deote the set of all cotiuous automorphisms o the torus. Propositio The above map A T A from GL(2, Z) to Aut(T 2 ) is surjective. Proof. let φ : T 2 T 2 be ay cotiuous toral automorphism. Sice φ is cotiuous at (0, 0) there exist δ > 0 such that φ(([0, δ) [0, δ))) [0, 1) [0, 1 ) ad such that 2 2 φ(x + Y ) = φ(x) + φ(y ) for all X, Y [0, δ) [0, δ), where + deotes the usual additio i R 2. Now, observe that φ(λx) = λφ(x) for all λ (0, 1) ad for all X, Y [0, δ) [0, δ), because the set {λ (0, 1) φ(λx) = λφ(x)} is a closed set cotais all dyadic ratioals (Sice the set cotais λ = 1, usig the additivity of φ, it cotais all umbers of form 2 λ = m 2 ). The set of all dyadic ratioals is dese i [0, 1] ad the by cotiuity the set cotais all λ (0, 1). Hece φ [0,δ) [0,δ) = L [0,δ) [0,δ) for some liear trasformatio L : R 2 R 2. This liear trasformatio iduces a iteger matrix A with determiat ±1 such that

3 CHAPTER 2. PERIODIC POINTS OF TORAL AUTOMORPHISMS 24 Ax = φ(x) for all x T 2 [ The kerel of a edomorphism (differet from the zero map), o a coected topological group caot have oempty iterior ]. Hece the proof. Remark I fact the 1-1 correspodece i the above propositio is a group isomorphism. For each A GL(2, Z), let P (T A ) deote the set of all periodic poits of T A. Propositio [21] For ay A GL(2, Z) the set P (T A ) is dese i [0, 1) [0, 1). Proof. Let A = a b GL(2, Z). c d We prove that P (T A ) Q 1 Q 1, where Q 1 = Q [0, 1). A geeral elemet i Q 1 Q 1 is of the form x = ( p 1 q, p 2 q ) where p 1, p 2, q Z with 0 p 1, p 2 < q. We ote that T A (X) = (fractioal part of ap 1 q + bp 2 q, fractioal part of cp 1 q + dp 2 q ) = a elemet of the form ( m, ) where 0 m, < q. Note that, for a fixed q N, the q q set {( m, )/0 m, < q; m, N} is ivariat ad fiite. Hece the orbit of x is q q fiite ad therefore evetually periodic. Now, the result follows from the fact that for ivertible maps the evetually periodic poits are periodic poits. Note that, for a toral automorphism T A, the periodic poits with period are solutios of the cogruet equatio A x = x(mod1). The followig propositio is i this directio. Lemma [28] If T : R 2 R 2 is a isomorphism the for every Riema mea-

4 CHAPTER 2. PERIODIC POINTS OF TORAL AUTOMORPHISMS 25 surable set ( havig Jorda cotet) S R 2, T (S) is Riema measurable ad Area(T (S)) = Det(T ) Area(S) Propositio [21] Let A GL(2, Z). The (1) The umber of solutios of A x = x i [0, 1) [0, 1), is Det(A I), provided Det(A I) 0. (2) If Det(A I) = 0 the A x = x has ifiitely may solutios i [0, 1) [0, 1). Proof. (1) Suppose Det(A I) 0. The ote that the umber of solutios of the equatio, A x = x i [0, 1) [0, 1) is equal to the umber of iteger poits i the image of [0, 1) [0, 1) uder A I, treated as a liear map from R 2 to R 2. Note also that the image of [0, 1) [0, 1) uder A I is a parallelogram ad hece the umber of iteger poits i it, is equal to its area, which is equal to Det(A I), by previous lemma. (2) Note that, whe Det(A I) = 0, the system (A I) x 1 has ifiitely may solutios i [0, 1) [0, 1). x 2 = 0 0 itself, Observe that for ay cotiuous toral automorphism T A the set P (T A ) is a subgroup of the torus. We ow ask: Which subgroups of [0, 1) [0, 1) arise i this way? 2.2 Automorphisms with determiat 1 ad trace 2 We start by listig all the matrices from GL(2, Z) with determiat 1 ad trace 2.

5 CHAPTER 2. PERIODIC POINTS OF TORAL AUTOMORPHISMS 26 Defiitio For m, Z we defie, m A m, = (m 1) m m if 0 if = 0 Note that Det(A m, ) = 1 ad T r(a m, ) = 2 for all m, Z. Propositio If A GL(2, Z) is such that Det(A) = 1 ad T race(a) = 2 the A = A m, for some m, Z such that divides (m 1) 2. Proof. Let A = a b GL(2, Z) be such that Det(A) = 1 ad T r(a) = 2. The c d we have a + d = 2 ad ad bc = 1. Hece bc = (a 1) 2 If b 0 the c = (a 1)2 b a iteger ad therefore A = A a,b. If b = 0 the a = d = 1 ad c ca be ay iteger. Hece A = A c+1,0. Remark Note that the characteristic polyomial of ay matrix of type A m, is (x 1) 2 ad hece 1 is a eige value. Therefore o matrix of type A m, ca be hyperbolic. We ow calculate the set of all periodic poits of the cotiuous toral automorphisms iduced by the matrices of form A m,. By iductio we ca prove that, for ay k N, A k m, = A km k+1,k for all m, Z. Let A GL(2, Z). The X T 2 is a periodic poit of T A if ad oly if it is a solutio of A k X = X for some k N. Thus, P (T A ) = k=1 {X T2 : A k X = X}. Notatio Let Q 1 be the set of all ratioal poits i [0, 1). Give r Q, we

6 CHAPTER 2. PERIODIC POINTS OF TORAL AUTOMORPHISMS 27 write S r = {(x, y) T 2 such that rx + y is ratioal } ad let S = Q 1 [0, 1). Note that S 0 = [0, 1) Q 1. Theorem For m, Z, the set of all periodic poits of the cotiuous toral automorphism T Am, is either the set S r for some r Q { } or T 2. Proof. Case: 1 Whe m 1 ad 0. The periodic poits with period k ca be obtaied by solvig the equatio A k m,x = X(mod1), which is equivalet to the system of liear equatios, (km k + 1)x 1 + kx 2 = x 1 + m 1 for some m 1, m 2 Z k(m 1)2 x 1 + (1 km + k)x 2 = x 2 + m 2 That is, (x 1, x 2 ) T 2 satisfies the equatio A X = X if ad oly if if ad oly if (km k + 1)x 1 + kx 2 = m 1 Z k(m 1)2 x 1 + (1 km + k)x 2 = m 2 Z which implies that, (km k + 1)x 1 + kx 2 = m 1 Z k(m 1)2 x 1 + (1 km + k)x 2 = m 2 Z (m 1)x 1 + x 2 Q (m 1)2 x 1 + (1 m)x 2 Q (2.1) (2.2)

7 CHAPTER 2. PERIODIC POINTS OF TORAL AUTOMORPHISMS 28 which reduces to solvig the sigle equatio, (m 1) x 1 + x 2 Q. Coversely, if (m 1) x 1 + x 2 Q holds. The (x 1, x 2 ) satisfies the equatio (2.2). The we ca prove that (x 1, x 2 ) satisfies the equatio (2.1) for some suitable k. P (T Am, ) = {(x 1, x 2 ) (m 1) x 1 + x 2 Q} = S (m 1) as desired. [ From the equatios (2.1) ad (2.2), it is observed that: (x 1, x 2 ) is a fixed poit of T Am, if ad oly if ( x 1 k, x 2 k ) is a fixed poit of T k A m,.] Case: 2 Whe m = 1 ad = 0. We get A 1,0 is the idetity matrix ad hece P (T Am, ) = F ix(t Am, ) = T 2. Case: 3 Whe m 1 ad = 0. We have A m,0 = A km k+1,0 1 0 m 1 1. Note that, for ay k N we have A k m,0 = Now, solvig the cogruece equatio A k m,0x = X(mod1) is equivalet to solvig the sigle coditio k(m 1)x 1 Z. This is implies that x 1 Q 1. Coversely, as before if x 1 Q 1 the we ca fid k Z such that k(m 1)x 1 Z. Hee P (T A ) = Q 1 [0, 1) = S. Case: 4 Whe m = 1 ad 0. I this case A 1, = The for ay k N we have A k 1, = A 1,k Now, solvig the cogruet equatio A k 1,X = X(mod1) is equivalet to solvig the sigle coditio kx 2 Z. This implies that x 2 Q 1..

8 CHAPTER 2. PERIODIC POINTS OF TORAL AUTOMORPHISMS 29 Coversely, if x 2 Q 1 the we ca fid some k Z such that k(m 1)x 2 Z. Hece P (T A ) = [0, 1) Q 1 = S 0. Remark Let A GL(2, Z) be of A m, type. The from the relatio, A k m, = A km k+1,k ( for all k N), it is clear that A ad its all powers pertai to the same case amog the four cases discussed i the previous propositio. Hece they have the same set of periodic poits. Remark The set S r ca be thought of as the poits o the lie through the origi with slope r ad its ratioal traslates i [0, 1) [0, 1). From above propositio, r = (m 1) whe m 1. O the other had give ay ratioal r = p q Q, we ca fid m, Z with (m 1) 2 such that p = m 1. For, choose m = pq + 1, = q q2. Hece every S r arises as P (A m, ) for some m, Z. Propositio The followig are equivalet for a subset of the torus. (1) It is P (T Am, ) for some A m, GL(2, Z). (2) It is S r for some r Q { }. Proof. Follows from the above remark. 2.3 Mai theorem Defiitio A cotiuous toral automorphism T A GL(2, Z) is said to be hyperbolic if A has o eige values with absolute value 1. Example The matrices of the type A m, are ot hyperbolic, because 1 is a eige value.

9 CHAPTER 2. PERIODIC POINTS OF TORAL AUTOMORPHISMS 30 It is already kow [21] that for a hyperbolic cotiuous toral automorphism, the periodic poits are precisely the ratioal poits. I this chapter, we calculate the set of periodic poits for other cotiuous toral automorphisms; this happes to be the subgroup of T 2 geerated by Q 1 Q 1 (a lie with ratioal slope). I fact, for all o-hyperbolic cotiuous toral automorphism, there are ucoutably may periodic poits. Lemma If A = 1)(d 1) bc 0. a c b d GL(2, Z) is hyperbolic the Det(A I) = (a Proof. Let A be hyperbolic. Now, suppose (a 1)(d 1) bc = 0 = (ad bc) (a+d)+1 Case: 1 If ad bc = 1, the a + d = 2. Which implies by propositio that A = A m, for some m,. Which is a cotradictio, by remark Case: 2 If ad bc = 1, the a + d = 0. Therefore the characteristic polyomial is x 2 1. Hece the eige values are ±1. Which cotradicts hypothesis. Remark If A is hyperbolic the so is A for all N. ( If λ is a eige value of A,the λ is eige value of A ). The Det(A I) 0 for all N. Hece the above lemma will apply to all the positive powers of A. Propositio If T A is hyperbolic the P (T A ) = Q 1 Q 1. Proof. The fixed poits of A are give by the cogruece equatio A X X(mod1). If we let, A = a c c x 1 (d 1)x 2 = k 2 Z. b d the we get, (a 1)x 1 b x 2 = k 1 Z ad Now the coditio c b (a 1)(d 1) 0 guaratees that the above system of liear equatios is cosistet ad its solutios are havig ratioal coordiates (by Cramer s rule). Hece P (Ã) = Q 1 Q 1.

10 CHAPTER 2. PERIODIC POINTS OF TORAL AUTOMORPHISMS 31 Eve though we assumed A to be hyperbolic i the above propositio, we have ot used its full stregth. What we eeded oly is that the determiat Det(A I) = c b (a 1)(d 1) is ozero for all N. Thus, i view of propositio the previous propositio ca be improved/restated as Propositio If T A is a toral automorphism such that for each N there are oly fiitely may periodic poits with period, the P (T A ) = Q 1 Q 1. We are ow i a positio to prove the mai theorem. Theorem For ay cotiuous toral automorphism T A, the set P (T A ) of periodic poits of T A is oe of the followig: 1. Q 1 Q S r for some r Q { }; where S r = {(x, y) T 2 rx + y is ratioal }. 3. T 2 Proof. Let A = a c b d GL(2, Z). For ay N we write A = Case: 1 a c b d where a, b, c, d are itegers. If Det(A I) = c b (a 1)(d 1) 0 for all N the proof follows from Propositio Case: 2 Suppose Det(A I) = c b (a 1)(d 1) = 0 for some N. Let S = {k Det(A k I) = c k b k (a k 1)(d k 1) = 0}. Subcase: (2a)

11 CHAPTER 2. PERIODIC POINTS OF TORAL AUTOMORPHISMS 32 If Det(A k ) = 1 for some k S, the T r(a k ) = 0. Therefore A k = Note that A 2k = Hece P (A) = T 2 = S 0 sice P (A) P (A k ) k N. a Subcase: (2b) If Det(A k ) = 1 for all k S the T r(a k ) = 2. Therefore A k = A m, for some m, Z From remark 2.2.6, it follows that A k ad its powers amely, A 2k, A 3k, A 3k,... share the same set of periodic poits. Note that, for ay j N the periodic poits of T A with period j are cotaied i P (A jk ). Hece, from theorem 2.2.5, P (T A ) = S r for some r Q { }. c b a We coclude this chapter with the followig remark. Remark Eve though there are apparetly four kids of subsets which ca appear as the set of periodic poits for some cotiuous toral automorphism, there are oly three upto homeomorphism.