The Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria

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1 ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Common Homoclinic Points of Commuting Toral Automorphisms Anthony Manning Klaus Schmidt Vienna, Preprint ESI 503 (1997) November 17, 1997 Supported by Federal Ministry of Science and Research, Austria Available via

2 COMMON HOMOCLINIC POINTS OF COMMUTING TORAL AUTOMORPHISMS ANTHONY MANNING AND KLAUS SCHMIDT Abstract. The points homoclinic to 0 under a hyperbolic toral automorphism form the intersection of the stable and unstable manifolds of 0. This is a subgroup isomorphic to the fundamental group of the torus. Suppose that two hyperbolic toral automorphisms commute so that they determine a Z 2 -action, which we assume is irreducible. We show, by an algebraic investigation of their eigenspaces, that they either have exactly the same homoclinic points or have no homoclinic point in common except 0 itself. We prove the corresponding result for a compact connected abelian group, and compare the two proofs. 1. Introduction Let n 2, and let A 2 GL(n; Z) be a linear ergodic automorphism of the n-torus X = T n = R n =Z n. A point x 2 X is homoclinic for A if lim jnj!1 (A n x; 0) = 0, where is the usual Euclidean metric on X. If the automorphism A is nonhyperbolic then A has no nonzero homoclinic points (cf. e.g. [4]). If A is hyperbolic we can describe the set A X of homoclinic points of A as follows: let A act linearly on R n and denote by S A and U A the stable (= contracting) and unstable (= expanding) subspaces of A, i.e. S A = fw 2 R n : lim A n w = 0g; n!1 U A = fw 2 R n : lim A?n w = 0g; n!1 R n = S A U A : If : R n 7?! T n is the quotient map, then A = f(s A \ (U A + m)) : m 2 Z n g; and it is well-known and easy to see that A is a dense subgroup of X. Now suppose that A; B are two commuting linear hyperbolic automorphisms of X = T n. We write : n = (n 1 ; n 2 ) 7! n = A n 1 B n 2 (1.1) for the Z 2 -action generated by A and B and conclude from Theorem 4.1 in [4] that has no nonzero homoclinic point, i.e. that x = 0 is the only point in X with lim n!1 n x = 0: 1991 Mathematics Subject Classication. Primary 28D15; Secondary 11R04, 11R32, 15A18, 22C05, 58F15. The second author would like to thank the Austrian Academy of Sciences and the Royal Society for partial support while this work was done. 1

3 2 ANTHONY MANNING AND KLAUS SCHMIDT It may happen, however, that there exist nonzero points x 2 X which are homoclinic both for A and B. In fact, if the action in (1.1) is irreducible in the sense that X has no proper subtorus which is invariant both under A and B, then we shall see that either A \ B = f0g or A = B ; more generally, if m; n 2 Z 2 are chosen so that m and n are hyperbolic, then either m = n or m \ n = f0g, and both possibilities occur for suitable choices of m and n. If the Z 2 -action is not irreducible then we can obviously not expect such a clear-cut dichotomy. In order to make the above statements more precise we x commuting linear hyperbolic automorphisms A; B 2 GL(n; Z) of T n and assume that the Z 2 -action generated by A; B via (1.1) is irreducible. Then it is easy to see that A and B are simultaneously diagonalisable over the algebraic closure Q of the eld Q of rational numbers; in particular there exists a basis w 1 ; : : :; w n of Q n consisting of common eigenvectors of A and B (cf. Lemma 2.1). For every k = 1; : : :; n we dene a homomorphism k from Z 2 into the multiplicative group of positive real numbers by setting k (n) = k n (w k )k=kw k k for every n 2 Z 2, where k k denotes the Euclidean norm on C n (in other words, k (n) is the absolute value of the eigenvalue of n for the eigenvector w k ). For k = 1; : : :; n we set H 0 k = fm 2 Z2 : j k (m)j = 1g and observe that H 0 k is the intersection of Z2 with a hyperplane (cf. Remark 3.3) H k R 2 and that E = Z 2 r n[ k=1 H 0 k = fm 2 Z2 : m is hyperbolicg (1.2) is therefore nonempty. We shall prove the following result. Theorem 1.1. Let A; B be commuting linear hyperbolic automorphisms of a nite-dimensional torus X = T n, and let be the Z 2 -action generated by A; B (cf. (1.1)). For every m = (m 1 ; m 2 ) 2 E (cf. (1.2)) we denote by m the group of homoclinic points of m. If m; n 2 E, then m \ n = 8 >< >: m f0g if either m; n or m;?n are not separated by any of the hyperplanes H k ; k = 1; : : :; m; otherwise: Theorem 1.1 is a special case of a more general result for whose statement we need a few denitions. Let : n 7! n be a Z 2 -action by continuous automorphisms of an innite compact, connected, abelian group X. The action is called ergodic if it is so with respect to the normalised Haar measure X of X, expansive if there exists an open neighbourhood N(0) of the identity element 0 2 X with T n2z 2 n (N(0)) = f0g, and irreducible or almost minimal if every closed, -invariant subgroup Y ( X is nite. Every expansive Z 2 -action by automorphisms of X is ergodic by Corollary 3.10 in [6]. Theorem 1.2. Let be an expansive, ergodic and irreducible Z 2 -action by automorphisms of a compact, connected, abelian group X. Put E = fm 2 Z 2 : m is expansiveg:

4 COMMON HOMOCLINIC POINTS OF COMMUTING TORAL AUTOMORPHISMS 3 Then the following is true. (1) There exist nitely many hyperplanes H 1 ; : : :; H m in R 2 Z 2 such that E = Z 2 r m[ k=1 H k = fm 2 Z 2 : m is expansiveg: (2) For every m 2 Z 2, the group of homoclinic points m of m is dense in X if and only if m 2 E, and m = f0g otherwise. (3) If m; n 2 E then m \ n = 8 >< >: m f0g if either m; n or m;?n are not separated by any of the hyperplanes H k ; k = 1; : : :; m; otherwise: If we call two elements m; n 2 E equivalent when they are not separated by any of the hyperplanes H k, then this relation partitions the set E into nitely many cones, each one of which is called a Weyl chamber of in [2]{[3] (cf. also [1]). In this terminology Theorems 1.1{1.2 say that m and n share their homoclinic points if and only if m; n or m;?n lie in the same Weyl chamber of E ; if not, then they have no nonzero common homoclinic point. Although Theorem 1.1 is a consequence of Theorem 1.2 we shall prove it separately in Section 2 with a direct argument. The proof of Theorem 1.2 uses a little more algebraic machinery and will occupy Section The proof of Theorem 1.1 Let A; B 2 GL(n; Z) be commuting linear hyperbolic automorphisms of X = T n such that the resulting Z 2 -action on X is irreducible. We denote by K C the smallest subeld containing all eigenvalues of A and B. Lemma 2.1. The matrices A and B are simultaneously diagonalisable over K, i.e. there exists a basis w 1 ; : : :; w n of K n consisting of common eigenvectors of A and B (considered as acting linearly on K n ). Proof. The assertion of the lemma is equivalent to the corresponding statement for the transposes A > and B > of A and B. The matrix A > has an eigenvector in K n with eigenvalue a 0, say, and the subspace W = fw 2 K n : Aw = a 0 wg is invariant under B >. Since B > has an eigenvector in W there exists a common eigenvector w 2 K n of A > and B > with eigenvalues a 0 ; b 0 2 K under A > and B >, respectively. We write? for the Galois group of the extension K : Q and let? act diagonally on K m for every m 1. For every 2?, (w) is a common eigenvector of A > and B > with eigenvalues (a; b) = ((a 0 ); (b 0 )). Put, for every (a; b) 2 E =?(a 0 ; b 0 ) K 2, and set V (a;b) = fw 2 K n : A > w = aw; B > w = bwg 6= f0g (2.1) V = M (a;b)2e V (a;b) : (2.2)

5 4 ANTHONY MANNING AND KLAUS SCHMIDT Then f0g 6= V K n, and the lemma is proved if we can show that V = K n : (2.3) Suppose that V 6= K n. We denote by R 2 = Z[u 1 ; u 2 ] the ring of polynomials with integral coecients in the variables u 1 ; u 2, write p = ff 2 R 2 : f(a; b) = 0g for the ideal in R 2 consisting of all polynomials vanishing at (a; b), and set Hilbert's Nullstellensatz implies that and hence that Hence p(a > ; B > ) = ff(a > ; B > ) : f 2 pg: V p = f(c 1 ; c 2 ) 2 C 2 : f(c 1 ; c 2 ) = 0 for every f 2 pg = E f0g 6= V = fw 2 K n : Cw = 0 for every C 2 p(a > ; B > )g 6= K n : f0g 6= fw 2 Q n : Cw = 0 for every C 2 p(a > ; B > )g 6= Q n ; and the annihilator of M = fw 2 Z n : Cw = 0 for every C 2 p(a > ; B > )g in X = c Zn is a proper subtorus of X which is invariant under A and B, contrary to our assumption of irreducibility. This contradiction proves both (2.3) and the lemma. Lemma 2.2. If an element 0 6= n 2 Z n is expressed as n = X (a;b)2e w (a;b) with w (a;b) 2 V (a;b) for every (a; b) 2 E (cf. (2.1){(2.3)), then w (a;b) 6= 0 for every (a; b) 2 E. P Proof. Suppose that (a;b)2e w (a;b) = n 2 Z n with w (a0 ;b 0 ) 6= 0 and w (a 00 ;b 00 ) = 0 for two eigenvalue pairs (a 0 ; b 0 ); (a 00 ; b 00 ) 2 E. We choose an element 2? with (a 0 ) = a 00 ; (b 0 ) = X b 00 and obtain that 0 = w (a;b)? (w (a;b) ) (a;b)2e has a nonzero component in V (a 00 ;b 00 ), which is absurd. This argument is a special case of that in [5]. Lemma 2.3. For every homoclinic point x 2 A T n there exists a point y 2 (R\ K) n with (y) = x, where : R n 7?! T n is the quotient map. Proof. Let a 1 ; : : :; a r and a r+1 ; : : :; a n be the expanding and contracting eigenvalues Q Q r of A, counted with multiplicity, and put A s = i=1 (A? a ii), n A u = i=r+1 (A? a ii), where I is the n n identity matrix. Then A s S A = A u U A = f0g. Every x 2 A is of the form x = (w) with fwg = S A \ (U A + m) for some m 2 Z n. If we write a 1 ; : : :a n 2 (R \ K) n and b 1 ; : : :; b n 2

6 COMMON HOMOCLINIC POINTS OF COMMUTING TORAL AUTOMORPHISMS 5 (R \ K) n for the row vectors of A s and A u, respectively, then w is the unique solution in R n of the inhomogeneous system of linear equations ha 1 ; wi = 0; : : :; ha n ; wi = 0; hb 1 ; w? mi = 0; : : :; hb n ; w? mi = 0; (2.4) where h; i denotes the Euclidean inner product in R n. As the coecients of the system (2.4) all lie in K, we conclude that the system (2.4) also has a unique solution in K n, i.e. that w 2 (R\ K) n. Proof of Theorem 1.1. We let A; B act linearly on R n and write R n = S A U A = S B U B for the splittings of R n into the stable (= contracting) and unstable (= expanding) subspaces of A and B. If S A = S B and U A = U B then it is clear that A = B. If S A = U B and U A = S B we can invert B and again arrive at the conclusion that A = B. In any other case (after replacing B by B?1 and interchanging A and B, if necessary) we have f0g 6= U A \ U B 6= U A ; U A + U B 6= R n : (2.5) If a point x 2 T n is homoclinic for both A and B then there exist elements p; q; r 2 Z n and w 2 R n such that fwg = S A \ (U A? p) = (S B? q) \ (U B? r): (2.6) By Lemma 2.3, w 2 (R\ K) n, so that we may replace the subspaces S A ; S B ; U A ; U B in (2.5){(2.6) by the subspaces S 0 A = S 0 B = M M f(a;b)2e:jaj<1g f(a;b)2e:jbj<1g V (a;b) ; U 0 A = V (a;b) ; U 0 B = M M f(a;b)2e:jaj>1g f(a;b)2e:jbj>1g V (a;b) ; V (a;b) : Then (2.6) shows that q 2 S 0 A +S0 B and p?r 2 U 0 A +U 0 B. However, according to (2.5) there exist elements (a; b); (a 0 ; b 0 ); (a 00 ; b 00 ) 2 E with V (a;b) U 0 A \U 0 B, V (a0 ;b 0 ) \ U 0 A = V (a 0 ;b 0 ) \ U 0 B = f0g, V (a 00 ;b 00 ) \ U 0 B = f0g, V (a 00 ;b 00 ) U 0 A and hence with V (a;b) \ (S 0 A + S0 B ) = f0g and V (a 0 ;b 0 ) \ (U 0 A + U 0 B ) = f0g, so that q = p?r = 0 by Lemma 2.2. Then w 2 S 0 A \S0 B, w+p 2 U 0 A \U 0 B, and hence p 2 (S 0 A \ S0 B ) + (U 0 A \ U 0 B ). Since V (a 00 ;b 00 ) \ ((S 0 A \ S0 B ) + (U 0 A \ U 0 B )) = f0g, this is an expression of p as a sum of common eigenvectors of A; B with zero component in V (a 00 ;b ). Lemma 2.2 yields that p = 0, so that w 2 S 0 00 A \ U 0 A = f0g. 3. The proof of Theorem 1.2 Let Q = Qrf0g, c = (c 1 ; c 2 ) 2 (Q ) 2, and let = Q(c) be the algebraic number eld generated by c 1 ; c 2. We write PK ; Pf K and PK 1 for the sets of places (= equivalence classes of valuations), nite places and innite places of K, choose for each v 2 PK a valuation j jv in v, and denote by K v the completion of K with respect to the valuation j j v. Proceeding as in Section 7 in [7] we set P (c) = P K 1 [ fv 2 P K f : jc i j v 6= 1 for some i = 1; 2g

7 6 ANTHONY MANNING AND KLAUS SCHMIDT and put R c = fa 2 K : jaj v 1 for every v 2 P K r P (c)g; Z c = Y v2p (c) K v ; Y c = Z c =i P (c) (R c ); (3.1) where i P (c) : R c 7?! Z c is the diagonal embedding a 7! (a v ; v 2 P (c)) 2 Z c with a v = a for every v 2 P (c). Then i P (c) (R c ) is a discrete and co-compact subgroup of the locally compact abelian group Z c and Y c is (isomorphic to) the Pontryagin dual c Rc of R c. We dene a Z 2 -action ^ on R c by setting ^ n (a) = c n a for every a 2 R c and n = (n 1 ; n 2 ) 2 Z 2, where c n = c n 1 1 cn 2 2. The Z2 -action c on Y c = c Rc dual to ^ is given by n c ((a v; v 2 P (c)) + i P (c) (R c )) = (c n a v ; v 2 P (c)) + i P (c) (R c ) (3.2) for every n 2 Z 2 and (a v ; v 2 P (c)) + i P (c) (R c ) 2 Y c. As an application of Theorem 29.2 and Corollary 7.4 in [7] we obtain the following lemma. Lemma 3.1. Let be an irreducible Z 2 -action by automorphisms of a compact, connected, abelian group X. Then there exist a point c = (c 1 ; c 2 ) 2 (Q ) 2 and continuous, surjective, nite-to-one group homomorphisms : Y c 7?! X, : X 7?! Y c such that n c = n and n = n c for every n 2 Z 2. Proof. The only assertion which goes beyond the statements of Theorem 29.2 and Corollary 7.4 in [7] is the existence of the homomorphism : X 7?! Y c with the required properties. The homomorphism : Y c 7?! X allows us to regard ^X as a ^-invariant subgroup of nite index in R c. Choose an integer L 1 with LR c ^X R c, denote by : X 7?! Y c the surjective homomorphism dual to the inclusion R c = LRc ^X, and note that is nite-to-one. Lemma 3.2. Let be an irreducible Z 2 -action by automorphisms of a compact, connected, abelian group X, and let c be the Z 2 -action dened in Lemma 3.1. The following conditions are equivalent. (1) is expansive; (2) c is expansive; (3) The orbit of c under the diagonal action on K 2 of the Galois group? of K : Q does not contain any point c 0 = (c 0 1 ; c0 2 ) with jc0 1 j = jc0 2 j = 1. Proof. This is Proposition 7.2 (5) in [7]. Proof of Theorem 1.2. Let be an expansive and irreducible Z 2 -action on a compact, connected abelian group X. Since the homomorphisms ; in Lemma 3.1 are nite-to-one, they are injective on the groups n c and n of homoclinic points for every n 2 Z 2 for which n or, equivalently, n c, is expansive. It follows that Theorem 1.2 holds for an expansive Z 2 -action if and only if it holds for a corresponding action c. This observation allows us to assume without loss of generality that the Z 2 -action in the statement of the theorem is of the form = c for some c = (c 1 ; ; c 2 ) 2 (Q ) 2.

8 COMMON HOMOCLINIC POINTS OF COMMUTING TORAL AUTOMORPHISMS 7 Equation (3.2) shows that an automorphism n = n c is expansive on X = Y c if and only if jc n j v 6= 1 for every v 2 P (c). We write : Z c 7?! Y c for the quotient map and denote by S n = f(a v ; v 2 P (c)) 2 Z c : a v = 0 for every v 2 P (c) with jc n j v > 1g; U n = f(a v ; v 2 P (c)) 2 Z c : a v = 0 for every v 2 P (c) with jc n j v < 1g the stable and unstable subspaces of n in Z c. An element x 2 X lies in n if and only if it is of the form x = (w) for some w 2 S n \ (U n + a) for some a 2 i P (c) (R c ). We complete the proof in the same way as that of Theorem 1.1. For every v 2 P (c), the map n 7! jc n j v is a homomorphism from Z 2 into the multiplicative group of positive real numbers, and we set H 0 v = fn 2 Z 2 : jc n j v = 1g. If c is expansive, Lemma 3.2 shows that each of these homomorphisms is nontrivial, so that H 0 v is the intersection with Z 2 of a hyperplane H v R 2. It follows that the set [ E = Z 2 r H 0 v = fn 2 Z 2 : n c is expansiveg v2p (c) is nonempty. If m; n 2 Z 2 satisfy that either m and n or m and?n lie on the same side of H v for every v 2 P (c), then the homoclinic points of m and n coincide. If m and n do not satisfy this condition then we can interchange m and n and replace n by?n, if necessary, and assume that S m + S n 6= Z c, S m + U n 6= Z c and U m + U n 6= Z c. If a point x 2 Y c is homoclinic both for m and n then there exist elements w 2 Z c and a; b; c 2 i P (c) (R c ) with (w) = x and fwg = S m \ (U m? a) = (S n? b) \ (U n? c) and, exactly as in the proof of Theorem 1.1, we obtain that a = b = c = w = 0. Remark 3.3. The group Y c in (3.1) is a nite-dimensional solenoid; in fact, Y c is a nite-dimensional torus if and only if each c i is an algebraic unit (cf. [6] and [7]). In the latter case Theorem 1.2 reduces to Theorem 1.1, and it may be useful to compare the two proofs in this case. Since each c i is an algebraic unit, P (c) = PK 1 and R c is the ring of algebraic integers in K = Q(c). Q Rewriting the covering space R n in the proof of Theorem 1.1 as Z c = v2p (c) K v amounts to a change of basis. In this new basis Lemma 2.3 is obvious, since every homoclinic point x 2 Y c is the image under of a point w = (a v ; v 2 P 1 (c)) 2 Z c with a v = ( a if jc n j v < 1; 0 otherwise for some a 2 R c. Lemma 2.2 is equally obvious, since every element of i P (c) (R c ) has a nonzero component in each K v ; v 2 P (c). The proof of Theorem 1.2 in this section yields an explicit description of the walls H v ; v 2 P (c), of the Weyl chambers of the Z 2 -action : they are the kernels of the homomorphism (t 1 ; t 2 ) 7! jc 1 j t 1 v jc 2 j t 2 v from R 2 to the

9 8 ANTHONY MANNING AND KLAUS SCHMIDT multiplicative group of positive real numbers, corresponding to the places v 2 P (c). We illustrate this with some examples. Examples 3.4. (1) Let A; B be commuting expansive automorphisms of the n-torus T n for which the Z 2 -action in (1.1) is irreducible. By comparing the proofs of the Theorems 1.1 and 1.2 we see that is a nite-to-one factor of a Z 2 -action c for some c = (c 1 ; c 2 ), where c 1 and c 2 are eigenvalues of A and B, respectively. We set K = Q(c), write E for the orbit of c under the diagonal action of the Galois group, and note that each (a; b) 2 E corresponds to a place v 2 P1 K with jc m 1 1 cm 2 2 j v = ja m 1 b m 2 j for every m = (m 1 ; m 2 ) 2 Z 2. The hyperplanes H v ; v 2 P (c) = P 1 (c) are the kernels of the homomorphisms (t 1 ; t 2 ) 7! jaj t 1 jbj t 2 from R 2 to the multiplicative group of positive real numbers, with (a; b) 2 E. Since these hyperplanes are lines with irrational slopes their intersection with Z 2 is equal to f0g, so that n is expansive whenever 0 6= n 2 Z 2. The automorphisms m ; n corresponding to two elements m; n 2 Z 2 have the same homoclinic points if and only if one of the line segments tm (1? t)n; 0 t 1, does not intersect any H v ; v 2 P (c). (2) Let c = (2; 3). Then K = Q, P (c) = f1; 2; 3g (where we are identifying the nite places of Q with the rational primes), Z c = R Q 2 Q 3 (where Q p denotes the p-adic integers), R c = Z[ 1 6 ], and Y c = Z c =i P (c) (R c ) is the 6-adic solenoid. The hyperplanes H v ; v 2 P (c), are given by H 1 = f(t 1 ; t 2 ) 2 R 2 : 2 t 1 3 t 2 = 1g; H 2 = f(t 1 ; t 2 ) 2 R 2 : t 1 = 0g; H 3 = f(t 1 ; t 2 ) 2 R 2 : t 2 = 0g; and the set of expansive elements of the action c is given by E = fn = (n 1 ; n 2 ) 2 Z 2 : n 1 n 2 6= 0g: In particular, any two elements m; n in the strictly positive quadrant of Z 2 lead to automorphisms m ; n with the same homoclinic points, whereas the only homoclinic point common to (1;1) and (1;?1) is 0. References [1] M. Boyle and D. Lind, Expansive subdynamics, Trans. Amer. Math. Soc. 349 (1997), 55{102. [2] A. Katok and R.J. Spatzier, First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity, Inst. Hautes Etudes Sci. Publ. Math. 79 (1994), 131{156. [3] A. Katok and R.J. Spatzier, Invariant measures for higher-rank hyperbolic abelian actions, Ergod. Th. & Dynam. Sys. 16 (1996), 751{778. [4] D. Lind and K. Schmidt, Homoclinic points of algebraic Z d -actions, Preprint Nr. 411, Erwin Schrodinger Institute, Vienna, [5] A. Manning, Irrationality of linear combinations of eigenvectors, Proc. Indian Acad. Sci. Math. Sci. 105 (1995), 269{271. [6] K. Schmidt, Automorphisms of compact abelian groups and ane varieties, Proc. London Math. Soc. 61 (1990), 480{496. [7] K. Schmidt, Dynamical systems of algebraic origin, Birkhauser Verlag, Basel-Berlin- Boston, 1995.

10 COMMON HOMOCLINIC POINTS OF COMMUTING TORAL AUTOMORPHISMS 9 Anthony Manning: Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK address: akm@maths.warwick.ac.uk Klaus Schmidt: Mathematics Institute, University of Vienna, Strudlhofgasse 4, A-1090 Vienna, Austria, and Erwin Schrodinger Institute for Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria address: klaus.schmidt@univie.ac.at