ESI The Erwin Schr�odinger International Boltzmanngasse 9Institute for Mathematical Physics A-1090 Wien, AustriaOn the Entropy Theory of Finitely GeneratedNilpotent Group ActionsValentin Ya. GolodetsSergey D. Sinel'shchikov

Vienna, Preprint ESI 692 (1999) April 14, 1999Supported by Federal Ministry of Science and Transport, AustriaAvailable via http://www.esi.ac.at

ON THE ENTROPY THEORY OF FINITELY GENERATEDNILPOTENT GROUP ACTIONSValentin Ya. GOLODETS, Sergey D. SINEL'SHCHIKOVInstitute for Low Temperature Physics & EngineeringUkrainian National Academy of Sciences47 Lenin Ave, 310164 Kharkov, UkraineAbstractLet G be a �nitely generated nilpotent torsionfree group. The entropy theory for G-actions is investigated. The Pinsker algebra of such actions is described explicitly. Thecomlpetley positive systems are shown to have a sort of 'asypmtotic independence prop-erty', just as in the case ofZd-actions. A notion of K-system for G-actions is considered.The properties of invariant partitions are used to prove that the property of completepositivity is equivalent to the properties of K-system and K-mixing. The relationshipbetween the K-systems and some spectral and mixing properties of G-actions is clari�ed.IntroductionEntropy is an important notion in statistical physics and information theory. Initially,A. Kolmogorov introduced the notion of entropy for automorphisms in ergodic theory[5]. A more general approach to entropy for automorphisms was suggested by Ya. Sinai[15]; in this setting the new invariant found a large number of applications in studyingthe dynamical systems. Also, A. Kolmogorov distinguished a class of dynamical sys-tems which possess a partition with several good properties of an algebraic nature [5].Those systems were later called K-systems after A. Kolmogorov by V. Rokhlin and Ya.Sinai [11]. They considered, in particular, the dynamical systems with trivial Pinskeralgebra (completely positive systems), and proved that this property is equivalent to theproperty of K-system. In order to do that, a techniques related to a class of invariantpartitions were developed.The K-systems possess a rather strong mixing property of (K-mixing [3]), which isequivalent to the complete positivity. It turns out that such systems are frequently usedin applications, in particular, in the models of mathematical physics [3].A deep result was obtained by D. Ornstein who proved the isomorphism of Bernoullishifts with the same entropy [7]. This inspired extensive activities on generalizing thisresult onto a larger class of groups (speci�cally, amenable groups). A more advancedresult in this sphere was obtained by D. Ornstein and B. Weiss [8]. A study of entropyof Zd-actions of an algebraic origin was made by K. Schmidt [14].A description of Pinsker algebra for actions of Zd, d < 1 was made by J. Conze[1], who also considered for the above group actions the completely positive systemsand the notion of K-system. The equivalence of the two latter properties was proved1

by B.Kami�nski [4], who developed the Rokhlin-Sinai's ([11]) theory of invariant parti-tions in the context of Zd. Recently a new rise of activities coming up in this sphere.In particular, the work of D. Rudolf and B. Weiss [12] contains a re�ned study ofstrong mixing properties of completely positive actions of countable amenable groups.E. Glasner, J. P. Thouvenot, and B. Weiss [2] investigated some problems related toPinsker algebras and completely positive systems. This work contains, in particular,some strong results on relative disjointness and quasi-factors.On the other hand, there is still a problem of describing a structure of Pinskeralgebra for actions of amenable groups, the properties of completely positive systemsand their relationship to K-systems.In this work, we approach this problem with considering the case of �nitely generatednilpotent torsionfree groups and their subgroups. In particular, this class of groupsincludes upper triangular unipotent matricial groups with integer entries. We alsoconsider some spectral and mixing properties for actions of such groups related to theentropy. We prove an analogue of the well known Pinsker formula (lemma 2.9) togetherwith the associated asymptotic relations, which allow then to describe explicitly thePinsker algebra (theorem 3.1). The next step is to consider the systems with trivialPinsker algebra (completely positive systems, see de�nition A in section 3) and toestablish that they are exactly the K-systems (de�nition C, section 5); the latter beingintroduced in a similar way as in [1, 4]. For doing this, we develop the approach ofV. Rokhlin, Ya. Sinai and B. Kami�nski [11, 4] in applying the techniques related toinvariant partitions in our setting. We also de�ne the notion of K-mixing for actionsof a nilpotent group and show that it is equivalent to complete positivity (see theorem6.5). On the other hand, the completely positive systems also possess a property of'asymptotic independence' (see proposition 3.2), just as in the case of Zd.To solve these problems, one has to overcome some di�culties. Unlike the case ofZd-actions, this requires a speci�c choice of a sequence of generators in the group withsome good properties (see the appendix), together with an order relation on those. Thepresence of such sequence of generators, if proved, already implies the existence of aninformation past in the sense of B. Pitzkel and A. Safonov [9, 13]. Another principalobstacle is that, while working with F�lner sequences, a translate of a rectangle in anilpotent group is no longer a rectangle; so, the problems of pavement arise in manyarguments, which requires some additional estimational techniques.To simplify our exposition, we consider the special case of Heizenberg group in alldetails, and then show how to transfer our techniques onto the general case.The outline of this paper is as follows. Section 1 contains a brief review of def-initions and basic properties related to the entropy of partitions and measure spacetransformations. Section 2 introduces the entropy for the Heizenberg group actionsand discusses its standard properties. Section 3 presents a description of the Pinskeralgebra for actions of the Heizenberg group. Section 4 show how to transfer the resultsconcerning the Pinsker algebra onto the case of general nilpotent group actions. Sec-tion 5 is devoted to studying the invariant and perfect partitions, complete positivityand K-systems. Section 6 clari�es the relationship between the K-systems and somespectral and mixing properties of the Heizenberg group actions. Section 7 exposes ageneralization of the results of sections 5 and 6 onto actions of general nilpotent groups.Finally, appendix is devoted to forming a special sequence of generators in a generaltorsionfree nilpotent group. 2

1 PreliminariesWe start with recalling the basic de�nitions. For a partition � with at most countablenumber of elements on a Lebesgue space (X;A; �) de�ne the entropy of � byH(�) = �Xi �(Ai) log �(Ai):Let L denote the collection of partitions � as above with H(�) < 1. Given anymeasurable partition �, the conditional entropy H(�j�) is de�ned in a standard way.In particular, H(�j�) = 0 , � � �, and H(�j�) is increasing in � and decreasing in�. The distance d on L is given byd(�; �) = H(�j�) +H(�j�):With d, L becomes a complete metric space. The basic property of the entropy forpartitions is 8�; � 2 L; 8 ; H(��j ) = H(�j ) +H(�j� ); (1.1)and hence the subadditivity of H:8�; � 2 L; 8 ; H(��j ) � H(�j ) +H(�j ):If f ng is an increasing sequence of partitions with = Wn n, then for all � 2 L onehas limn H(�j n) = H(�j ):In a similar way, for a decreasing sequence of partitions f ng with = Vn n one hasfor all � 2 L limn H(�j n) = H(�j ):Let T be an automorphism of (X;A; �). For any partition �, we use the notation�nT = n�1_0 T k�; ��T = 1_1 T�k�; �T = 1_�1 T k�:The sequence 1nH(�nT ) has a limit which is denoted by h(�; T ); it can be alsorepresented as a conditional entropy H(�j��T ). More generally, if is a T -invariantmeasurable partition, the sequence 1nH(�nT j ) has a limit equal to H(�j��T ), whichis denoted by h(�; T; ). In particular, h(�; T ) = h(�; T; �), with � being the trivialpartition.The entropy of the automorphism T is de�ned byh(T ) = sup�2L h(�; T ):We remind also a very important Pinsker formula:h(��; T ) = h(�; T ) +H(�j�T _ ��T );and, in a more general setting,h(��; T; ) = h(�; T; ) +H(�j�T _ ��T _ )for any T -invariant measurable partition .3

2 The entropy for actions of the Heizenberg group and itspropertiesConsider the two step nilpotent countable matrix groupG = 8<:0@ 1 n3 n10 1 n20 0 1 1A : ni 2Z9=; :We �x the generatorsT1 = 0@ 1 0 10 1 00 0 11A ; T2 =0@ 1 0 00 1 10 0 11A ; T3 =0@ 1 1 00 1 00 0 11A :Suppose G acts on a Lebesgue space (X;A; �) with �nite invariant measure �. Forany �nite subset E � G and a measurable partition �, let S(E) stand for the numberof elements of E, and use the notation �E = Wg2E g�. Among the �nite subsets, wedistinguish the rectangles fT i33 T i22 T i11 : m1 � i1 �M1; m2 � i2 �M2; m3 � i3 �M3g.Given a sequence of such rectangles f�ng, we say that the modulus of �n tends to in�nityif mink=1;2;3(Mk(n)�mk(n))!1 as n!1, M1(n)�m1(n)M2(n)�m2(n) !1, M1(n)�m1(n)M3(n)�m3(n) !1as n!1.Proposition 2.1 Let f�ng be a sequence of rectangles in G whose modulus tends toin�nity. Then f�ng is a F�lner sequence of sets in G.Proof. It follows from the commutation relation�T i33 T i22 T i11 ��1 T k33 T k22 T k11 = T k3�i33 T k2�i22 T k1+i2(k3�i3)�i11that for any g = �T i33 T i22 T i11 ��1 and � given by 0 � kj � pj , j = 1; 2; 3,S(g�n��n) � 2ji3jp1p2 + 2ji2jp1p3 + 2(ji1j+ ji2i3j+ ji2jp3)p2p3;and so S(g�n��n)S(�n) � 2ji3jp3 + 2ji2jp2 + 2(ji1j+ ji2i3j+ ji1jp3)p1 ! 0as n!1. �It follows from proposition 2.1 and [9, theorem 1] that for any sequence f�ng ofrectangles in G whose modulus tends to in�nity and any � 2 L, there exists a limit1S(�n)H(��n). This limit is independent of the choice of �n. It is called the entropy of �with respect to G, and is denoted by h(�;G). However, it admits di�erent expressions.Now introduce the concept of the past for a partition � with respect to the G-action. Speci�cally, we set up ��G = WT k33 T k22 T k11 �, where the join is taken over alltriples (k3; k2; k1) 2 Z3 which are lexicographically less than (0; 0; 0). In more details,��G = ��T1(�T1)�T2(�T1T2)�T3. 4

Theorem 2.2 ([9, 13]) For any partition � 2 L, h(�;G) = H(�j��G).We list here some properties of h(�;G).Proposition 2.3 If � 2 L, h(�;G) � H(�):Proof. This is a straightforward conclusion from theorem 2.2. �Proposition 2.4 If �; � 2 L,h(� _ �;G) � h(�;G) + h(�;G):Proof. This follows from the sub-additivity of the functional H(�j�) in �. �Proposition 2.5 If �; � 2 L,jh(�;G)� h(�;G)j � d(�; �):Proof. See [1] for a proof. �Let Gp be a subgroup of G of index p. We consider only the subgroups generatedby T p11 ; T p22 ; T p33 with p2p3p1 2Z. Consider a 'fundamental domain' �p in G with respectto Gp which contains zero of G. Given a partition �, we denote for simplicity's sake by�p the partition ��p = Wg2�p g � �.Lemma 2.6 For a subgroup Gp � G of index p,h(�p; Gp) = ph(�;G): (2.1)Proof. Note that the 'fundamental domain' �p can be chosen to be a rectangle; letit be fT i33 T i22 T i11 : 0 � i1 < p1; 0 � i2 < p2; 0 � i3 < p3g. In this case the index of Gp isp = p1p2p3. Let also � = fT k3p33 T k2p22 T k1p11 : 0 � k1 � m1; 0 � k2 � m2; 0 � k3 � m3gbe a rectangle in Gp. It follows from the commutation relation T i33 T i22 T i11 T k33 T k22 T k11 =T i3+k33 T i2+k22 T i1+k1�i2k31 that the set � = fT j33 T j22 T j11 = fg; f 2 �p; g 2 �g is inside therectangle �0 = fT j33 T j22 T j11 : �p2m3p3 � j1 < m1p1; 0 � j2 < m2p2; 0 � j3 < m3p3g.Observe that S(�) = S(�)S(�p) = m1m2m3p1p2p3, S(�0) = (m1p1 + p2m3p3)m2m3p2p3,and hence S(�0)S(�) = 1 + p2p3p1 � m3m1 .Let us assume that the modulus of � tends to in�nity. This means that m2m1 ! 0.We impose also the additional assumption that m3m1 ! 0. Note that under the latterassumption the modulus of � tends to in�nity if and only if the modulus of �0 tends toin�nity, and S(�0)S(�) ! 1. In view of the above observation we havelim 1S(�0)H(��0) = lim 1S(�0)H(��_��0n�) � lim 1S(�)H(��) + lim 1S(�0)H(��0n�) �5

� lim 1S(�)H(��) + lim S(�0 n �)S(�0) H(�);and since the latter limit is zero, we deduce thatlim 1S(�0)H(��0) � lim 1S(�)H(��):What remains is to observe that the converse inequality is obvious, so �nally wehaveh(�;G) = lim 1S(�0)H(��0) = lim 1S(�)H(��) = lim 1pS(�)H((��p)�) = 1ph(�p; Gp): �Proposition 2.7 For any transformation T 2 G,h(�;G) � h(�; T ):Proof. This is an easy consequence of theorem 2.2. �Proposition 2.8 If �; � 2 L,h(�;G) � h(�;G) +H(�j�G): (2.2)Proof. Consider the rectangles �n = fT i33 T i22 T i11 : 0 � i1 < n2; 0 � i2; i3 < ng,�p = fT k33 T k22 T k11 : 0 � k1 < p2; 0 � k2; k3 < pg. It follows from the commutationrelation T i33 T i22 T i11 T k33 T k22 T k11 = T i3+k33 T i2+k22 T i1+k1�i2k31 that the set fT j33 T j22 T j11 = fg :f 2 �n; g 2 �pg is inside the rectangle �n;p = fT j33 T j22 T j11 : �(n � 1)(p � 1) � j1 �p2 + n2 � 2; 0 � j2; j3 < n+ p� 2g. Thus we obtain:H(��n) � H(��n��n;p) = H(��n;p) +H(��nj��n;p) �� H(��n;p) +Xf2�nH(f�j��n;p) � H(��n;p) +Xf2�nH(f�j�f�p) == H(��n;p) + S(�n)H(�j��p):After dividing out by S(�n) = n4 and observing via a routine calculation that S(�n;p)S(�n) !1 as n!1, we get for any �xed ph(�;G) � h(�;G) +H(�j��p):Now what remains is to send p to in�nity and to observe that H(�j��p)! H(�j�G) inorder to get the desired relation. �Lemma 2.9 (Pinsker formula). If �; � 2 L,h(��;G) = h(�;G) +H(�j��G�G): (2.3)6

Proof. The proof of this formula is just the same as that one can �nd in [1]. It isbased on (2.1) and (2.2). �In a similar way one can prove a relativized version of the above Pinsker formula.Speci�cally, if �; � 2 L and � is any G-invariant measurable partition, thenh(��;G; �) = h(�;G; �) +H(�j��G�G�): (2.4)Lemma 2.10 If �; � 2 L,limn!1H(�jT�n3 (��G)��G) = H(�j��G): (2.5)Proof. Apply the Pinsker formula (2.3) to � and T�n3 �:h(� _ T�n3 �;G) = h(�;G) +H(T�n3 �j(T�n3 �)�G _ �G) = h(�;G) +H(�j��G _ �G);with the latter equality being due to the obvious relation (T�n3 �)�G = T�n3 (��G).On the other hand, an application of (1.1) yieldsh(� _ T�n3 �;G) = H(� _ T�n3 �j��G _ (T�n3 �)�G) == H(�j��G _ T�n3 (��G)) +H(T�n3 �j� _ ��G _ T�n3 (��G)) =H(�j��G _ T�n3 (��G)) +H(�jT n3 (� _ ��G) _ ��G)It is easy to see that T n3 (� _ ��G) is an increasing sequence of partitions whose join is�G, and hence H(�jT n3 (� _ ��G) _ ��G) ! H(�j�G _ ��G). Now the comparison of thetwo expressions for h(� _ T�n3 �;G) proves our statement. �There is also a relativized version of (2.5). Let �; � 2 L and � is a G-invariantmeasurable partition, thenlimn!1H(�jT�n3 (��G)��G�) = H(�j��G�): (2.6)The proof is similar to that of (2.5).3 A description of Pinsker algebra for actions of the Heizen-berg groupAssociate to any measurable partition � the partition �(�) which is de�ned to be theleast upper bound of all those partitions � 2 L that � � �G and h(�;G) = 0. Inparticular, with � = ", one has �(") = �(G), the Pinsker partition of the dynamicalsystem (X;G). We are going to describe �(�) more explicitly.For � 2 L we set up�̂ = n̂ (T�n1 ��T1 _ T�n2 (�T1)�T2 _ T�n3 (�T1T2)�T3):To make this partition invariant, set up �1 = Wn �̂�n with �n being a sequence ofrectangles in G whose modulus tends to in�nity.7

Theorem 3.1 For any partition � 2 L, �(�) = �1.Proof. First we suppose that � 2 L, and � � �1. Note that the set of partitionswhich are less than some �̂�nis dense in the set of partitions which are less than �1.In this context, it su�ces to show that h(�;G) = 0 for � 2 L, � � �̂�n. Actually, weconsider here the case � � �̂; the general case can be treated similarly. One has for allk1; k2; k3H(�jT�k11 ��T1 _ T�k22 (�T1)�T2 _ T�k33 (�T1T2)�T3 _ ��T1 _ (�T1)�T2 _ (�T1T2)�T3) = 0:Let k1 !1, then an application of an analogue of (2:6) for G =Z[1] with a generatorT1 and � = T�k22 (�T1)�T2 _ T�k33 (�T1T2)�T3 _ (�T1)�T2 _ (�T1T2)�T3 yieldsH(�jT�k22 (�T1)�T2 _ T�k33 (�T1T2)�T3 _ ��T1 _ (�T1)�T2 _ (�T1T2)�T3) = 0;and henceH(�jT�k22 (��T1 _ (�T1)�T2) _ T�k33 (�T1T2)�T3 _ ��T1 _ (�T1)�T2 _ (�T1T2)�T3) = 0:Apply again an analogue of (2:6) for G =Z2 [1] with � = T�k33 (�T1T2)�T3 _ (�T1T2)�T3 andthe generators T1; T2 to getH(�jT�k33 (�T1T2)�T3 _ ��T1 _ (�T1)�T2 _ (�T1T2)�T3) = 0;which also impliesH(�jT�k33 (��T1 _ (�T1)�T2 _ (�T1T2)�T3) _ ��T1 _ (�T1)�T2 _ (�T1T2)�T3) = 0:Finally, an application of (2:6) assures H(�j��G) = 0, which was to be proved.Conversely, let � 2 L, � � �G, and h(�;G) = 0. Recall that for any subgroup Gpof a �nite index p of the form described above, h(�;Gp) � h(�p; Gp) = ph(�;G) = 0.Given any partition 2 L, the double application of the Pinsker formula (2.5) yieldsH( j �Gp) +H(�j��Gp � Gp) = H(�j��Gp) +H( j �Gp � �Gp);and since H(�j��Gp � Gp) = H(�j��Gp) = 0, we obtainH( j �Gp) = H( j �Gp � �Gp);whence H( j �Gp) = H( j �Gp � �).Let Gp be generated by T n1 ; T n2 ; T n3 , then �Gp � T�n1 �T1 � T�n2 ( T1)�T2 � T�n3 ( T1T2)�T3.Thus one hasH( j�) � H( j �Gp � �) = H( j �Gp) � H � jT�n1 �T1 � T�n2 ( T1)�T2 � T�n3 ( T1T2)�T3� :Pass in this relation to a limit as n ! 1 to get H( j�) � H( j ̂). In the case when � �� for some rectangle � we also deduceH( j�) � H( j ̂) � H( j�̂�) � H( j�1):It follows from the condition � � �G that � can be represented as a limit in themetric d of some sequence of partitions n with n � ��n for some sequence of rectangles8

�n whose modulus tends to in�nity. Thus a passage to a limit in the above inequalitya�ords 0 = H(�j�) = H(�j�1), that is, � � �1, and hence �(�) � �1. �Definition A. The dynamical system (X;G) is said to have a completely positiveentropy if �(G) = �(") = �; equivalently, for any partition � 2 L, h(�;G) = 0 implies� = �.Proposition 3.2 A dynamical system (X;G) has completely positive entropy if andonly if for any partition � 2 L one has H(�) = supGp h(�;Gp), where the sup is takenover some sequence of subgroups Gp of �nite indices in G.Proof. If (X;G) has a completely positive entropy, it was shown at the end of theproof of theorem 3.1 that VGp ��Gp � �(G), that is, under our assumptions, ��Gp ! � forsome sequence of �nite index subgroups Gp whose index tends to in�nity. This impliesH(�) � h(�;Gp) = H(�j��Gp)! H(�);which was to be proved.Conversely, if h(�;G) = 0, it follows from (2.1) that h(�;Gp) = 0 for all �nite indexsubgroups Gp, that is, H(�) = 0, and hence the complete positivity. �4 Entropic theory and Pinsker algebra: the general caseThe results of sections 2, 3 admit a generalization onto a much broader class of trans-formation groups. Let G be a �nitely generated nilpotent torsionfree group. It is shownin Appendix that there exists a sequence of generators of G submitted to its structure.Reverse the sequence of generators as in lemma A.3 and rewrite it in the formT1; : : : ; Tn1; Tn1+1; : : : ; Tn2; Tn2+1; : : : ; TnN�1; TnN�1+1; : : : ; TnNwith n0 = 0, Tni+1; : : : ; Tni+1 2 Zi+1, and T jk =2 Zi for k > ni, j 2 Zn f0g, and thesubgroups Zi being chosen as in the Appendix.De�ne the linear order relation on the above generators by setting T1 < T2 < : : : <TM , (nN = M), together with the associated lexicographic linear order relation on G:T jMM � : : : �T j11 < T kMM � : : : � T k11 i� (jM ; : : : ; j1) is lexicographically less than (kM ; : : : ; k1).It is easy to verify that this order relation is invariant with respect to the left shifts onG. So we can apply the results of [9, 13].It follows from our construction that the multiplication law in G can be describedby the relation as follows:T iMM � : : : � T i11 � T kMM � : : : � T k11 = T iM+kM+fMM � : : : � T i1+k1+f11 ; (4.1)where fM = 0 and for j = 1; : : : ;M�1, the integer-valued (in fact, polynomial) functionfj = fj(ij+1; : : : ; iM ; kj+1; : : : ; kM ) is independent of i1; : : : ; ij, k1; : : : ; kj.Given an increasing sequence of �nite dimensional rectangles�n = fT kMM � : : : � T k11 : l1(n) � k1 � L1(n); : : : ; lM � kM � LM (n)g;9

we say that the modulus of �n tends to in�nity if Sn �n = G, (Lj(n) � lj(n)) !1 foreach j as n!1, and each j = 1; : : : ;M , withFj(n) = maxfjfj(ij+1; : : : ; iM ; kj+1; : : : ; kM )j : �n � ij+1 � n; : : : ;�n � iM � n;lj+1(n) � kj+1 � Lj+1(n); : : : ; lM(n) � kM � LM(n)g;one has Fj(n)Lj(n) � lj(n) ! 0 (4.2)as n!1.Just as in proposition 2.1, we claim that the sequence of rectangles �n whose modulustends to in�nity, forms a F�lner sequence of sets. Let �n = fT kMM � : : : � T k11 : l1(n) �k1 � L1(n); : : : ; lM(n) � kM � LM (n)g and g = T iMM � : : : � T i11 be a �xed element of G.To verify as in the proof of proposition 2.1 that S(g�n��n)S(�n) !1, it su�ces to observethat g �T kMM � : : : �T k11 = T kM+iM+�M (k1;:::;kM )M � : : : �T k1+i1+�1(k1;:::;kM )1 , with �j being derivedfrom fj as in (4.1) by �xing i1; : : : ; iM (in particular �j is independent of k1; : : : ; kj).Now our statement is an obvious consequence of the condition (4.2) as above.In this setting, one can reproduce the results of sections 2 and 3, using in their proofsjust the same ideas. We restrict ourselves to rewriting here only those de�nitions andstatements whose analogues are not completely identical with what one can �nd above.Let us demonstrate the possibility of transferring the results of the previous sectionsby sketching the proof of an analogue of lemma 2.6.Let Gp be a subgroup of G of index p. We consider only the subgroups generatedby T p11 ; : : : ; T pMM with the powers pi being chosen so that T p011 ; : : : ; T p0MM never generatethe same subgroup when p0i < pi for at least one i. Consider a 'fundamental domain' �pin G with respect to Gp which contains zero of G. This 'fundamental domain' �p canbe chosen to be a rectangle; let it be fT iMM � : : : � T i11 : 0 � i1 < p1; : : : ; 0 � iM < pMg.In this case the index of Gp is p = p1 � : : : � pM . Given a partition �, we denote, just asin section 2, by �p the partition ��p = Wg2�p g � �.Lemma 4.1 (analogue of lemma 2.6) For a subgroup Gp as above, the relation (2.1) isvalid.Proof. Let � = fT kMpMM � : : :�T k1p11 : 0 � k1 < m1; : : : ; 0 � kM < mMg be a rectan-gle in Gp. It is a routine veri�cation that the set � = fT jMM �: : :�T j11 = fg; f 2 �p; g 2 �gis inside the rectangle �0 = fT jMM � : : : � T j11 : jj1j � m1p1 + �1(m1; : : : ;mM); : : : ; jjM j �mMpM + �M(m1; : : : ;mM)g, with �j = maxfjfj(ij+1; : : : ; iM ; kj+1pj+1; : : : ; kMpM )j :0 � ij+1 < pj+1; : : : ; 0 � iM < pM ; 0 � k1 < m1; : : : ; 0 � kM < mMg and fj being asin (4.1).Observe that S(�) = S(�)S(�p) = m1 � : : : �mMp1 � : : : � pM . Let us assume that themodulus of � tends to in�nity as above. Then the modulus of �0 tends to in�nity, and itfollows from our de�nitions that S(�0)S(�) ! 1. In view of the above observation we havelim 1S(�0)H(��0) = lim 1S(�0)H(��_��0n�) � lim 1S(�)H(��) + lim 1S(�0)H(��0n�) �10

� lim 1S(�)H(��) + lim S(�0 n �)S(�0) H(�);and since the latter limit is zero, we deduce thatlim 1S(�0)H(��0) � lim 1S(�)H(��):What remains is to observe that the converse inequality is obvious, so �nally wehaveh(�;G) = lim 1S(�0)H(��0) = lim 1S(�)H(��) = lim 1pS(�)H((��p)�) = 1ph(�p; Gp): �Another example we demonstrate here is in applying very similar ideas to reproduc-ing proposition 2.8 in the more general setting of this section.Proposition 4.2 (analogue of proposition 2.8) For �; � 2 L and G a �nitely generatednilpotent torsionfree group, (2.2) is valid.Proof. Consider the rectangles �n = fT iMM � : : : �T k11 : 0 � k1 < q1(n); : : : ; 0 � kM <qM(n)g, �p = fT iMM � : : : �T i11 : 0 � i1 < r1(p); : : : ; 0 � iM < rM(p)g. One can verify thatthe set fT jMM �: : :�T j11 = fg : f 2 �n; g 2 �pg is inside the rectangle �n;p = fT jMM �: : :�T j11 :jj1j < q1(n) + �1(q1(n); : : : ; qM(n)); : : : ; jjM j < qM(n) + �M(q1(n); : : : ; qM(n))g, with�j = maxfjfj(ij+1; : : : ; iM ; kj+1; : : : ; kM )j :0 � ij+1 < rj(p); : : : ; 0 � iM < rM(p); 0 � k1 < q1(n); : : : ; 0 � kM < qM(n)g:Thus we obtain: H(��n) � H(��n��n;p) = H(��n;p) +H(��nj��n;p) �� H(��n;p) +Xf2�nH(f�j��n;p) � H(��n;p) +Xf2�nH(f�j�f�p) == H(��n;p) + S(�n)H(�j��p):Assume that the modulus of �n tends to in�nity as n ! 1 as above. Then �n;ptends to in�nity too, and S(�n;p)S(�n) ! 1 as n!1. Thus after dividing out by S(�n) weget for any �xed p h(�;G) � h(�;G) +H(�j��p):Now what remains is to send p to in�nity and to observe that H(�j��p)! H(�j�G) inorder to get the desired relation. �Lemma 4.3 (analogue of lemma 2.10) If �; � 2 L,limn!1H(�jT�nM (��G)��G) = H(�j��G):11

5 K-systems and invariant partitionsThis section is intended to reproduce in an appropriate form the theory of B. Kami�nsky[4] in the case of actions of the Heizenberg group as in section 2. Let G be such a group.We need a notion of �-relative entropy for a G-invariant measurable partition � which isde�ned by h�(G) = sup�2L h(�;G; �). Also, by the Pinsker closure � of G with respect to� we mean the join of all partitions � 2 L such that h(�;G; �) = 0. Clearly � = �(G).We list here some straightforward properties of �:(1) � � �,(2) � is G-invariant,(3) � = �,(4) for any automorphism T commuting with G one has T� = �.There exists a �-relative version of the Rokhlin-Sinai theorem concerning the exis-tence of perfect partitions for a single automorphism S [11] to be generalized below.Speci�cally, we have:Lemma 5.1 There exists a measurable partition � such that(i) � � S�1� � �,(ii) 1Wn=0Sn� = ",(iii) 1Vn=0 S�n� = �,(iv) h(�; S; �) = h(S; �).Lemma 5.2 For every partition � 2 L and G-invariant measurable partition �1; �2one has h(�;G; �1 _ �2) = h(�;G; �1 _ �2).Proof is just the same as that of [4, lemma 2].De�ne some special invariance properties related to the lexicographic order relationin G. From now on we replace the ordinary notion of G-invariance (g� = � for allg 2 G) by the term total invariance.A measurable partition � is said to be G-invariant if T�11 � � �, T�12 �T1 � �, andT�13 �T1T2 � �. Note that � is G-invariant i� T k33 T k22 T k11 � � � for those triples (k1; k2; k3)which are lexicographically less than (0; 0; 0). Another important observation is thatfor � G-invariant ��G = T�11 �.A measurable partition � is said to be strongly invariant i� T�11 � � �, 1Vn=0 T�n1 � =T�12 �T1, and 1Vn=0 T�n2 �T1 = T�13 �T1T2. Of course this notion is stronger than invariance.A measurable partition � is said to be G-exhaustive if � is strongly invariant and�G = ".Lemma 5.3 Suppose that a measurable partition � is strongly invariant, � 2 L, � �T p3 �T1T2 for some positive integer p, and h(�;G) = 0. Then � � 1Vn=0T�n3 �T1T2.12

Proof. Let us �rst prove our statement under the assumption � 2 L, � � T p3T q2 �T1for some integer q. It follows from h(�;G) = 0 and (2.1) that for the �nite indexsubgroup Gk generated by T k1 ; T k2 ; T3 one has h(�;Gk) = 0. An application of ananalogue of the relativized Pinsker formula (2:4) for the case G = Zgenerated by T k1[1] yields for any partition 2 LH(� _ j��T k1 _ �T k1 _ (�T k1 )�T k2 _ ( T k1 )�T k2 _ T p�13 �T1T2) == H(�j��T k1 _(�T k1 )�T k2 _( T k1 )�T k2 _T p�13 �T1T2)+H( j �T k1 _�T k1 _(�T k1 )�T k2 _( T k1 )�T k2 _T p�13 �T1T2)= H( j �T k1 _(�T k1 )�T k2 _( T k1 )�T k2 _T p�13 �T1T2)+H(�j��T k1 _ T k1 _(�T k1 )�T k2 _( T k1 )�T k2 _T p�13 �T1T2)It follows from our assumptions on � and the strong invariance of � that (�T1T2)�T3 �T p�13 �T1T2, and so H(�j��T k1 _(�T k1 )�T k2 _( T k1 )�T k2 _T p�13 �T1T2) = H(�j��T k1 _ T k1 _(�T k1 )�T k2 _( T k1 )�T k2 _ T p�13 �T1T2) = 0, whenceH( j �T k1 _�T k1 _(�T k1 )�T k2 _( T k1 )�T k2 _T p�13 �T1T2) = H( j �T k1 _(�T k1 )�T k2 _( T k1 )�T k2 _T p�13 �T1T2)for all k. If we assume � T p3 T q2T r1 � for some integer r we getH( j� _ T p�13 �T1T2) � H( j �T k1 _ �T k1 _ (�T k1 )�T k2 _ ( T k1 )�T k2 _ T p�13 �T1T2) == H( j �T k1 _(�T k1 )�T k2 _( T k1 )�T k2 _T p�13 �T1T2) � H( jT p3T q2T r�k1 � _T p3 T q�k2 �T1_T p�13 �T1T2):If we let in this relation k go to in�nity and apply the strong invariance of �, weobtain H( j� _ T p�13 �T1T2) � H( jT p3T q�12 �T1 _ T p�13 �T1T2) = H( jT p3T q�12 �T1):Since this is valid for a class of partitions which is d-dense in the class of partitions� 2 L, � � T p3 T q2 �T1, we can replace with � in the latter inequality to getH(�jT p3 T q�12 �T1) = 0;and thus we have deduced � � T p3T q�12 �T1 from � � T p3T q2 �T1. If we proceed this wayin�nitely many times, we obtain � � Vq T p3T q2 �T1 = T p�13 �T1T2. Now we can get rid of ourmore subtle assumption � � T p3T q2 �T1 as compared with that written in the statementof the lemma via an approximation argument since Wq T p3 T q2 �T1 = T p3 �T1T2.Thus we have deduced � � T p�13 �T1T2 from � � T p3 �T1T2. Now proceed by inductionin p to get the desired statement. �A relativised version of lemma 5.3 is also valid. Speci�cally we haveLemma 5.4 Let � be a G-totally invariant partition. Suppose also that a measurablepartition � is strongly invariant, � � �, � 2 L, � � � � T p3 �T1T2 for some positiveinteger p, and h�(�;G) = 0. Then � � 1Vn=0 T�n3 �T1T2.13

Proof. One can reproduce literally the proof of lemma 5.3 with functionals likeH(�j�) being replaced by H(�j � _�) and �(G) by �. �Lemma 5.5 For every G-exhaustive partition �, 1Vn=0 T�n3 �T1T2 � �(G).Proof. Let � 2 L and � � �(G), that is h(�;G) = 0, and assume also � 2 L with� � T p3 �T1T2 for some positive integer p. Denote also by Gk the �nite index subgroupgenerated by T k1 ; T k2 ; T k3 , then certainly h(�;Gk) = 0. Since this subgroup has quite thesame structure as G itself, we can apply the Pinsker formula (2.3) as follows:h(� _ �;Gk) = h(�;Gk) +H(�j��Gk _ �Gk) = h(�;Gk) +H(�j��Gk _ �Gk):Since h(�;Gk) = H(�j��Gk _ �Gk) = 0, we deduce that h(�;Gk) = H(�j��Gk _ �Gk ) forall k, whenceH(�j�) � H(�j��Gk_�Gk) = h(�;Gk) � H(�jT�k+11 ��T1_T�k+12 (�T1)�T2_T�k+13 (�T1T2)�T3)If we let here k go to in�nity, we get H(�j�) � H(�j�̂) with �̂ being as in the de�nitionof �(�). Now our purpose is to prove that �̂ � 1Vn=0 T�n3 �T1T2.Let 2 L and � �̂. It was shown in the proof of theorem 3.1 that �̂ � �(G),that is h( ;G) = 0. Since � � T p3 �T1T2 and � is G-invariant we have �̂ � T p3 �T1T2 andhence � T p3 �T1T2. Now by lemma 5.3 we deduce that � 1Vn=0 T�n3 �T1T2, and hence ourstatement.This, together with our previous observations implyH(�j�) � H(�j 1̂n=0 T�n3 �T1T2):Since �G = ", we observe that the latter inequality is valid for � in a dense subsetof L, and hence it is also valid for �. Thus we obtain0 = H(�j�) � H(�j 1̂n=0 T�n3 �T1T2);that is, � � 1Vn=0 T�n3 �T1T2, which was to be proved. �We also need a relativized version of lemma 5.5.Lemma 5.6 Let � be a G-totally invariant partition. For every G-exhaustive partition�, � � �, one has 1Vn=0 T�n3 �T1T2 � �.Proof is just the same as that of lemma 5.5, with some minor changes indicated inthe proof of lemma 5.4. �14

Lemma 5.7 There exists a measurable partition � with the properties:(a) T�11 � � �, T�12 �T1 � �, T�13 �T1T2 � �,(b) �G = ",(c) 1Vn=0 T�n3 �T1T2 � �(G),(d) h(G) = H(�j��G) = H(�jT�11 �).Proof. Let �k 2 L be an increasing sequence whose join is ". We claim thatthere exists an increasing sequence of positive integers n1 < n2 < : : : such that for�p = Wpk=1 T�nk3 �k one hasH(�pj(�p)�G)�H(�pj(�p+s+1)�G) � 1p; s � 0: (5.1)To form such sequence of nk, it su�ces to assure the propertyH(�pj(�q�1)�G)�H(�pj(�q)�G) < 1p 12q�p ; p < q: (5.2)In fact, if we take a sum in (5.2), we get H(�pj(�p)�G) �H(�pj(�q)�G) < 1p as desired. Inorder to obtain (5.2), we have to proceed by induction. Speci�cally, suppose n1; : : : ; nq�1have already been chosen, then nq should be selected to be so large that (5.2) is validwith p = 1; : : : ; q � 1. This is possible by a virtue of (2.5).Let � = 1Wp=1 �p, and � = � _ ��G . It is easy to verify that (a) and (b) are satis�ed for�. Let us prove that (c) holds for �. Since (�p)�G is an increasing sequence of partitionswhose join is ��G = ��G , we can pass to a limit in (5.1) as s!1 to getH(�pj(�p)�G)�H(�pj��G) � 1p ; p � 1: (5.3)Now let � 2 L and � � 1Vn=0 T�n3 �T1T2. Since the latter partition is totally invariantwith respect to G, we have �G � 1Vn=0 T�n3 �T1T2 = 1Vn=0 T�n3 �T1T2. Hence �G � T�n3 �T1T2for all n � 1, and so �G � (�T1T2)�T3.It follows from the Pinsker formula (2.3) thath(� _ �p; G) = h(�p; G) +H(�j��G _ (�p)G) = h(�;G) +H(�pj(�p)�G _ �G);and hence h(�;G) = h(�p; G) �H(�pj(�p)�G _ �G) +H(�j��G _ (�p)G): (5.4)It follows from lemma 5.2 thatH(�pj(�p)�G _ �G) � H(�pj(�p)�G _ (�T1T2)�T3) = H(�pj(�p)�G _ (�T1T2)�T3) �� H(�pj(�p)�G _ ��G) = H(�pj��G);15

and so (5.4) impliesh(�;G) � h(�p; G)�H(�pj��G) +H(�j��G _ (�p)G) � 1p +H(�j��G _ (�p)G):Since 1Wp=1(�p)G = ", we can pass to a limit in the latter inequality as p!1 to geth(�;G) = 0, that is � � �(G).To prove (d), observe that since �k increases up to ", H(�pj(�p)�G) = h(�p; G) =h(�p; G) ! h(G). On the other hand, since �p is increasing up to �, one has alsoH(�pj��G) ! H(�j��G ). Now it follows from (5.3) that H(�pj(�p)�G) and H(�pj��G) =H(�pj��G) have the same limit, and so h(G) = H(�j��G ) = H(�j��G) = H(�j��G) =H(�jT�11 �). �Definition B. A measurable partition � is said to be G-perfect if(i) T�11 � � �, T�12 �T1 � �, T�13 �T1T2 � �,(ii) �G = ",(iii) 1Vn=0 T�n1 � = T�12 �T1 , 1Vn=0 T�n2 �T1 = T�13 �T1T2,(iv) 1Vn=0 T�n3 �T1T2 = �(G),(v) h(G) = H(�j��G ) = H(�jT�11 �).Theorem 5.8 For any G-action there exists a G-perfect partition.Proof. Let � be a measurable partition which satis�es the properties (a) { (d) oflemma 5.7. The property (a) implies T�13 �T1T2 � �T1T2. Now a relativized version of thistheorem for the case of Z2 generated by T1; T2 on the quotient space X=�T1T2 impliesthat there exists a measurable partition � with the properties:T�13 �T1T2 � � � �T1T2; (5.5)T�11 � � �; T�12 �T1 � �; (5.6)�T1T2 = �T1T2; (5.7)1̂n=0 T�n1 � = T�12 �T1; 1̂n=0 T�n2 �T1 = T�13 �T1T2 = T�13 �T1T2; (5.8)hT�13 �T1T2 (�; (T1; T2)) = hT�13 �T1T2 (T1; T2) = H(�jT�11 � _ T�13 �T1T2): (5.9)To prove this, one has to repeat the proof of [4, theorem 4], with some slight changesbeing introduced like those in the proof of lemma 5.4 and � = T�13 �T1T2.It follows from (5.5) that T�13 �T1T2 � T�13 �T1T2 � �; this together with (5.6) im-plies (i). An application of (5.5), (5.7), and (b) yields 1Wn=0 T n3 �T1T2 � 1Wn=0 T n3 �T1T2 �1Wn=0 T n3 �T1T2 = ", i. e. (ii) is true. (5.7), (5.8) imply (iii). Thus we have checked that� is G-exhaustive, and so lemma 5.5 implies 1Vn=0 T�n3 �T1T2 � �(G). It also follows from16

(c) that 1Vn=0 T�n3 �T1T2 � �(G), so in view of (5.7) we get (iv). Now what remains is toverify (v).Let �k 2 L be an increasing sequence of partitions with Wk �k = �. Hence (Wk �k)�G =��G = T�11 �, and soH(�kjT�11 �) � H(�kj(�k)�G) = h(�k; G) � h(G); k � 1:Pass to a limit as k !1 to get H(�jT�11 �) � h(G). Rewrite (5.9) in the formH(�j��(T1T2) _ T�13 �T1T2) = sup�2L H(�j��(T1T2) _ T�13 �T1T2): (5.10)In view of (5.5) one has T�13 �T1T2 � ��(T1T2) � �T1T2, and soH(�j��(T1T2) _ T�13 �T1T2) = H(�j��(T1T2)) = H(�jT�11 �): (5.11)It follows from lemma 5.2 that H(�j��(T1T2) _ T�13 �T1T2) = H(�j��(T1T2) _ T�13 �T1T2), andso sup�2L H(�j��(T1T2) _ T�13 �T1T2) � sup�2L��� H(�j��(T1T2) _ T�13 �T1T2) �sup�2L��� H(�j��(T1T2) _ T�13 �T1T2) = sup�2L��� H(�jT�11 �) = H(�jT�11 �):A comparison of this result to (d), (5.10), (5.11) yields h(G) = H(�jT�11 �) � H(�jT�11 �)which completes the proof. �Lemma 5.9 The following conditions are equivalent:(a) h(G) = 0,(b) the only G-exhaustive partition is ",(c) each G-strongly invariant partitions is totally invariant.Proof. We start with demonstrating the equivalence of (a) and (b). Let h(G) = 0,and � a G-exhaustive partition. it follows from lemma 5.5 that 1Vn=0 T�n3 �T1T2 � �(G).Since h(G) = 0, we have �(G) = ", and then �T1T2 = ". On the other hand zeta hasthe property 1Vn=0T�n2 �T1 = T�13 �T1T2, and hence �T1 = ". In a similar way, apply theproperty 1Vn=0 T�n1 � = T�12 �T1 to get � = ", so (a) implies (b).Now let us suppose that (b) holds, and � be a perfect partition. In particular, � isG-exhaustive, and hence � = ". On the other hand the property (iv) implies �(G) = ",that is h(G) = 0. Thus we have the equivalence of (a) and (b).Check the equivalence of (a) and (c). Let h(G) = 0 and � be a strongly G-invariantpartition. We have 0 = H(�j��G ) = H(�jT�11 �), and so � � T�11 �. Thus the stronginvariance of � implies T�11 � = �, hence � = 1Vn=0 T�n1 � = T�12 �T1. Another application ofthe strong invariance yields � = 1Vn=0 T�n2 �T1 = T�13 �T1T2, and hence � is totally invariant.17

Conversely, let us suppose that (c) is valid, and � be a perfect partition. In particular,� is strongly invariant and h(G) = H(�jT�11 �). It follows from (c) that T�11 � = �, andhence h(G)=0, which completes the proof. �Definition C. G is said to be a K-group if there exists a measurable partition �such that(i) T�11 � � �, T�12 �T1 � �, T�13 �T1T2 � �,(ii) �G = ",(iii) 1Vn=0 T�n1 � = T�12 �T1 , 1Vn=0 T�n2 �T1 = T�13 �T1T2,(iv) 1Vn=0 T�n3 �T1T2 = �.Theorem 5.10 G is a K-group i� it has completely positive entropy.Proof. Let G be a K-group and � be a partition as in the de�nition above. Inparticular, � is G-exhaustive, and thus by lemma 5.5 �(G) = �, that is G has completelypositive entropy. The converse is an easy consequence of theorem 5.8. �The results of this section can be literally reformulated in the case of a general�nitely generated nilpotent torsionfree group.6 Relation between entropy, spectral and mixing propertiesfor the Heizenberg group actionsRemind that a representation of G is said to have a countable Lebesgue spectrum if itis equivalent to a countable direct sum of the regular representation of G.Lemma 6.1 ([10]). Let � be a measurable partition such that T�11 � � �, and (X=�; ��)is nonatomic, then dim(L2(�) L2(T�11 �)) =1.Theorem 6.2 If h(G) > 0 then G has a countable Lebesgue spectrum in L2(X;�) L2(�(G)).Proof. Let � be a G-perfect partition. It follows from �G = " and Vn T n3 �(T1;T2) =�(G) that L2(X;�) L2(�(G)) = +1Mn=�1 L2(T n3 �(T1;T2)) L2(T n�13 �(T1;T2)) == +1Mn=�1 UnT3(L2(�(T1;T2)) L2(T�13 �(T1;T2))):In a similar way, the relation Vn T n2 �T1 = T�13 �(T1;T2) impliesL2(�(T1;T2)) L2(T�13 �(T1;T2)) = +1Mn=�1L2(T n2 �T1) L2(T n�12 �T1) =18

= +1Mn=�1UnT2(L2(�T1) L2(T�12 �T1));and, �nally,L2(�T1) L2(T�12 �T1) = +1Mn=�1 L2(T n1 �) L2(T n�11 �) = +1Mn=�1 UnT1(L2(�) L2(T�11 �)):Thus we obtainL2(X;�) L2(�(G)) = M(k;m;n)2Z3UkT3UmT2UnT1(L2(�) L2(T�11 �)): (6.1)We claim that T�11 � � �. In fact, if one assumes the contrary T�11 � = �, one candeduce from the properties (ii), (iii), (iv) of a perfect partition that �(G) = ", that ish(G) = 0, which contradicts to our assumptions on the G-action. Thus T�11 � � �. Nowthe properties (i) and (ii) imply that the space (X=�; ��) is nonatomic, and so by lemma6.1 one has dim(L2(�)L2(T�11 �)) =1. Let f1; f2; : : : be a basis in L2(�)L2(T�11 �).It follows from (6.1) that fg;i = Ugfi, 1 2 N, g 2 G, form an orthonormal basis inL2(X;�) L2(�(G)), and Ugfh;i = fgh;i. This means that G has a countable Lebesguespectrum in L2(X;�) L2(�(G)). �As a consequence of theorems 5.10 and 6.2 we obtainCorollary 6.3 If G is a K-group then it has a countable Lebesgue spectrum.Corollary 6.4 If G has a singular spectrum or a spectrum with �nite multiplicity thenh(G) = 0. In particular, every group G with discrete spectrum has zero entropy.We describe here the relationship between the K-property and di�erent mixing prop-erties of G-actions. Given a �nite collection A1; : : : ; Ar of sets of positive measure, weassociate a �nite partition { corresponding to a (�nite) �-algebra of sets generatedby the above sets. Let further �n be a sequence of rectangles in G whose modulustends to in�nity, and S1mn(A1; : : : ; Ar) be the sigma algebra of sets corresponding tothe partition {̂mn = T�m1 ({�n)�T1 _ T�m2 (({�n)T1)�T2 _ T�m3 (({�n)T1T2)�T3:Definition D. A G-action is said to have the property of K-mixing (or brie yK-mixing) if for any sets A0; A1; : : : ; Ar, r � 0, and for each �xed nlimm!1 supB2S1mn(A1;:::;Ar) j�(A0 \B)� �(A0)�(B)j = 0: (6.2)Theorem 6.5 The following conditions are equivalent:(i) the G-action is a K-system;(ii) the G-action is K-mixing. 19

Proof. Suppose the G-action is a K-system, and A0; A1; : : : ; Ar, r � 0, be a �-nite collection of sets involved into the de�nition of the K-mixing property. Let �mnbe a projection of X onto the quotient space X={̂mn, and consider the associated de-composition of � with respect to this projection: � = R �ymnd(� � ��1)(y). Given anyB 2 S1mn(A1; : : : ; Ar), one hasj�(A0 \B)� �(A0)�(B)j = ������� Z�(B) �ymn(A0)d(� � ��1)(y)� Z�(B) �(A0)d(� � ��1)(y)������� �� Z�(B) j�ymn(A0)� �(A0)jd(� � ��1)(y):We know from theorems 3.1 and 5.10 that {1 = �({) = �, and henceTm S1mn(A1; : : : ; Ar) may contain only sets of measure 0 or 1. Now an application ofDoob's theorem on convergence of conditional expectations shows that j�ymn(A0) ��(A0)j ! 0 as m!1, and hence the limit relation (6.2).Conversely, suppose the G-action is K-mixing, and let { = fA1; : : : ; Arg be a �nitepartition such that h({; G) = 0. Since clearly { � {G one can deduce from theorem3.1 that { � �({) = {1.Let Ai be any of the sets that constitute {. It follows from { � {1 = Wn {̂�n thatone can select a sequence of sets Bn 2 Tm S1mn(A1; : : : ; Ar) such that �(Ai�Bn) ! 0as n ! 1. Since one also has Bn 2 S1mn(A1; : : : ; Ar) for all m, we deduce from (6.2)that j�(Ai \Bn)� �(Ai)�(Bn)j = 0 for all n, and after passage to a limit in n one has�(Ai) = �(Ai)�(Ai). Thus, �(Ai) = 0 or 1, which means that � = �. Thus we haveproved the complete positivity of the G-action, and theorem 5.10 implies that this is aK-system. �Definition E. A G-action is said to have the property of r-mixing if for all mea-surable sets A0; A1; : : : ; Arlimg1;:::;gr!�1 �(A0 \ g1A1 \ : : : \ gr � : : : � g1Ar) = �(A0) � : : : � �(Ar):Here gp = T lp3 Tmp2 T np1 ! �1 means that for any h = T k33 T k22 T k11 there exists N suchthat for all p > N (lp;mp; np) is lexicographically less than (k1; k2; k3).Theorem 6.6 If a G-action is a K-system, then it is r-mixing for all r � 1.Proof. Our argument is based on the property of K-mixing which is equivalent tothe K-property by theorem 6.5. We proceed by induction. If r = 1, our statement isstraightforward from the de�nition of K-mixing. Suppose the G-action is r-mixing andwe are given sets A0; A1; : : : ; Ar+1 and an " > 0. It follows from the K-mixing propertythat there exists some h 2 G such that for all g1; : : : ; gr+1 which are lexicographicallyless than hj�(A0\g1A1\ : : :\g1 � : : :�gr+1Ar+1)��(A0)�(g1A1\ : : :\gr+1 � : : :�g1Ar+1)j < ": (6.3)20

Replace in (6.3) �(g1A1 \ : : :\ gr+1 � : : : � g1Ar+1) by �(A1 \ : : :\ g�11 gr+1 � : : : � g1Ar+1).Observe that g�11 gr+1 � : : : � g1 coincides with gr+1 � : : : � g2 modulo center of G (which isof the lowest lexicographical order), and hence we may apply the induction hypothesis(r-mixing) to obtain the desired resultj�(A0 \ g1A1 \ : : : \ g1 � : : : � gr+1Ar+1)� �(A0) � : : : � �(Ar+1)j < ": �7 Perfect partitions and K-systems: the general caseDescribe brie y how to extend the results of sections 5, 6 onto actions of �nitely gener-ated nilpotent torsionfree groups. The reader is referred to section 4 and Appendix fora notation. We again restrict ourselves to those statements which are reproduced notliterally.A measurable partition � is said to be G-invariant if T�11 � � �, : : :, T�1M �T1:::TM�1 � �.A measurable partition � is said to be strongly invariant i� T�11 � � �, 1Vn=0 T�n1 � =T�12 �T1, : : :, 1Vn=0 T�nM�1�T1:::TM�2 = T�1M �T1:::TM�1.Lemma 7.1 (analogue of lemma 5.3) Suppose that a measurable partition � is stronglyinvariant, � 2 L, � � T pM�T1T2 for some positive integer p, and h(�;G) = 0. Then� � 1Vn=0 T�nM �T1:::TM�1.Lemma 7.2 (analogue of lemma 5.5) For every G-exhaustive partition � one has1Vn=0 T�nM �T1:::TM�1 � �(G).Lemma 7.3 (analogue of lemma 5.7) There exists a measurable partition � with theproperties:(a) T�11 � � �, : : :, T�1M �T1:::TM�1 � �,(b) �G = ",(c) 1Vn=0 T�nM �T1:::TM�1 � �(G),(d) h(G) = H(�j��G) = H(�jT�11 �).Definition F (analogue of De�nition B) A measurable partition � is said to beG-perfect if(i) T�11 � � �,: : :, T�1M �T1:::TM�1 � �,(ii) �G = ",(iii) 1Vn=0 T�n1 � = T�12 �T1, : : :, 1Vn=0 T�nM�1�T1:::TM�2 = T�1M �T1:::TM�1,(iv) 1Vn=0 T�nM �T1:::TM�1 = �(G),(v) h(G) = H(�j��G ) = H(�jT�11 �). 21

Definition G (analogue of de�nition C) G is said to be a K-group if there exists ameasurable partition � such that(i) T�11 � � �, : : :, T�1M �T1:::TM�1 � �,(ii) �G = ",(iii) 1Vn=0 T�n1 � = T�12 �T1, : : :, 1Vn=0 T�nM�1�T1:::TM�2 = T�1M �T1:::TM�1,(iv) 1Vn=0 T�nM �T1:::TM�1 = �.With the above background, the results of section 6 concerning the spectral andmixing properties can be literally reformulated in the case of a general �nitely generatednilpotent torsionfree group.AppendixLet G be a countable nilpotent �nitely generated torsionfree group andG = GN � GN�1 � GN�2 � : : : � G2 � G1 � G0 = feg (A:1)its lower central series. Speci�cally, GN�1 = [G;G], the commutant subgroup, andGi = [G;Gi+1]. Our �rst step is to verify that each Gi is �nitely generated. For that,we proveLemma A.1. Let G be a nilpotent �nitely generated group. There exists a �nite(ordered) sequence of generatorsS1; : : : ; Sn1 ; Sn1+1; : : : ; Sn2; Sn2+1; : : : ; SnN�1 ; SnN�1+1; : : : ; SnN 2 Gsuch that n0 = 0, Sni+1; : : : ; SnN 2 GN�i, and Sjk =2 GN�i for k � ni, j 2Zn f0g.Proof. We proceed by induction in the nilpotence order N . In the case N = 1 ourstatement is trivial. Now suppose that this statement is true for any �nitely generatedN � 1-step nilpotent group, and consider N -step nilpotent G.Consider the quotient group G=G1. We can apply the induction hypothesis to G=G1in order to choose in it a sequence of generatorsS1; : : : ; Sn1 ; Sn1+1; : : : ; Sn2 ; Sn2+1; : : : ; SnN�1+1; : : : ; SnNwith the required properties. Let it be a projection of a (ordered) sequenceS1; : : : ; Sn1; Sn1+1; : : : ; Sn2; Sn2+1; : : : ; SnN�2+1; : : : ; SnN�1of elements of G. We are to complete it with a �nite sequence of generatorsSnN�1+1; : : : ; SnN of G1. For that, we consider the �nite subset of G1f[S�1i ; S�1j ] : 1 � i � nN�1; nN�2 + 1 � j � nN�1g: (A:2)We claim this set generates G1. This is an easy consequence of the de�nition G1 =[G;G2], together with the following observation, based on the fact that all the abovecommutors are central in G. Given f1; f2 2 G, g 2 G2, one has[f1f2; g] = f1(f2gf�12 g�1)gf�11 g�1 = [f1; g][f2; g];22

and in a similar way, for f 2 G, g1; g2 2 G2,[f; g1g2] = [f; g1][f; g2]:What remains is to replace the set (A.2) with a �nite minimal subset which still gener-ates G1. �Corollary A.2. The commutant subgroup [G;G] is �nitely generated.Now consider the upper central series of G. Speci�cally, let Z1 be the center ofG. De�ne the subgroup Z2 in such a way that Z2=Z1 = center(G=Z1). Continue thisprocedure by choosing at step i the subgroup Zi so that Zi=Zi�1 = center(G=Zi�1). Itis easy to verify that after �nitely many steps we get ZN = G, with N being exactlythe same as in (A.1) [6]. Thus we have a �nite sequence of subgroupsG = ZN � ZN�1 � ZN�2 � : : : � Z2 � Z1 � Z0 = feg (A:3)We start with observing that the property of being torsionfree is inherited by thequotient groups in (A.3).Proposition A.3. For each i = 1; : : : ; N , n > 0, x 2 Zi, and xn 2 Zi�1 impliesx 2 Zi�1.Proof. We proceed by an induction in i. In the case i = 1 our statement reducesto observing that Z0 = feg and Z1(� G) is torsionfree.Now suppose that our proposition is valid for i� 1, and prove it for i. That is, weassume x 2 Zi and xn 2 Zi�1. For the sake of simplicity we stick to the case n = 2; thegeneral case can be treated in a similar way. Suppose the contrary: x =2 Zi�1. SinceZi�1=Zi�2 = center(G=Zi�2), we deduce that there is some y 2 G with [x; y] =2 Zi�2.On the other hand, of course [x; y] 2 Zi�1 since Zi=Zi�1 = center(G=Zi�1). In a similarway one can verify that [x2; y] 2 Zi�2. Since also[x2; y] = xxyx�1x�1y�1 = x[x; y]yx�1y�1 2 [x; y]2Zi�2;we deduce that [x; y]2 2 Zi�2. Now the induction hypothesis implies [x; y] 2 Zi�2, whichcontradicts to our choice of y. This proves our statement. �Corollary A.4. Let x 2 G be such that xn 2 Z1 for some n > 0. Then x 2 Z1.Proof. Choose i so that x 2 Zi n Zi�1. If we assume i > 1, then Zi�1 � Z1, and soxn 2 Zi�1. Now an application of proposition A.1 yields x 2 Zi�1. This contradicts toour choice of i. Thus i = 1, which is exactly our statement. �Lemma A.5. Let G be a nilpotent �nitely generated torsionfree group. There existsa �nite (ordered) sequence of generatorsT1; : : : ; Tn1; Tn1+1; : : : ; Tn2; Tn2+1; : : : ; TnN ; TnN�1+1; : : : ; TnN 2 Gsuch that n0 = 0, Tni+1; : : : ; TnN 2 ZN�i, and T jk =2 ZN�i for k � ni, j 2Zn f0g.Proof. We proceed by induction in N . In the case N = 1 our statement is trivial.Now suppose that this statement is true for any nilpotent �nitely generated torsionfreegroup with the length N � 1 series (A.3), and consider G with (A.3) of length N .Consider the quotient group G=Z1. By corollary A.4, it is torsionfree. We can applythe induction hypothesis to G=Z1 in order to choose in it a sequence of generatorsT1; : : : ; Tn1 ; Tn1+1; : : : ; T n2; T n2+1; : : : ; TnN�2+1; : : : ; TnN�123

with the required properties. Let it be a projection of a (ordered) sequenceT1; : : : ; Tn1 ; Tn1+1; : : : ; Tn2; Tn2+1; : : : ; TnN�2+1; : : : ; TnN�1of elements of G. We are to complete it with a sequence of generators TnN�1+1; : : : ; TnNof Z1 = center(G). For this sequence to be �nite, we claim that Z1 is always �nitelygenerated. This is accessible by an induction in N .In the case N = 1 (G Abelian) our statement is trivial. Suppose it is valid forN�1-step nilpotent �nitely generated torsionfree groups, and letG be N -step nilpotent.Consider Zc = [G;G]\Z1. Since [G;G] is a �nitely generated by corollary A.2, it followsfrom the induction hypothesis that center([G;G]) is �nitely generated, and hence so isits subgroup Zc. On the other hand, since Zc � [G;G], the quotient group Z1=Zc isembeddable into an Abelian �nitely generated group G=[G;G], and so one can form a�nite sequence of generators for Z1 via merging those for Zc and Z1=Zc, after raisingthe latter up to elements of Z1. �References[1] Conze J. P., Entropie d'un groupe ab�elien de transformations, Z. Wahr. Theor.Verw. Geb., 25, 11 { 30 (1972).[2] E. Glasner, J.-P. Thouvenot, B. Weiss. Entropy theory without past. PreprintESI-612.[3] Cornfeld I. P., Fomin S. V., Sinai Ya. G.,, Ergodic Theory, Ney York: Springer,1980.[4] Kami�nski B., The theory of invariant partitions for Zd-actions, Bull. del'Academie Polonaise des Sciences, Ser. Sci. Math., 29, 349 { 362 (1981).[5] Kolmogorov A. N., A new metric invariant for transitive automorphisms ofLebesgue spaces, Dokl. Acad. Sci. USSR, 119, 861 { 864 (1958) (in Russian).[6] Kurosh A. G.. Group theory, Moscow, Nauka, 1967 (in Russian).[7] Ornstein D., Two Bernoulli shifts with �nite entropy are isomorphic, Adv. inMath., 5, 339 { 348 (1970).[8] Ornstein D., Weiss B., Entropy and isomorphism theorems for actions of amenablegroups, J. d'Analyse Math., 48, 1 { 141 (1987).[9] Pitzkel B. S., On information futures of amenable groups, Dokl. Acad. Sci. USSR,223, 1067 { 1070 (1975) (in Russian).[10] Rokhlin V. A. , Lectures on the entropy theory of transformations with invariantmeasure, Uspehi Mat. Nauk, 22, 4 { 54 (1967) (in Russian).[11] Rokhlin V. A., Sinai Ya. G., Construction and properties of invariant measurablepartitions, Dokl. Acad. Sci. USSR, 141, 1038 { 1041 (1961) (in Russian).[12] D. J. Rudolf, B. Weiss, Entropy and mixing for amenable group actions, Preprint.24

[13] Safonov A. V., Information pasts in groups, Izvestiya Acad. Sci. USSR, 47, 421{ 426 (1983) (in Russian).[14] Schmidt K. Dynamical systems of algebraic origin, Warwick, preprint 36/1994.[15] Sinai Ya. G., On the notion of entropy for dynamical systems, Dokl. Acad. Sci.USSR, 124, 768 { 771 (1959) (in Russian).

25