ESI The Erwin Schr�odinger International Boltzmanngasse 9Institute for Mathematical Physics A-1090 Wien, AustriaRigidity of Measurable Structurefor Zd{Actions by Automorphisms of a TorusAnatole KatokSvetlana KatokKlaus Schmidt

Vienna, Preprint ESI 850 (2000) March 3, 2000Supported by Federal Ministry of Science and Transport, AustriaAvailable via http://www.esi.ac.at

RIGIDITY OF MEASURABLE STRUCTURE FORZd{ACTIONS BY AUTOMORPHISMS OF A TORUSANATOLE KATOK, SVETLANA KATOK, AND KLAUS SCHMIDTAbstract. We show that for certain classes of actions ofZd; d �2, by automorphisms of the torus any measurable conjugacy hasto be a�ne, hence measurable conjugacy implies algebraic conju-gacy; similarly any measurable factor is algebraic, and algebraicand a�ne centralizers provide invariants of measurable conjugacy.Using the algebraic machinery of dual modules and informationabout class numbers of algebraic number �elds we consruct var-ious examples of Zd-actions by Bernoulli automorphisms whosemeasurable orbit structure is rigid, including actions which areweakly isomorphic but not isomorphic. We show that the struc-ture of the centralizer for these actions may or may not serve as adistinguishing measure{theoretic invariant.1. Introduction; description of resultsIn the course of the last decade various rigidity properties have beenfound for two di�erent classes of actions by higher{rank abelian groups:on the one hand, certain Anosov and partially hyperbolic actions of Zdand Rd; d � 2, on compact manifolds ([9, 10, 12]) and, on the other,actions of Zd; d � 2, by automorphisms of compact abelian groups(cf. e.g. [8, 16]). Among these rigidity phenomena is a relative scarcityof invariant measures which stands in contrast with the classical cased = 1 ([11]).In this paper we make the �rst step in investigating a di�erent albeitrelated phenomenon: rigidity of the measurable orbit structure withrespect to the natural smooth invariant measure.In the classical case of actions by Z or R there are certain natu-ral classes of measure{preserving transformations which possess suchrigidity: ergodic translations on compact abelian groups give a ratherDate: March 22, 2000.The research of the �rst author was partially supported by NSF grant DMS-9704776. The �rst two authors are grateful to the Erwin Schr�odinger Institute,Vienna, and the third author to the Center for Dynamical Systems at Penn StateUniversity, for hospitality and support while some of this work was done.1

2 ANATOLE KATOK, SVETLANA KATOK, AND KLAUS SCHMIDTtrivial example, while horocycle ows and other homogeneous unipo-tent systems present a much more interesting one [20, 21, 22]. In con-trast to such situations, individual elements of the higher{rank actionsmentioned above are Bernoulli automorphisms. The measurable or-bit structure of a Bernoulli map can be viewed as very \soft". Recallthat the only metric invariant of Bernoulli automorphisms is entropy([19]); in particular, weak isomorphism is equivalent to isomorphismfor Bernoulli maps since it implies equality of entropies. Furthermore,description of centralizers, factors, joinings and other invariant objectsassociated with a Bernoulli map is impossible in reasonable terms sinceeach of these objects is huge and does not possess any discernible struc-ture.In this paper we demonstrate that some very natural actions ofZd; d � 2, by Bernoulli automorphisms display a remarkable rigidity oftheir measurable orbit structure. In particular, isomorphisms betweensuch actions, centralizers, and factor maps are very restricted, and alot of algebraic information is encoded in the measurable structure ofsuch actions (see Section 5).All these properties occur for broad subclasses of both main classesof actions of higher{rank abelian groups mentioned above: Anosov andpartially hyperbolic actions on compact manifolds, and actions by au-tomorphisms of compact abelian groups. However, at present we areunable to present su�ciently de�nitive general results due to variousdi�culties of both conceptual and technical nature. Trying to presentthe most general available results would lead to cumbersome nota-tions and inelegant formulations. To avoid that we chose to restrict ourpresent analysis to a smaller class which in fact represents the intersec-tion of the two, namely the actions of Zd; d � 2, by automorphisms ofthe torus. Thus we study the measurable structure of such actions withrespect to Lebesgue (Haar) measure from the point of view of ergodictheory.Our main purpose is to demonstrate several striking phenomena bymeans of applying to speci�c examples general rigidity results which arepresented in Section 5 and are based on rigidity of invariant measuresdeveloped in [11] (see [7] for further results along these lines includingrigidity of joinings). Hence we do not strive for the greatest possiblegenerality even within the class of actions by automorphisms of a torus.The basic algebraic setup for irreducible actions by automorphisms ofa torus is presented in Section 3. Then we adapt further necessary alge-braic preliminaries to the special but in a sense most representative caseof Cartan actions, i.e. toZn�1{actions by hyperbolic automorphisms ofthe n{dimensional torus (see Section 4).

MEASURE-THEORETIC RIGIDITY 3The role of entropy for a smooth action of a higher{rank abeliangroup G on a �nite-dimensional manifold is played by the entropy func-tion onG whose values are entropies of individual elements of the action(see Section 2.2 for more details) which is naturally invariant of isomor-phism and also of weak isomorphism and is equivariant with respect toa time change.In Section 6 we produce several kinds of speci�c examples of actionsby ergodic (and hence Bernoulli) automorphisms of tori with the sameentropy function. These examples provide concrete instances when gen-eral criteria developed in Section 5 can be applied. Our examples in-clude:(i) actions which are not weakly isomorphic (Section 6.1),(ii) actions which are weakly isomorphic but not isomorphic, such thatone action is a maximal action by Bernoulli automorphisms andthe other is not (Section 6.2),(iii) weakly isomorphic, but nonisomorphic, maximal actions (Section6.3).Once rigidity of conjugacies is established, examples of type (i) ap-pear in a rather simple{minded fashion: one simply constructs actionswith the same entropy data which are not isomorphic over Q. This isnot surprising since entropy contains only partial information abouteigenvalues. Thus one can produce actions with di�erent eigenvaluestructure but identical entropy data.Examples of weakly isomorphic but nonisomorphic actions are moresophisticated. We �nd them among Cartan actions (see Section 4).The centralizer of a Cartan action in the group of automorphisms ofthe torus is (isomorphic to) a �nite extension of the acting group, andin some cases Cartan actions isomorphic over Q may be distinguishedby looking at the index of the group in its centralizer (type (ii); see Ex-amples 2a and 2b). The underlying cause for this phenomenon is theexistence of algebraic number �elds K = Q(�), where � is a unit, suchthat the ring of integers OK 6= Z[�]. In general �nding even simplestpossible examples for n = 3 involves the use of data from algebraicnumber theory and rather involved calculations. For examples of type(ii) one may use some special tricks which allow to �nd some of theseand to show nonisomorphism without a serious use of symbolic manip-ulations on a computer.A Cartan action � ofZn�1 on Tn is called maximal if its centralizer inthe group of automorphisms of the torus is equal to �(Zn�1)�f�Idg. Amaximal Cartan action turns out to me maximal in the above sense: it

4 ANATOLE KATOK, SVETLANA KATOK, AND KLAUS SCHMIDTcannot be extended to any action of a bigger abelian group by Bernoulliautomorphisms.Examples of maximal Cartan actions isomorphic over Q but not iso-morphic (type (iii)) are the most remarkable. Conjugacy over Q guar-antees that the actions by automorphisms of the torus Tn arising fromtheir centralizers are weakly isomorphic with �nite �bres. The mech-anism providing obstructions for algebraic isomorphism in this caseinvolves the connection between the class number of an algebraic num-ber �eld and GL(n;Z){conjugacy classes of matrices in SL(n;Z) whichhave the same characteristic polynomial (see Example 3). In �ndingthese examples the use of computational number{theoretic algorithms(which in our case were implemented via the Pari-GP package) hasbeen essential.One of our central conclusions is that for a broad class of actions ofZd; d � 2, (see condition (R) in Section 2.2) the conjugacy class ofthe centralizer of the action in the group of a�ne automorphisms ofthe torus is an invariant of measurable conjugacy. Let Zmeas(�) be thecentralizer of the action � in the group of measurable automorphisms.As it turns out in all our examples but Example 3b, the conjugacy classof the pair (Zmeas (�); �) is a distinguishing invariant of the measurableisomorphism. Thus, in particular, Example 3b shows that there areweakly isomorphic, but nonisomorphic actions for which the a�ne andhence the measurable centralizers are isomorphic as abstract groups.We would like to acknowledge a contribution of J.-P. Thouvenot tothe early development of ideas which led to this paper. He made animportant observation that rigidity of invariant measures can be usedto prove rigidity of isomorphisms via a joining construction (see Section5.1). 2. Preliminaries2.1. Basic ergodic theory. Any invertible (over Q) integral n � nmatrix A 2 M(n;Z) \ GL(n;Q) determines an endomorphism of thetorus Tn = Rn=Zn which we denote by FA. Conversely, any endomor-phism of Tn is given by a matrix from A 2 M(n;Z) \ GL(n;Q). If,in addition, detA = �1, i.e. if A is invertible over Z, then FA is anautomorphism of Tn (the group of all such A is denoted by GL(n;Z)).The map FA preserves Lebesgue (Haar) measure �; it is ergodic withrespect to � if and only if there are no roots of unity among the eigen-values of A, as was �rst pointed out by Halmos ([6]). Furthermore, inthis case there are eigenvalues of absolute value greater than one and(FA; �) is an exact endomorphism. If FA is an automorphism it is in factBernoulli ([14]). For simplicity we will call such a map FA an ergodic

MEASURE-THEORETIC RIGIDITY 5toral endomorphism (respectively, automorphism, if A is invertible). Ifall eigenvalues of A have absolute values di�erent from one we will callthe endomorphism (automorphism) FA hyperbolic.When it does not lead to a confusion we will not distinguish betweena matrix A and corresponding toral endomorphism FA.Let �1; : : : ; �n be the eigenvalues of the matrix A, listed with theirmultiplicities. The entropy h�(FA) of FA with respect to Lebesgue mea-sure is equal to Xfi:j�ij>1g log j�ij:In particular, entropy is determined by the conjugacy class of the ma-trix A over Q (or over C ). Hence all ergodic toral automorphisms whichare conjugate over Q are measurably conjugate with respect to Lebesguemeasure.Classi�cation, up to a conjugacy over Z, of matrices in SL(n;Z),which are irreducible and conjugate over Q is closely related to the no-tion of class number of an algebraic number �eld. A detailed discussionrelevant to our purposes appears in Section 4.2. Here we only mentionthe simplest case n = 2 which is not directly related to rigidity. Inthis case trace determines conjugacy class over Q and, in particular,entropy. However if the class number of the corresponding number �eldis greater than one there are matrices with the given trace which arenot conjugate over Z. This algebraic distinctiveness is not re ected inthe measurable structure: in fact, in the case of equal entropies theclassical Adler{Weiss construction of the Markov partition in [1] yieldsmetric isomorphisms which are more concrete and speci�c than in thegeneral Ornstein isomorphism theory and yet not algebraic.2.2. Higher rank actions. Let � be an action by commuting toralautomorphisms given by integral matrices A1; : : : ; Ad. It de�nes an em-bedding �� :Zd! GL(n;Z) by�n� = An11 : : : Andd ;where n = (n1; : : : ; nd) 2Zd, and we have�n = F�n�:Similarly, we write �� : Zd+ ! M(n;Z) \ GL(n;Q) for an action byendomorphisms. Conversely, any embedding � : Zd ! GL(n;Z) (re-spectively, � : Zd+ ! M(n;Z) \ GL(n;Q)) de�nes an action by auto-morphisms (respectively, endomorphisms) of Tn denoted by ��.Sometimes we will not explicitly distinguish between an action andthe corresponding embedding, e.g. we may talk about \the centralizerof an action in GL(n;Z)" etc.

6 ANATOLE KATOK, SVETLANA KATOK, AND KLAUS SCHMIDTDe�nitions. Let � and �0 be two actions ofZd (Zd+) by automorphisms(endomorphisms) of Tn and Tn0, respectively. The actions � and �0are measurably (or metrically, or measure{theoretically) isomorphic (orconjugate) if there exists a Lebesgue measure{preserving bijection ' :Tn! Tn0 such that ' � � = �0 � '.The actions � and �0 are measurably isomorphic up to a time changeif there exist a measure{preserving bijection ' : Tn ! Tn0 and a C 2GL(d;Z) such that ' � � � C = �0 � '.The action �0 is a measurable factor of � if there exists a Lebesguemeasure{preserving transformation ' : Tn ! Tn0 such that ' � � =�0 � '. If, in particular, ' is almost everywhere �nite{to{one, then �0is called a �nite factor or a factor with �nite �bres of �.Actions � and �0 are weakly measurably isomorphic if each is a mea-surable factor of the other.A joining between � and �0 is a measure � on Tn � Tn0 = Tn+n0invariant under the Cartesian product action �� �0 such that its pro-jections into Tn and Tn0 are Lebesgue measures. As will be explainedin Section 5, conjugacies and factors produce special kinds of joinings.These measure{theoretic notions have natural algebraic counterparts.De�nitions. The actions � and �0 are algebraically isomorphic (orconjugate) if n = n0 and if there exists a group automorphism ' :Tn! Tn such that ' � � = �0 � '.The actions � and �0 are algebraically isomorphic up to a time changeif there exists an automorphism ' : Tn ! Tn and C 2 GL(d;Z) suchthat ' � � � C = �0 � '.The action �0 is an algebraic factor of � if there exists a surjectivehomomorphism ' : Tn! Tn0 such that ' � � = �0 � '.The actions � and �0 are weakly algebraically isomorphic if each isan algebraic factor of the other. In this case n = n0 and each factormap has �nite �bres.Finally, we call a map ' : Tn ! Tn0 a�ne if there is a surjectivecontinuous group homomorphism : Tn ! Tn0 and x0 2 Tn0 s.t.'(x) = (x) + x0 for every x 2 Tn.As already mentioned, we intend to show that under certain con-dition for d � 2, measure theoretic properties imply their algebraiccounterparts.We will say that an algebraic factor �0 of � is a rank{one factor if �0is an algebraic factor of � and �0(Zd+) contains a cyclic sub{semigroupof �nite index.The most general situation when certain rigidity phenomena appearis the following :

MEASURE-THEORETIC RIGIDITY 7(R0): The action � does not possess nontrivial rank{one algebraicfactors.In the case of actions by automorphisms the condition (R0) is equiv-alent to the following condition (R) (cf. [27]):(R): The action � contains a group, isomorphic toZ2, which consistsof ergodic automorphisms.By Proposition 6.6 in [25], Condition (R) is equivalent to saying thatthe restriction of � to a subgroup isomorphic to Z2 is mixing.A Lyapunov exponent for an action � of Zd is a function � :Zd! Rwhich associates to each n 2 Zd the logarithm of the absolute value ofthe eigenvalue for �n� corresponding to a �xed eigenvector. Any Lya-punov exponent is a linear function; hence it extends uniquely to Rd.The multiplicity of an exponent is de�ned as the sum of multiplicitiesof eigenvalues corresponding to this exponent. Let �i; i = 1; : : : ; k, bethe di�erent Lyapunov exponents and let mi be the multiplicity of �i.Then the entropy formula for a single toral endomorphism implies thath�(n) = h�(�n�) = Xfi:�i(n)>0gmi�i(n):The function h� :Zd! R is called the entropy function of the action�. It naturally extends to a symmetric, convex piecewise linear functionof Rd. Any cone inRd where all Lyapunov exponents have constant signis called a Weyl chamber. The entropy function is linear in any Weylchamber.The entropy function is a prime invariant of measurable isomor-phism; since entropy does not increase for factors the entropy functionis also invariant of a weak measurable isomorphism. Furthermore itchanges equivariantly with respect to automorphisms of Zd.Remark. it is interesting to point out that the convex piecewise linearstructure of the entropy function persists in much greater generality,namely for smooth actions on di�erentiable manifolds with a Borelinvariant measure with compact support.2.3. Finite algebraic factors and invariant lattices. Every alge-braic action has many algebraic factors with �nite �bres. These factorsare in one{to{one correspondence with lattices � � Rn which containthe standard lattice �0 = Zn, and which satisfy that ��(�) � �. Thefactor{action associated with a particular lattice � � �0 is denoted by��. Let us point out that in the case of actions by automorphisms suchfactors are also invertible: if � � �0 and ��(�) � �, then ��(�) = �.

8 ANATOLE KATOK, SVETLANA KATOK, AND KLAUS SCHMIDTLet � � �0 be a lattice. Take any basis in � and let S 2 GL(n;Q)be the matrix which maps the standard basis in �0 to this basis. Thenobviously the factor{action �� is equal to the action �S��S�1 . In par-ticular, �� and ��� are conjugate over Q, although not necessarily overZ. Notice that conjugacy over Q is equivalent to conjugacy over R orover C .For any positive integer q, the lattice 1q�0 is invariant under anyautomorphism in GL(n;Z) and gives rise to a factor which is conjugateto the initial action: one can set S = 1q Id and obtains that �� = �� 1q �0 .On the other hand one can �nd, for any lattice � � �0, a positiveinteger q such that 1q�0 � � (take q the least common multiple ofdenominators of coordinates for a basis of �). Thus � 1q�0 appears as afactor of ��. Summarizing, we have the following properties of �nitefactors.Proposition 2.1. Let � and �0 be Zd{actions by automorphism of thetorus Tn. The following are equivalent.1. �� and ��0 are conjugate over Q;2. there exists an action �00 such that both � and �0 are isomorphicto �nite algebraic factors of �00;3. � and �0 are weakly algebraically isomorphic, i.e. each of them isisomorphic to a �nite algebraic factor of the other.Obviously, weak algebraic isomorphism implies weak measurable iso-morphism. For Z{actions by Bernoulli automorphisms, weak isomor-phism implies isomorphism since it preserves entropy, the only isomor-phism invariant for Bernoulli maps. In Section 5 we will show that,for actions by toral automorphisms satisfying Condition (R), measur-able isomorphism implies algebraic isomorphism. Hence, existence ofsuch actions which are conjugate over Q but not over Zprovides ex-amples of actions by Bernoulli maps which are weakly isomorphic butnot isomorphic.2.4. Dual modules. For any action � of Zd by automorphisms ofa compact abelian group X we denote by �̂ the dual action on thediscrete group X̂ of characters of X. For an element � 2 X̂ we denoteX̂�;� the subgroup of X̂ generated by the orbit �̂�.De�nition. The action � is called cyclic if X̂�;� = X̂ for some � 2 X̂ .Cyclicity is obviously an invariant of algebraic conjugacy of actionsup to a time change.More generally, the dual group X̂ has the structure of a moduleover the ring Z[u�11 ; : : : ; u�1d ] of Laurent polynomials in d commuting

MEASURE-THEORETIC RIGIDITY 9variables. Action by the generators of �̂ corresponds to multiplicationsby independent variables. This module is called the dual module ofthe action � (cf. [24, 25]). Cyclicity of the action corresponds to thecondition that this module has a single generator. The structure of thedual module up to isomorphism is an invariant of algebraic conjugacyof the action up to a time change.In the case of the torus X = Tn which concerns us in this paperone can slightly modify the construction of the dual module to make itmore geometric. A Zd-action � by automorphisms of the torus Rn=Znnaturally extends to an action on Rn (this extension coincides with theembedding �� if matrices are identi�ed with linear transformations).This action preserves the latticeZn and furnishesZn with the structureof a module over the ring Z[u�11 ; : : : ; u�1d ]. This module is | in anobvious sense | a transpose of the dual module de�ned above. Inparticular, the condition of cyclicity of the action does not depend onwhich of these two de�nitions of dual module one adopts.2.5. Algebraic and a�ne centralizers. Let � be an action ofZd bytoral automorphisms, and let ��(Zd) = f�n� : n 2 Zdg. The centralizerof � in the group of automorphisms of Tn is denoted by Z(�) and isnot distinguished from the centralizer of ��(Zd) in GL(n;Z).Similarly, the centralizer of � in the semigroup of all endomorphismsofTn (identi�ed with the centralizer of ��(Zd) in the semigroupM(n;Z)\GL(n;Q)) is denoted by C(�).The centralizer of � in the group of a�ne automorphisms of Tn willbe denoted by ZA� (�).The centralizer of � in the semigroup of surjective a�ne maps of Tnwill be denoted by CA� (�).3. Irreducible actions3.1. De�nition. The action � on Tn is called irreducible if any non-trivial algebraic factor of � has �nite �bres.The following characterization of irreducible actions is useful (cf. [2]).Proposition 3.1. The following conditions are equivalent:1. � is irreducible;2. �� contains a matrix with characteristic polynomial irreducibleover Q;3. �� does not have a nontrivial invariant rational subspace or, equiv-alently, any �{invariant closed subgroup of Tn is �nite.Corollary 3.2. Any irreducible action � of Zd+; d � 2, satis�es con-dition (R0).

10 ANATOLE KATOK, SVETLANA KATOK, AND KLAUS SCHMIDTProof. A rank one algebraic factor has to have �bres of positive dimen-sion. Hence the pre{image of the origin under the factor map is a unionof �nitely many rational tori of positive dimension and by Proposition3.1 � cannot be irreducible.3.2. Uniqueness of cyclic actions. Cyclicity uniquely determines anirreducible action up to algebraic conjugacy within a class of weaklyalgebraically conjugate actions.Proposition 3.3. If � is an irreducible cyclic action of Zd; d � 1, onTn and �0 is another cyclic action such that �� and ��0 are conjugateover Q, then � and �0 are algebraically isomorphic.For the proof of Proposition 3.3 we need an elementary lemma.Lemma 3.4. Let � :Zd! GL(n;Z) be an irreducible embedding. Thecentralizer of � in GL(n;Q) acts transitively on Zn n f0g.Proof. By diagonalizing � over C and taking the real form of it, oneimmediately sees that the centralizer of � in GL(n;R) acts transitivelyon vectors with nonzero projections on all eigenspaces and thus has asingle open and dense orbit. Since the centralizer over R is the closureof the centralizer overQ, the Q-linear span of the orbit of any integer orrational vector under the centralizer is an invariant rational subspace.Hence any integer point other than the origin belongs to the singleopen dense orbit of the centralizer of � in GL(n;R). This implies thestatement of the lemma.Proof of Proposition 3.3. Choose C 2 M(n;Z) such that C��0C�1 =��. Let k; l 2Zn be cyclic vectors for ��jZn and ��0jZn, respectively.Now consider the integer vector C(l) and �nd D 2 GL(n;Q) com-muting with �� such that DC(l) = k. We have DC��0C�1D�1 = ��.The conjugacy DC maps bijectively theZ{span of the ��0{orbit of l toZ{span of the ��{orbit of k. By cyclicity both spans coincide with Zn,and hence DC 2 GL(n;Z).3.3. Centralizers of integer matrices and algebraic number�elds. There is an intimate connection between irreducible actionson Tn and groups of units in number �elds of degree n. Since this con-nection (in the particular case where the action is Cartan and hencethe number �eld is totally real) plays a central role in the constructionof our principal examples (type (ii) and (iii) of the Introduction), wewill describe it here in detail even though most of this material is fairlyroutine from the point of view of algebraic number theory.Let A 2 GL(n;Z) be a matrix with an irreducible characteristicpolynomial f and hence distinct eigenvalues. The centralizer of A in

MEASURE-THEORETIC RIGIDITY 11M(n;Q) can be identi�ed with the ring of all polynomials in A withrational coe�cients modulo the principal ideal generated by the polyno-mial f(A), and hence with the �eld K = Q(�), where � is an eigenvalueof A, by the map : p(A) 7! p(�)(1)with p 2 Q[x]. Notice that if B = p(A) is an integer matrix then (B)is an algebraic integer, and if B 2 GL(n;Z) then (B) is an algebraicunit (converse is not necessarily true).Lemma 3.5. The map in (1) is injective.Proof. If (p(A)) = 1 for p(A) 6= Id, then p(A) has 1 as an eigenvalue,and hence has a rational subspace consisting of all invariant vectors.This subspace must be invariant under A which contradicts its irre-ducibility.Denote by OK the ring of integers in K, by UK the group of unitsin OK, by C(A) the centralizer of A in M(n;Z) and by Z(A) thecentralizer of A in the group GL(n;Z).Lemma 3.6. (C(A)) is a ring in K such thatZ[�]� (C(A)) � OK,and (Z(A)) = UK \ (C(A)).Proof. (C(A)) is a ring because C(A) is a ring. As we pointed outabove images of integer matrices are algebraic integers and images ofmatrices with determinant �1 are algebraic units. Hence (C(A)) �OK. Finally, for every polynomial p with integer coe�cients, p(A) isan integer matrix, hence Z[�]� (C(A)).Notice that Z(�) is a �nite index subring of OK; hence (C(A)) hasthe same property.Remark. The groups of units in two di�erent rings, say O1 � O2, maycoincide. Examples can be found in the table of totally real cubic �eldsin [4].Proposition 3.7. Z(A) is isomorphic to Zr1+r2�1�F where r1 is thenumber the real embeddings, r2 is the number of pairs of complex con-jugate embeddings of the �eld K into C , and F is a �nite cyclic group.Proof. By lemma 3.6, Z(A) is isomorphic to the group of units in theorder O, the statement follows from the Dirichlet Unit Theorem ([3],Ch.2, x3).Now consider an irreducible action � of Zd on Tn. Denote ��(Zd) by�, and let � be an eigenvalue of a matrixA 2 � with an irreducible char-acteristic polynomial. The centralizers of � in M(n;Z) and GL(n;Z)

12 ANATOLE KATOK, SVETLANA KATOK, AND KLAUS SCHMIDTcoincide with C(A) and Z(A) correspondingly. The �eld K = Q(�)has degree n and we can consider the map as above. By Lemma 3.6 (�) � UK.For the purposes of purely algebraic considerations in this and thenext section it is convenient to consider actions of integer n�n matriceson Qn rather than on Rn and correspondingly to think of � as an actionby automorphisms of the rational torus TnQ= Qn=Zn.Let v = (v1; : : : ; vn) be an eigenvector of A with eigenvalue � whosecoordinates belong to K. Consider the \projection" � : Qn ! K de-�ned by �(r1; : : : rn) =Pni=1 rivi. It is a bijection ([29], Prop. 8) whichconjugates the action of the group � with the action on K given bymultiplication by corresponding eigenvalues Qdi=1 �kii ; k1; : : : ; kd 2 Z.Here A1; : : : ; Ad 2 � are the images of the generators of the action �,and Aiv = �iv; i = 1; : : : ; d. The lattice �Zn � K is a module over thering Z[�1; : : : ; �d].Conversely, any such data, consisting of an algebraic number �eldK = Q(�) of degree n, a d-tuple �� = (�1; : : : ; �d) of multiplicativelyindependent units in K, and a lattice L � K which is a module overZ[�1; : : : ; �d], determine an Zd-action ���;L by automorphisms of Tn upto algebraic conjugacy (corresponding to a choice of a basis in thelattice L). This action is generated by multiplications by �1; : : : ; �d(which preserve L by assumption). The action ���;L diagonalizes over Cas follows. Let �1 = id; �2; : : : ; �n be di�erent embeddings of K into C .The multiplications by �i; i = 1; : : : ; d, are simultaneously conjugateover C to the respective matrices �i 0 ::: 00 �2(�i) ::: 0::: ::: ::: :::0 0 ::: �n(�i)! ; i = 1; : : : ; d:We will assume that the action is irreducible which in many inter-esting cases can be easily checked.Thus, all actions ���;L with �xed �� are weakly algebraically isomor-phic since the corresponding embeddings are conjugate over Q (Propo-sition 2.1). Actions produced with di�erent sets of units in the same�eld, say �� and �� = (�1; : : : ; �d), are weakly algebraically isomorphicif and only if there is an element g of the Galois group of K such that�i = g�i; i = 1; : : : ; d. By Proposition 3.3 there is a unique cyclic action(up to algebraic isomorphism) within any class of weakly algebraicallyisomorphic actions: it corresponds to setting L =Z[�1; : : : ; �d]; we willdenote this action by �min�� . Cyclicity of the action �min�� is obvious sincethe whole lattice is obtained from its single element 1 by the action ofthe ring Z[��11 ; : : : ; ��1d ].Let us summarize this discussion.

MEASURE-THEORETIC RIGIDITY 13Proposition 3.8. Any irreducible action � of Zd by automorphismsof Tn is algebraically conjugate to an action of the form ���;L. It isweakly algebraically conjugate to the cyclic action �min�� . The �eld K =Q[�1; : : : ; �d] has degree n, and the vector of units �� = (�1; : : : ; �d) isde�ned up to action by an element of the Galois group of K : Q.Apart from the cyclic model �min�� there is another canonical choiceof the lattice L, namely the ring of integers OK. We will denote theaction ���;OK by �max�� . More generally, one can choose as the lattice Lany subring O such that Z[�1; : : : ; �d] � O � OK.Proposition 3.9. Assume that O ) Z[�1; : : : ; �d]. Then the action���;O is not algebraically isomorphic up to a time change to �min�� . Inparticular, if OK 6= Z[�1; : : : ; �d], then the actions �max�� and �min�� arenot algebraically isomorphic up to a time change.Proof. Let us denote the centralizers in M(n;Z) of the actions ���;Oand �min�� by C1 and C2, respectively. The centralizer C1 contains mul-tiplications by all elements of O. For, if one takes any basis in O, themultiplication by an element � 2 O takes elements of the basis intoelements of O, which are linear combinations with integral coe�cientsof the basis elements; hence the multiplication is given by an integermatrix. On the other hand any element of each centralizer is a multi-plication by an integer in K (Lemma 3.6).Now assume that the multiplication by � 2 OK belongs to C2. Thismeans that this multiplication preserves Z[�1; : : : ; �d]; in particular,� = ��1 2Z[�1; : : : ; �d]. Thus C2 consists of multiplication by elementsof Z[�1; : : : ; �d]. An algebraic isomorphism up to a time change has topreserve both the module of polynomials with integer coe�cients in thegenerators of the action and the centralizer of the action in M(n;Z),which is impossible.The central question which appears in connection with our examplesis the classi�cation of weakly algebraically isomorphic Cartan actionsup to algebraic isomorphism.Proposition 3.9 is useful in distinguishing weakly algebraically iso-morphic actions when OK 6= Z[�1; : : : ; �d]. Cyclicity also can serve asa distinguishing invariant.Corollary 3.10. The action ���;O is cyclic if and only if O =Z[�1; : : : ;�d].Proof. The action �min�� corresponding to the ringZ[�1; : : : ; �d] is cyclicby de�nition since the ring coincides with the orbit of 1. By Proposition

14 ANATOLE KATOK, SVETLANA KATOK, AND KLAUS SCHMIDT3.3, if ���;O were cyclic, it would be algebraically conjugate to �min�� ,which, by Proposition 3.9, implies that O =Z[�1; : : : ; �d].The property common to all actions of the ���;O is transitivity of theaction of the centralizer C(���;O) on the lattice. Similarly to cyclicitythis property is obviously an invariant of algebraic conjugacy up to atime change.Proposition 3.11. Any irreducible action � of Zd by automorphismsof Tn whose centralizer C(�) in M(n;Z) acts transitively on Zn isalgebraically isomorphic to an action ���;O, where O � OK is a ringwhich contains Z[�1; : : : ; �d].Proof. By Proposition 3.8 any irreducible action � of Zd by automor-phisms of Tn is algebraically conjugate to an action of the form ���;L fora lattice L � K. Let C be the centralizer of ���;L in the semigroup oflinear endomorphisms of L. We �x an element � 2 L with C(�)� = Land consider conjugation of the action ���;L by multiplication by ��1;this is simply ���;��1L. The centralizer of ���;��1L acts on the element1 2 ��1L transitively. By Lemma 3.6 the centralizer consists of all mul-tiplications by elements of a certain subring O � OK which containsZ[�1; : : : ; �d]. Thus 1 2 ��1L = O.3.4. Structure of algebraic and a�ne centralizers for irreducibleactions. By Lemma 3.6, the centralizer C(�), as an additive group, isisomorphic toZn and has an additional ring structure. In the terminol-ogy of Proposition 3.7, the centralizer Z(�) for an irreducible action �by toral automorphisms is isomorphic to Zr1+r2�1 � F .An irreducible action � has maximal rank if d = r1 + r2 � 1. In thiscase Z(�) is a �nite extension of �.Notice that any a�ne map commuting with an action � by toralautomorphisms preserves the set Fix(�) of �xed points of the action.This set is always a subgroup of the torus and hence, for an irreducibleaction, always �nite. The translation by any element of Fix(�) com-mutes with � and thus belongs to ZA� (�). Furthermore, the a�necentralizers ZA� (�) and CA� (�) are generated by these translationsand, respectively, Z(�) and C(�).Remark. Most of the material of this section extends to general irre-ducible actions of Zd by automorphisms of compact connected abeliangroups; a group possessing such an action must be a torus or a solenoid([25, 26]). In the solenoid case, which includes natural extensions ofZd{actions by toral endomorphisms, the algebraic numbers �1; : : : ; �dwhich appear in the constructions are not in general integers. As we

MEASURE-THEORETIC RIGIDITY 15mentioned in the introduction we restrict our algebraic setting heresince we are able to exhibit some of the most interesting and strik-ing new phenomena using Cartan actions and certain actions directlyderived from them. However, other interesting examples appear for ac-tions on the torus connected with not totally real algebraic number�elds, actions on solenoids, and actions on zero-dimensional abeliangroups (cf. e.g. [16, 24, 25, 26]).One can also extend the setup of this section to certain classes of re-ducible actions. Since some of these satisfy condition (R) basic rigidityresults still hold and a number of further interesting examples can beconstructed. 4. Cartan actions4.1. Structure of Cartan actions. Of particular interest for ourstudy are abelian groups of ergodic automorphisms of Tn of maximalpossible rank n � 1 (in agreement with the real rank of the Lie groupSL(n;R)).De�nition. An action of Zn�1 on Tn for n � 3 by ergodic automor-phisms is called a Cartan action.Proposition 4.1. Let � be a Cartan action on Tn.1. Any element of �� other than identity has real eigenvalues and ishyperbolic and thus Bernoulli.2. � is irreducible.3. The centralizer of Z(�) is a �nite extension of ��(Zn�1).Proof. First, let us point out that it is su�cient to prove the propositionfor irreducible actions. For, if � is not irreducible, it has a nontrivialirreducible algebraic factor of dimension, say, m � n � 1. Since ev-ery factor of an ergodic automorphism is ergodic, we thus obtain anaction of Zn�1 in Tm by ergodic automorphisms. By considering a re-striction of this action to a subgroup of rank m� 1 which contains anirreducible matrix, we obtain a Cartan action on Tm. By Statement 3.for irreducible actions, the centralizer of this Cartan action is a �niteextension of Zm�1, and thus cannot contain Zn�1, a contradiction.Now assuming that � is irreducible, take a matrixA 2 ��(Zn�1) withirreducible characteristic polynomial f . Such a matrix exists by Propo-sition 3.1. It has distinct eigenvalues, say � = �1; : : : ; �n. Consider thecorrespondence de�ned in (1). By Lemma 3.6 for every B 2 ��(Zn�1)we have (B) 2 UK , hence the group of units UK in K contains a sub-group isomorphic to Zn�1. By the Dirichlet Unit Theorem the rank ofthe group of units in K is equal to r1 + r2 � 1, where r1 is the number

16 ANATOLE KATOK, SVETLANA KATOK, AND KLAUS SCHMIDTof real embeddings and r2 is the number of pairs of complex conjugateembeddings of K into C . Since r1 + 2r2 = n we deduce that r2 = 0,so the �eld K is totally real, that is all eigenvalues of A, and hence ofany matrix in ��(Zn�1), are real. The same argument gives Statement3, since any element of the centralizer of ��(Zn�1) in GL(n;Z) corre-sponds to a unit in K. Hyperbolicity of matrices in ��(Zn�1) is provedin the same way as Lemma 3.5.Lemma 4.2. Let A be a hyperbolic matrix in SL(n;Z) with irreduciblecharacteristic polynomial and distinct real eigenvalues. Then every el-ement of the centralizer Z(A) other than f�1g is hyperbolic.Proof. Assume that B 2 Z(A) is not hyperbolic. As B is simultane-ously diagonalizable with A and has real eigenvalues, it has an eigen-value +1 or �1. The corresponding eigenspace is rational and A{invariant. Since A is irreducible, this eigenspace has to coincide withthe whole space and hence B = �1.Corollary 4.3. Cartan actions are exactly the maximal rank irreducibleactions corresponding to totally real number �elds.Corollary 4.4. The centralizer Z(�) for a Cartan action � is isomor-phic to Zn�1� f�1g.We will call a Cartan action � maximal if � is an index two subgroupin Z(�).Let us point out that ZA� (�) is isomorphic Z(�) � Fix(�). Thus,the factor of ZA� (�) by the subgroup of �nite order elements is alwaysisomorphic to Zn�1. If � is maximal, this factor is identi�ed with �itself. In the next Section we will show (Corollary 5.4) that for a Cartanaction � on Tn; n � 3 the isomorphism type of the pair (ZA� (�); �) isan invariant of the measurable isomorphism. Thus, in particular, for amaximal Cartan action the order of the group Fix(�) is a measurableinvariant.Remark. An important geometric distinction between Cartan actionsand general irreducible actions by hyperbolic automorphisms is theabsence of multiple Lyapunov exponents. This greatly simpli�es proofsof various rigidity properties both in the di�erentiable and measurablecontext.4.2. Algebraically nonisomorphic maximal Cartan actions. InSection 3.3 we described a particular class of irreducible actions ���;Owhich is characterized by the transitivity of the action of the centralizerC(���;O) on the lattice (Proposition 3.11). In the case OK =Z[�] thereis only one such action, namely the cyclic one (Corollary 3.10). Now

MEASURE-THEORETIC RIGIDITY 17we will analyze this special case for totally real �elds in detail andshow how information about the class number of the �eld helps toconstruct algebraically nonisomorphic maximal Cartan actions. Thiswill in particular provide examples of Cartan actions not isomorphicup to a time change to any action of the form ���;O.It is well{known that for n = 2 there are natural bijections be-tween conjugacy classes of hyperbolic elements in SL(2;Z) of a giventrace, ideal classes in the corresponding real quadratic �eld, and con-gruence classes of primitive integral inde�nite quadratic forms of thecorresponding discriminant. This has been used by Sarnak [23] in hisproof of the Prime Geodesic Theorem for surfaces of constant negativecurvature (see also [13]). It follows from an old Theorem of Latimerand MacDu�ee (see [17], [28], and a more modern account in [29]),that the �rst bijection persists for n > 2. Let A a hyperbolic matrixA 2 SL(n;Z) with irreducible characteristic polynomial f , and hencedistinct real eigenvalues, K = Q(�), where � is an eigenvalue of A,and OK = Z[�]. To each matrix A0 with the same eigenvalues, we as-sign the eigenvector v = (v1; : : : ; vn) with eigenvalue �: A0v = �v withall its entries in OK, which can be always done, and to this eigenvec-tor, an ideal in OK with the Z{basis v1; : : : ; vn. The described mapis a bijection between the GL(n;Z){conjugacy classes of matrices inSL(n;Z) which have the same characteristic polynomial f and the setof ideal classes inOK. Moreover, it allows us to reach conclusions aboutcentralizers as well.Theorem 4.5. Let A 2 SL(n;Z) be a hyperbolic matrix with irre-ducible characteristic polynomial f and distinct real eigenvalues, K =Q(�) where � is an eigenvalue of A, and OK =Z[�]. Suppose the num-ber of eigenvalues among �1; : : : ; �n that belong to K is equal to r. Ifthe class number h(K) > r, then there exists a matrix A0 2 SL(n;Z)having the same eigenvalues as A whose centralizer Z(A0) is not con-jugate in GL(n;Z) to Z(A). Furthermore, the number of matrices inSL(n;Z) having the same eigenvalues as A with pairwise nonconjugate(in GL(n;Z)) centralizers is at least [h(K)r ] + 1, where [x] is the largestinteger < x.Proof. Suppose the matrix A corresponds to the ideal class I1 with theZ{basis v(1). Then Av(1) = �v(1):Since h(K) > 1, there exists a matrix A2 having the same eigenvalueswhich corresponds to a di�erent ideal class I2 with the basis v(2), and

18 ANATOLE KATOK, SVETLANA KATOK, AND KLAUS SCHMIDTwe have A2v(2) = �v(2):The eigenvectors v(1) and v(2) are chosen with all their entries in OK.Now assume that Z(A2) is conjugate to Z(A). Then Z(A2) containsa matrix B2 conjugate to A. Since B2 commutes with A2 we haveB2v(2) = �2v(2), and since B2 is conjugate to A, �2 is one of the rootsof f . Moreover, since B2 2 SL(n;Z) and all entries of v(2) are in K,�2 2 K. Thus � is one of r roots of f which belongs to K.From B2 = S�1AS (S 2 GL(n;Z)) we deduce that �2(Sv(2)) =A(Sv(2)). Since I1 and I2 belong to di�erent ideal classes, Sv(2) 6= kv(1)for any k in the quotient �eld of OK, and since � is a simple eigenvaluefor A, we deduce that �2 6= �, and thus �2 can take one of the r � 1remaining values.Now assume that A3 corresponds to the third ideal class, i.eA3v(3) = �v(3);and B3 commutes with A3 and is conjugate to A, and hence to B2.Then B3v(3) = �3v(3) where �3 is a root of f belonging to the �eldK. By the previous considerations, �3 6= � and �3 6= �2. An inductionargument shows that if the class number of K is greater than r, thereexists a matrix A0 such that no matrix in Z(A0) is conjugate to A, i.e.Z(A0) and Z(A) are not conjugate in GL(n;Z).Since A0 has the same characteristic polynomial as A, continuingthe same process, we can �nd not more that r matrices representingdi�erent ideal classes having centralizers conjugate to Z(A0), and therequired estimate follows.5. Measure{theoretic rigidity of conjugacies,centralizers, and factors5.1. Conjugacies. Suppose � and �0 are measurable actions of thesame group G by measure{preserving transformations of the spaces(X;�) and (Y; �), respectively. If H : (X;�) ! (Y; �) is a metric iso-morphism (conjugacy) between the actions then the lift of the measure� onto the graphH � X �Y coincides with the lift of � to graphH�1.The resulting measure � is a very special case of a joining of � and�0: it is invariant under the diagonal (product) action � � �0 and itsprojections to X and Y coincide with � and �, respectively. Obviouslythe projections establish metric isomorphism of the action � � �0 on(X � Y; �) with � on (X;�) and �0 on (Y; �) correspondingly.Similarly, if an automorphism H : (X;�) ! (X;�) commutes withthe action �, the lift of � to graphH � X � X is a self-joining of

MEASURE-THEORETIC RIGIDITY 19�, i.e. it is � � �{invariant and both of its projections coincide with�. Thus an information about invariant measures of the products ofdi�erent actions as well as the product of an action with itself maygive an information about isomorphisms and centralizers.The use of this joining construction in order to deduce rigidity ofisomorphisms and centralizers from properties of invariant measures ofthe product was �rst suggested in this context to the authors by J.-P.Thouvenot.In both cases the ergodic properties of the joining would be knownbecause of the isomorphism with the original actions. Very similarconsiderations apply to the actions of semi{groups by noninvertiblemeasure{preserving transformations. We will use the following corol-lary of the results of [11].Theorem 5.1. Let � be an action of Z2 by ergodic toral automor-phisms and let � be a weakly mixing �{invariant measure such that forsome m 2Z2, �m is a K-automorphism. Then � is a translate of Haarmeasure on an �{invariant rational subtorus.Proof. We refer to Corollary 5.2' from ([11], \Corrections..."). Accord-ing to this corollary the measure � is an extension of a zero entropymeasure for an algebraic factor of smaller dimension with Haar condi-tional measures in the �ber. But since � contains a K-automorphismit does not have non{trivial zero entropy factors. Hence the factor inquestion is the action on a single point and � itself is a Haar measureon a rational subtorus.Conclusion of Theorem 5.1 obviously holds for any action ofZd; d �2 which contains a subgroupZ2 satisfying assumptions of Theorem 5.1.Thus we can deduce the following result which is central for our con-structions.Theorem 5.2. Let � and �0 be two actions of Zd by automorphismsof Tn and Tn0 correspondingly and assume that � satis�es condition(R). Suppose that H : Tn! Tn0 is a measure{preserving isomorphismbetween (�; �) and (�0; �), where � is Haar measure. Then n = n0 andH coincides (mod 0) with an a�ne automorphism on the torus Tn, andhence � and �0 are algebraically isomorphic.Proof. First of all, condition (R) is invariant under metric isomor-phism, hence �0 also satis�es this condition. But ergodicity with respectto Haar measure can also be expressed in terms of the eigenvalues;hence ���0 also satis�es (R). Now consider the joining measure � ongraphH � Tn+n0. The conditions of Theorem 5.1 are satis�ed for theinvariant measure � of the action �� �0. Thus � is a translate of Haar

20 ANATOLE KATOK, SVETLANA KATOK, AND KLAUS SCHMIDTmeasure on a rational ���0{invariant subtorus T0 � Tn+n0 = Tn�Tn0.On the other hand we know that projections of T0 to both Tn and Tn0preserve Haar measure and are one{to{one. The partitions of T0 intopre{images of points for each of the projections are measurable parti-tions and Haar measures on elements are conditional measures. Thisimplies that both projections are onto, both partitions are partitionsinto points, and hence n = n0 and T0 = graph I, where I : Tn ! Tnis an a�ne automorphism which has to coincide (mod 0) with themeasure{preserving isomorphism H.Since a time change is in a sense a trivial modi�cation of an action weare primarily interested in distinguishing actions up to a time change.The corresponding rigidity criterion follows immediately from Theorem5.2.Corollary 5.3. Let � and �0 be two actions of Zd by automorphismsof Tn and Tn0, respectively, and assume that � satis�es condition (R).If � and �0 are measurably isomorphic up to a time change then theyare algebraically isomorphic up to a time change.5.2. Centralizers. Applying Theorem 5.2 to the case � = �0 we im-mediately obtain rigidity of the centralizers.Corollary 5.4. Let � be an action of Zd by automorphisms of Tnsatisfying condition (R). Any invertible Lebesgue measure{preservingtransformation commuting with � coincides (mod 0) with an a�neautomorphism of Tn.Any a�ne transformation commuting with � preserves the �nite setof �xed points of the action. Hence the centralizer of � in a�ne auto-morphisms has a �nite index subgroups which consist of automorphismsand which corresponds to the centralizer of ��(Zd) in GL(n;Z).Thus, in contrast with the case of a single automorphism, the cen-tralizer of such an action � is not more than countable, and can beidenti�ed with a �nite extension of a certain subgroup of GL(n;Z). Asan immediate consequence we obtain the following result.Proposition 5.5. For any d and k, 2 � d � k, there exists a Zd{action by hyperbolic toral automorphisms such that its centralizer in thegroup of Lebesgue measure{preserving transformations is isomorphic tof�1g �Zk.Proof. Consider a hyperbolic matrix A 2 SL(k+1;Z) with irreduciblecharacteristic polynomial and real eigenvalues such that the origin is theonly �xed point of FA. Consider a subgroup of Z(A) isomorphic to Zdand containing A as one of its generators. This subgroup determines an

MEASURE-THEORETIC RIGIDITY 21embedding � :Zd! SL(k+1;Z). Since d � 2 and by Proposition 4.2,all matrices in �(Zd) are hyperbolic and hence ergodic, condition (R)is satis�ed. Hence by Corollary 5.4, the measure{theoretic centralizerof the action �� coincides with its algebraic centralizer, which, in turn,and obviously, coincides with centralizer of the single automorphismFA isomorphic to f�1g �Zk.5.3. Factors, noninvertible centralizers and weak isomorphism.A small modi�cation of the proof of Theorem 5.2 produces a resultabout rigidity of factors.Theorem 5.6. Let � and �0 be two actions of Zd by automorphismsof Tn and Tn0 respectively, and assume that � satis�es condition (R).Suppose that H : Tn ! Tn0 is a Lebesgue measure{preserving trans-formation such that H � � = �0 � H. Then �0 also satis�es (R) andH coincides (mod 0) with an epimorphism h : Tn ! Tn0 followed bytranslation. In particular, �0 is an algebraic factor of �.Proof. Since �0 is a measurable factor of �, every element which isergodic for � is also ergodic for �0. Hence �0 also satis�es condition(R). As before consider the product action � � �0 which now by thesame argument also satis�es (R). Take the � � �0 invariant measure� = (Id�H)�� on graphH. This measure provides a joining of � and�0. Since (�� �0; (Id�H)��) is isomorphic to (�; �) the conditions ofCorollary 5.1 are satis�ed and � is a translate of Haar measure on aninvariant rational subtorus T0. Since T0 projects to the �rst coordinateone-to-one we deduce that H is an algebraic epimorphism (mod 0)followed by a translation.Similarly to the previous section the application of Theorem 5.6 tothe case � = �0 gives a description of the centralizer of � in the groupof all measure{preserving transformations.Corollary 5.7. Let � be an action ofZd by automorphisms of Tn satis-fying condition (R). Any Lebesgue measure{preserving transformationcommuting with � coincides (mod 0) with an a�ne map on Tn.Now we can obtain the following strengthening of Proposition 2.1 foractions satisfying condition (R) which is one of the central conclusionsof this paper.Theorem 5.8. Let � be an action ofZd by automorphisms of Tn satis-fying condition (R) and �0 another Zd-action by toral automorphisms.Then (�; �) is weakly isomorphic to (�0; �0) if and only if �� and ��0are isomorphic over Q, i.e. if � and �0 are �nite algebraic factors ofeach other.

22 ANATOLE KATOK, SVETLANA KATOK, AND KLAUS SCHMIDTProof. By Theorem 5.6, � and �0 are algebraic factors of each other.This implies that �0 acts on the torus of the same dimension n andhence both algebraic factor{maps have �nite �bres. Now the statementfollows from Proposition 2.1.5.4. Distinguishing weakly isomorphic actions. Similarly we cantranslate criteria for algebraic conjugacy of weakly algebraically conju-gate actions to the measurable setting.Theorem 5.9. If � is an irreducible cyclic action of Zd; d � 2, on Tnand �0 is a non{cyclic Zd-action by toral automorphisms. Then � and�0 are not measurably isomorphic up to a time change.Proof. Since action � satis�es condition (R) (Corollary 3.2) we canapply Theorem 5.8 and conclude that we only need to consider thecase when �� and ��0 are isomorphic over Q up to a time change. Butthen, by Proposition 3.3, � and �0 are not algebraically isomorphic upto a time change and hence, by Corollary 5.3, they are not measurablyisomorphic up to a time change.Combining Proposition 3.9 and Corollary 5.3 we immediately obtainrigidity for the minimal irreducible models.Corollary 5.10. Assume that O ) Z[�1; : : : ; �d]. Then the action���;O is not measurably isomorphic up to a time change to �min�� . Inparticular, if OK ) Z[�1; : : : ; �d], then the actions �max�� and �min�� arenot measurably isomorphic up to a time change.6. ExamplesNow we proceed to produce examples of actions for which the entropydata coincide but which are not algebraically isomorphic, and hence byTheorem 5.2 not measure{theoretically isomorphic.6.1. Weakly nonisomorphic actions. In this section we consideractions which are not algebraically isomorphic over Q (or, equivalently,over R) and hence by Theorem 5.8 are not even weakly isomorphic.The easiest way is as follows.Example 1a. Start with any action � of Zd; d � 2, by ergodic auto-morphisms of Tn. We may double the entropies of all its elements intwo di�erent ways: by considering the Cartesian square � � � actingon T2n, and by taking second powers of all elements: �n2 = �2n for alln 2Zd. Obviously ��� is not algebraically isomorphic to �2, since, forexample, they act on tori of di�erent dimension. Hence by Theorem 5.2(�� �; �) is not metrically isomorphic to (�2; �) either.

MEASURE-THEORETIC RIGIDITY 23Now we assume that � contains an automorphism FA where A ishyperbolic with an irreducible characteristic polynomial and distinctpositive real eigenvalues. In this case it is easy to �nd an invariant dis-tinguishing the two actions, namely, the algebraic type of the centralizerof the action in the group of measure{preserving transformations. ByCorollary 5.4, the centralizer of � in the group of measure{preservingtransformations coincides with the centralizer in the group of a�nemaps, which is a �nite extension of the centralizer in the group of auto-morphisms. By the Dirichlet Unit Theorem, the centralizer of Z(�2) inthe group of automorphisms of the torus is isomorphic to f�1g�Zn�1,whereas the centralizer of ��� contains theZ2(n�1){action by producttransformations �n1 � �n2 ; n1;n2 2 Zn�1. In fact, the centralizer of�� � can be calculated explicitly:Proposition 6.1. Let � be an eigenvalue of A. Then K = Q(�) is atotally real algebraic �eld. If its ring of integers OK is equal to Z[�]then the centralizer of � � � in GL(2n;Z) is isomorphic to the groupGL(2;OK), i.e. the group of 2 � 2 matrices with entries in OK whosedeterminant is a unit in OK.Proof. First we notice that a matrix in block form B = (X YZ T ) withX;Y;Z; T 2 M(n;Z) commutes with ( A 00 A ) if an only if X;Y;Z; Tcommutewith A and can thus be identi�ed with elements ofOK. In thiscase B can be identi�ed with a matrix inM(2;OK). Since det ( X YZ T ) =det(XT �Y Z) = �1 (cf. [5]), the norm of the determinant of the 2�2matrix corresponding to B is equal �1. Hence this determinant is aunit in OK, and we obtain the desired isomorphism.It is not di�cult to modify Example 1a to obtain weakly nonisomor-phic actions with the same entropy on the torus of the same dimension.Example 1b. For a natural number k de�ne the action �k similarlyto �2: �nk = �kn for all n 2Zd.The actions �3�� and �2��2 act on T2n, have the same entropiesfor all elements and are not isomorphic.As before, we can see that centralizers of these two actions are notisomorphic. In particular, the centralizer of �3 � � is abelian since ithas simple eigenvalues, while the centralizer of �2 � �2 is not.6.2. Cartan actions distinguished by cyclicity or maximality.We give two examples which illustrate the method of Section 3.3. Theyprovide weakly algebraically isomorphic Cartan actions of Z2 on T3which are not algebraically isomorphic even up to a time change (i.e. alinear change of coordinates inZ2) by Proposition 3.9. These examplesutilize the existence of number �elds K = Q(�) and units �� = (�1; �2)

24 ANATOLE KATOK, SVETLANA KATOK, AND KLAUS SCHMIDTin them for which OK 6= Z[�1; �2]. In each example one action has aform �min�� and the other �max�� . Hence by Corollary 5.10 they are notmeasurably isomorphic up to a time changeIn other words, in each example one action, namely, �min�� , is a cyclicCartan action, and the other is not.We will aslo show that in these examples the conjugacy type of thepair (Z(�); �) distinguishes weakly isomorphic actions. Let us point outthat a noncylic action for example �max�� may be maximal, for examplewhen fundamental units lie in a proper subring of OK. However in ourexamples centralizers for the cyclic actions will be dirrefern and thuswill serve as a distuinguishing invariant.The information about cubic �elds is either taken from [4] or obtainedwith the help of the computer package Pari-GP. Some calculations weremade by Arsen Elkin during the REU program at Penn State in summerof 1999.We construct twoZ2{actions, �, generated by commuting matricesAandB, and �0, generated by commutingmatricesA0 and B 0 inGL(3;Z).These actions are weakly algebraically isomorphic by Proposition 3.8since they are produced with the same set of units on two di�erentorders, Z[�] and OK, but not algebraically isomorphic by Proposition3.9. In these examples the action � is cyclic by Corollary 3.10 and willbe shown to be a maximal Cartan action. Thus Z(�) = � � f�Idg.The action �0 is not maximal, speci�cally, Z(�0)=f�Idg is a nontrivial�nite extension of �0.Example 2a. LetK be a totally real cubic �eld given by the irreduciblepolynomial f(x) = x3 + 3x2 � 6x + 1, i.e. K = Q(�) where � is oneof its roots. The discriminant of K is equal to 81, hence its Galoisgroup is cyclic, and [OK :Z[�]] = 3. The algebraic integers �1 = � and�2 = 2� 4�� �2 are units with f(�1) = f(�2) = 0. The minimal orderin K containing �1 and �2 is Z[�1; �2] = Z[�], and the maximal orderis OK. A basis in fundamental units is � = �2+5�+13 and ��1, hence UKis not contained in Z[�].With respect to the basis f1; �; �2g in Z[�], multiplications by �1and �2 are given by the matricesA = � 0 1 00 0 1�1 6 �3 � ; B = � 2 �4 �11 �4 �11 �5 �1� ;respectively (if acting from the right on row{vectors). A direct calcu-lation shows that this action is maximal.

MEASURE-THEORETIC RIGIDITY 25With respect to the basis f�23 + 53� + 13�2;�13 + 73� + 23�2g in OK,multiplications by �1 and �2 are given by the matricesA0 = � 1 2 �1�1 �2 22 5 �2� ; B 0 = � 1 �1 �1�1 �2 �1�1 �4 �2� :We have A0 = V AV �1, B 0 = V BV �1 for V = � 2 �2 �10 �3 01 �4 �2�. Since A is acompanion matrix of f , � = hA;Bi has a cyclic element in Z3. If A0also had a cyclic elementm = (m1;m2;m3) 2Z3, then the vectorsm=(m1;m2 ;m3); mA0=(m1�m2+2m3;2m1�2m2+5m3 ;�m1+2m2�2m3)m(A0)2=(�3m1+5m2�7m3;�7m1+12m2�16m3;5m1�7m2+12m3);would have to generate Z3 or, equivalentlydet� m1 m2 m3m1�m2+2m3 2m1�2m2+5m3 �m1+2m2�2m3�3m1+5m2�7m3 �7m1+12m2�16m3 5m1�7m2+12m3 �= 3m31 + 18m21m3 � 9m1m22 � 9m1m2m3+ 27m1m23 + 3m32 � 9m2m23 + 3m33 = 1:This contradiction shows that A0 has no cyclic vector, and since B 0 =2� 4A0�A02 , the action �0 is not cyclic. In this example both actions� and �0 have a single �xed point (0; 0; 0), hence their linear and a�necentralizers coincide, and by Corollary 5.3 � and �0 are not measurablyisomorphic up to a time change.The action �0 is not maximal beacuse Z(�0) contains fundamentalunits.Example 2b. Let us consider a totally real cubic �eld K given bythe irreducible polynomial f(x) = x3� 7x2+11x� 1. Thus K = Q(�)where � is one of its roots. In this �eld the ring of integersOK has basisf1; �; 12�2+ 12g and hence [OK :Z[�]] = 2. The fundamental units inOKare f12�2�2�+ 12 ; ��2g. We choose the units � = �1 = (12�2�2�+ 12)2and �2 = � � 2 which are contained in both orders, OK and Z[�].InZ[�] we consider the basis f1; �; �2g relative to which the multipli-cation by �1 is represented by the companion matrix A = � 0 1 00 0 11 �11 7 �and multiplication by �2 is represented by the matrixB = ��2 1 00 �2 11 �11 5�.For OK with the basis f1; �; 12�2 + 12g multiplications by �1 and �2are represented by the matricesA0 = � 0 1 0�1 0 2�3 �5 7� and B0 = ��2 1 0�1 �2 2�3 �5 5�.It can be seen directly that � and �0 are not algebraically conjugateup to a time change since A0 is a square of a matrix from SL(3;Z):A0 =� 0 �2 1�1 �5 3�2 �9 6�2, while A is not a square of a matrix in GL(3;Z), which ischecked by reducing modulo 2. In this case it is also easily seen that the

26 ANATOLE KATOK, SVETLANA KATOK, AND KLAUS SCHMIDTaction �0 is not cyclic since the corresponding determinant is divisibleby 2. The action � has 2 �xed points on T3: (0; 0; 0) and (12 ; 12; 12),while the action �0 has 4 �xed points: (0; 0; 0), (12 ; 12 ; 12), (12 ; 12; 0), and(0; 0; 12). Hence the a�ne centralizer of � is Z(�)�Z=2Z, and the a�necentralizer of �0 is Z(�0)� (Z=2Z�Z=2Z).By Lemma 4.2, the group of elements of �nite order in ZA� (�) isZ=2Z�Z=2Zand in ZA� (�0) it isZ=2Z�Z=2Z�Z=2Z. The indices ofeach action in its a�ne centralizer are [ZA� (�) : �] = 4 and [ZA� (�0) :�0] = 16.This gives two alternative arguments that the actions are not mea-surably isomorphic up to a time change.6.3. Nonisomorphic maximal Cartan actions. We �nd examplesof weakly algebraically isomorphic maximal Cartan actions which arenot algebraically isomorphic up to time change. For such an action �the structure of the pair (Z(�); �) is always the same: Z(�) is iso-morphic as a group to � � f�Idg. The algebraic tool which allows todistinguish the actions is Theorem 4.5.Example 3a. An example for n = 3 can be obtained from a totallyreal cubic �eld with class number 2 and the Galois group S3. Thesmallest discriminant for such a �eld is 1957 ([4], Table B4), and itcan be represented as K = Q(�) where � is a unit in K with minimalpolynomial f(x) = x3 � 2x2 � 8x � 1. In this �eld the ring of integersOK = Z[�] and the fundamental units are �1 = � and �2 = � + 2.Two actions are constructed with this set of units (fundamental, hencemultiplicatively independent) on two di�erent lattices, OK with thebasis f1; �; �2g, representing the principal ideal class, and L with thebasis f2; 1+�; 1+�2g representing to the second ideal class. Notice thatthe units �1 and �2 do not belong to L, but L is a Z[�]-module. The�rst action � is generated by the matricesA = � 0 1 00 0 11 8 2 � and B = � 2 1 00 2 11 8 4 �which represent multiplication by �1 and �2, respectively, on OK. Thesecond action �0 is generated by matrices A0 = ��1 2 0�1 1 1�5 9 2� and B 0 =� 1 2 0�1 3 1�5 9 5� which represent multiplication by �1 and �2, respectively,on L in the given basis. By Proposition 3.8 these actions are weaklyalgebraically isomorphic. By Theorem 4.5 they are not algebraicallyisomorphic. Since the Galois group is S3 there are no nontrivial timechanges which produce conjugacy over Q. Therefore, but Theorem 5.2the actions are not measurably isomorphic.It is interesting to point out that for actions � and �0 the a�necentralizers ZA� (�) and ZA� (�0) are not isomorphic as abstract groups.

MEASURE-THEORETIC RIGIDITY 27The action � has 2 �xed points on T3: (0; 0; 0) and (12 ; 12 ; 12), while theaction �0 has a single �xed point (0; 0; 0). Hence ZA� (�) is isomorphicto Z(�) �Z=2Z, ZA� (�0) is isomorphic to Z(�0). As abstract groups,ZA� (�) �Z2�Z=2Z�Z=2Zand ZA� (�0) �Z2�Z=2Z.Hence by Corollary 5.4 the measurable centralizers of � and �0 arenot conjugate in the group of measure{preserving transformation pro-viding a distinguishing invariant of measurable isomorphism.Example 3b. This example is obtained from a totally real cubic �eldwith class number 3, Galois group S3, and discriminant 2597. It canbe represented as K = Q(�) where � is a unit in K with minimalpolynomial f(x) = x3 � 2x2 � 8x + 1. In this �eld the ring of integersOK = Z[�] and the fundamental units are �1 = � and �2 = � + 2.Three actions are constructed with this set of units on three di�erentlattices, OK with the basis f1; �; �2g, representing the principal idealclass, L with the basis f2; 1 + �; 1 + �2g representing the second idealclass, and L2 with the basis f4; 3 + �; 3 + �2g representing the thirdideal class.Multiplications by �1 and �2 generate the following three weaklyalgebraically isomorphic actions which are not algebraically isomorphicby Theorem 4.5 even up to a time change, and therefore not measurablyisomorphic: A = � 0 1 00 0 1�1 8 2 � and B = � 2 1 00 2 1�1 8 4 � ;A0 = ��1 2 0�1 1 1�6 9 2� and B 0 = � 1 2 0�1 3 1�6 9 4� ;A00 = � �3 4 0�3 3 1�10 11 2� and B 00 = � �1 4 0�3 5 1�10 11 4� :Each action has 2 �xed point in T3, (0; 0; 0) and (12 ; 12; 12). Hence alla�ne centralizers are isomorphic as abstract groups to Z2 �Z=2Z�Z=2Z.Example 3c Finally we give an example of two nonisomorphic maxi-mal Cartan actions which come from the vector of fundamental units�� = (�1; �2) in a totally real cubic �eld K such that Z(�1; �2) 6= OK.Thus the whole group of units does not generate the ring OK. Bothactions �min�� and �max�� of the group Z2 are maximal Cartan actions byLemma 3.6. However by Corollary 3.10 the former is cyclic and thelatter is not and hence they are not measurably isomorphic up to atime change by Corollary 5.10.

28 ANATOLE KATOK, SVETLANA KATOK, AND KLAUS SCHMIDTFor a speci�c example we pick the totally real cubic �eld K = Q(�)with class number 1 discriminant 1304 given by the polynomial x3 �x2 � 11x � 1. For this �led we have [OK : Z(�)] = 2. Generatorsin OK can be taken to be f1; �; � = �2+12 g. Fundamental units are�1 = ��; �2 = �5 + 14� + 10� = 14� + 5�2 2 Z[�]. Thus thewhole group of units lies in Z[�]. To construct the generators for twonon{isomorphic action �min�� and �max�� we write multiplications by �1and �2 in bases f1; �; �2g and f1; �; �g, correspondingly. The resultingmatrices are: A = � 0 �1 01 0 �11 11 1� B = � 0 14 55 55 1919 214 74 � ;A0 = � 0 �1 01 0 �20 �6 �1� B = � �5 14 10�14 55 38�30 114 79� :The �rst action has only one �xed point, the origin; the second hasfour �xed points (0; 0; 0), (12 ; 12; 12), (12; 12 ; 0), and (0; 0; 12). Thus we havean example of two maximal Cartan actions of Z2 which have noniso-morphic a�ne and hence measurable centralizers.References[1] R.L. Adler and B. Weiss Entropy, a complete metric invariant for automor-phisms of the torus, Proc. Nat. Acad. Sci. 57 (1967), 1573{1576.[2] D. Berend, Multi-invariant sets on tori, Trans. Amer. Math. Soc. 280 (1983),509{532.[3] Z.I. Borevich and I.R. Shafarevich, Number Theory, Academic Press, NewYork, 1966.[4] H. Cohen, A course in computational algebraic number theory Springer, Berlin{Heidelberg{New York, 1996.[5] F.R. Gantmacher, Theory of matrices, vol. 1, Chelsea, New York, 1959; vol. 2,Chelsea, New York, 1960.[6] P.R. Halmos, On automorphisms of compact groups, Bull. Amer. Math. Soc.49 (1943), 619{624.[7] B. Kalinin and A. Katok, Rigidity of invariant measures, factors and joiningsfor actions of higher{rank abelian groups, Preprint (2000).[8] A. Katok and K.Schmidt, The cohomology of expansive Zd actions by auto-morphisms of compact abelian groups Paci�c Jour. Math. 170 (1995), 105{142.[9] A. Katok and R.J. Spatzier, First cohomology of Anosov actions of higherrank abelian groups and applications to rigidity Publ. Math. IHES 79 (1994),131{156.[10] A. Katok and R.J. Spatzier, Subelliptic estimates of polynomial di�erentialoperators and applications to rigidity of abelian actions Math. Res. Lett. 1(1994), 193{202.[11] A. Katok and R.J. Spatzier, Invariant measures for higher rank hyperbolicabelian actions, Ergod. Th. & Dynam. Sys. 16 (1996), 751{778; Corrections:Ergod. Th. & Dynam. Sys. 18 (1998), 503{507.

MEASURE-THEORETIC RIGIDITY 29[12] A. Katok and R.J. Spatzier, Di�erential rigidity of Anosov actions of higherrank abelian groups and algebraic lattice actions, Proc. Steklov Inst. Math. 216(1997), 287{314.[13] S. Katok, Coding of closed geodesics after Gauss and Morse, Geom. Dedicata63 (1996), 123{145.[14] I. Katznelson, Ergodic automorphisms of Tn are Bernoulli shifts, Israel J.Math. 10 (1971), 186{195.[15] B. Kitchens and K. Schmidt, Automorphisms of compact groups, Ergod. Th.& Dynam. Sys. 9 (1989), 691{735.[16] B. Kitchens and K. Schmidt, Isomorphism rigidity of simple algebraic Zd{actions, preprint (1999).[17] Latimer and C.C. MacDu�ee, A correspondence between classes of ideals andclasses of matrices, Ann. Math. 74 (1933), 313{316.[18] D. Lind, K. Schmidt and T. Ward,Mahler measure and entropy for commutingautomorphisms of compact groups, Invent. Math. 101 (1990), 593{629.[19] D. Ornstein, Bernoulli shifts with the same entropy are isomorphic, Adv. Math.4 (1970), 337{352.[20] M. Ratner, Rigidity of horocycle ows, Ann. Math. 115, (1982), 597{614.[21] M. Ratner, Factors of horocycle ows, Ergod. Th. & Dynam. Sys. 2 (1982),465{489.[22] M. Ratner, Horocycle ows, joinings and rigidity of products Ann. Math. 118(1983), 277-313.[23] P. Sarnak, Prime geodesic theorem, Ph.D. thesis, Stanford, 1980.[24] K. Schmidt, Automorphisms of compact abelian groups and a�ne varieties,Proc. London Math. Soc. 61 (1990), 480{496.[25] K. Schmidt, Dynamical systems of algebraic origin, Birkh�auser Verlag, Basel-Berlin-Boston, 1995.[26] K. Schmidt, Measurable rigidity of algebraic Zd-actions, Preprint (2000).[27] A. Starkov, First cohomology group, mixing and minimal sets of commutativegroup of algebraic action on torus, preprint.[28] O. Taussky, Introduction into connections between algebraic number theory andintegral matrices, Appendix to: H. Cohn, A classical invitation to algebraicnumbers and class �eld, Springer, New{York, 1978.[29] D.I. Wallace, Conjugacy classes of hyperbolic matrices in SL(n;Z) and idealclasses in an order. Trans. Amer. Math. Soc. 283 (1984), 177{184.

30 ANATOLE KATOK, SVETLANA KATOK, AND KLAUS SCHMIDTA. Katok: Department of Mathematics, Pennsylvania State Univer-sity, University Park, PA 16802, USAE-mail address: katok a@math.psu.eduS. Katok: Department of Mathematics, Pennsylvania State Univer-sity, University Park, PA 16802, USAE-mail address: katok s@math.psu.eduK. Schmidt: Mathematics Institute, University of Vienna, Strudl-hofgasse 4, A-1090 Vienna, Austria,andErwin Schr�odinger Institute for Mathematical Physics, Boltzmann-gasse 9, A-1090 Vienna, AustriaE-mail address: klaus.schmidt@univie.ac.at