ALGEBRAIC POLYMORPHISMS KLAUS SCHMIDT AND ANATOLY VERSHIK. Dedicated to the memory of our colleague and friend William Parry

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1 ALGEBRAIC OLYMORHISMS KLAUS SCHMIDT AND ANATOLY VERSHIK Dedicated to the memory of our colleague ad fried William arry Abstract. I this paper we cosider a special class of polymorphisms with ivariat measure, the algebraic polymorphisms of compact groups. A geeral polymorphism is by defiitio a may-valued map with ivariat measure, ad the cojugate operator of a polymorphism is a Markov operator (i.e., a positive operator o L 2 of orm 1 which preserves the costats). I the algebraic case a polymorphism is a correspodece i the sese of algebraic geometry, but here we ivestigate it from a dyamical poit of view. The most importat examples are the algebraic polymorphisms of torus, where we itroduce a parametrizatio of the semigroup of toral polymorphisms i terms of ratioal matrices ad describe the spectra of the correspodig Markov operators. A toral polymorphism is a automorphism of T m if ad oly if the associated ratioal matrix lies i GL(m, Z). We characterize toral polymorphisms which are factors of toral automorphisms. 1. Algebraic polymorphisms Defiitio 1.1. Let G be a compact group with Borel field B G, ormalized Haar measure λ G ad idetity elemet 1 = 1 G. A closed subgroup G G is a (algebraic) correspodece of G if π 1 () = π 2 () = G, where π i : G G G, i = 1, 2, are the coordiate projectios (which are obviously group homomorphisms). Every correspodece G G defies a map Π from G to the set of all oempty closed subsets of G by Π (x) = {y : (x, y) } (1.1) for every x G. Clearly, π i seds the Haar measure o to Haar measure o G; i the termiology of [1], the correspodece defies a (algebraic) polymorphism of G (more exactly, determies a polymorphism of the measure space (G, B G, λ G ) to itself ). 1 The correspodece G G ad the polymorphism Π obviously determie each other. Algebraic polymorphisms from oe compact group to aother are defied similarly. A correspodece G G is a factor of a correspodece G G (ad the polymorphism Π is a factor of Π ) if there exists a surjective group homomorphism φ: G G with (φ φ)() =. If φ ca be chose to be a group isomorphism the ad (resp. Π ad Π ) are isomorphic. This otio of factors is cosistet with the termiology i [1]: if Π is a measurepreservig polymorphism of a probability space (X, S, µ) determied by a self-couplig ν 2000 Mathematics Subject Classificatio. 37A05,37A45. Key words ad phrases. Algebraic polymorphisms, toral automorphisms. The first author would like to thak the Departmet of Mathematics, Ohio State Uiversity, Columbus, Ohio, for hospitality while some of this work was doe. The secod author would like to thak the Erwi Schrödiger Istitute i Viea for hospitality ad support while some of this work was doe, ad gratefully ackowledges partial support by the grats RFBR ad NSh I geeral, a measure-preservig polymorphism Π of a probability space (X, S, µ) is determied by a probability measure ν o X X with π i ν = µ for i = 1, 2, i.e., by a couplig of µ with itself. 1

2 ALGEBRAIC OLYMORHISMS 2 of µ, ad if T S is a sub-sigma-algebra, the the factor polymorphism Π T of (X, T) is determied by the restrictio of ν to the sigma-algebra T T S S. Let G G be a correspodece (sice we oly cosider algebraic correspodeces ad polymorphisms we drop the term algebraic from ow o). The subgroup = {(y, x) : (x, y) } (1.2) correspods to the cojugate (or iverse) polymorphism of Π. If 1, 2 are two correspodeces of G, their product 1 2 is the correspodece Clearly, 1 2 = {(x, z) G G : (x, y) 2 ad (y, z) 1 Π 1 2 (x) = Π 1 Π 2 (x) = for at least oe y G}. y Π 2 (x) Π 1 (y) (1.3) for every x G. With respect to the compositio (1.3) the set of all correspodeces (or, equivaletly, the set of all polymorphisms) of G is a semigroup, deoted by (G), with ivolutio, idetity elemet 1 = {(g, g), g G} ad zero elemet 0 = G G. For later use we itroduce also the higher powers of, 2, defied recursively by = 1. (1.4) If G G is a correspodece such that the group homomorphisms π i : G, i = 1, 2, are ijectios, the is (the graph of) a automorphism of G, ad the cojugate correspodece yields the iverse automorphism. If π 2 is a ijectio the is (the graph of) a edomorphism (i.e., of a surjective group homomorphism), ad if π 1 is a ijectio the is (the graph of) a exomorphism (i.e., is the graph of a edomorphism). The group of automorphisms as well as semigroups of edo- ad exomorphisms are subsemigroups of the semigroup of (G) of correspodeces of G. We ote i passig that the product of the algebraic polymorphisms is a special case of the geeral otio of the product of measure-preservig polymorphisms i [1]. Defiitio 1.2. For the followig defiitios we fix a correspodece of a compact group G. We write B ad λ for the Borel field ad the Haar measure of. (1) Algebraic factor polymorphisms. Let H G be a closed subgroup, ad let H = /(H H) (G/H G/H) be the associated factor correspodece. The subgroup H G is ivariat, co-ivariat or doubly ivariat uder the polymorphism Π if H is a edomorphism, exomorphism or a automorphism. Examples will be give i Sectio 3. (2) The Markov operator. ut B (i) = π 1 i (B G ) B, i = 1, 2, ad let F i L 2 (G, B G, λ G ) be the subspace of fuctios measurable with respect to B (i), i = 1, 2. Let r i be the orthogoal projectio i L 2 (G, B G, λ G ) oto F i, i = 1, 2. We defie the Markov operator V : L 2 (G, B G, λ G ) L 2 (G, B G, λ G ) as follows: if f L 2 (G, B G, λ G ), we defie h L 2 (, B, λ ) by for every (x, y) ad set h(x, y) = f(x) V f = E λ (h B (2) ), (1.5)

3 ALGEBRAIC OLYMORHISMS 3 where E λ ( ) stads for coditioal expectatio with respect to λ. The ad V = r 2 r 1, V = r 1 r 2, (1.6) (cf. (1.2)). Note that V preserves positivity ad has orm 1. V = V (1.7) (3) The Markov process X. The closed, shift-ivariat subgroup X = {(x ) G Z : (x, x +1 ) for every Z} (1.8) is the Markov process of, ad the correspodig Markov shift σ : X X is defied by (σ x) = x +1 for every x = (x ) X. Note that σ is a automorphism of the compact group X which preserves the ormalized Haar measure λ X of X, ad that the Markov shift σ : X X correspodig to is the time reversal of σ. Motivated by cosiderig the various tail sigma-algebras (past, future ad two-sided) of the Markov process X we call the polymorphism Π right (left, or totally) odetermiistic if there there is o closed ivariat (co-ivariat, or doubly ivariat) proper subgroup H G (cf. Theorem 2.4). (4) Ergodicity. The polymorphism Π is ergodic if the costats are the oly V -ivariat fuctios. ropositio 1.3. Let G be a compact group ad G G a correspodece. The there exist closed ormal subgroups K (i) G, i = 1, 2, ad a cotiuous group isomorphism η : G/K (1) G/K(2) such that = {(g 1, g 2 ) G G : η (g 1 K (1) ) = g 2K (2) }. (1.9) roof. We set K (1) = {g G : (g, 1) }, K (2) = {g G : (1, g) } ad observe that K (1) ad K (2) are ormal subgroups of G, sice π 1 () = π 2 () = G. Sice = {(g 1 p 1, g 2 p 2 ) : (g 1, g 2 ), p 1 K (1) }, we may view as a subset G/K (1) G/K(2), p 2 K (2), ad the defiitio of the groups K(i) implies that is the graph of a cotiuous group isomorphism η : G/K (1) Remark 1.4. The triples (K (1) η : G/K (1) polymorphisms of G. G/K(2) G/K(2)., K(2), η ), where K (i), i = 1, 2, are subgroups of G ad is a group isomorphism, form a parametrizatio of the algebraic Defiitio 1.5. A correspodece G G is fiite-to-oe (ad defies a polymorphism of discrete type) if the groups K (i) i (1.9) are both fiite. The fiite-to-oe correspodeces of G form a subsemigroup f (G) (G) of the semigroup of all correspodeces of G. For the otatio i the followig characterizatio of (co-)ivariace we agai refer to (1.9). Theorem 1.6. Let G G be a correspodece ad H G a closed ormal subgroup. (1) H is ivariat uder the polymorphism Π if ad oly if η (HK (1) ) H; (2) H is co-ivariat uder Π if ad oly if η 1 (HK(2) ) H; (3) H is bi-ivariat uder Π if ad oly if K (1) H ad η (H) = H (i which case we also have that K (2) H).

4 ALGEBRAIC OLYMORHISMS 4 roof. Clearly, K (2) η (HK (1) ). If η (HK (1) ) H the ivariace follows from Defiitio 1.2 (1). Coversely, if K (2) η (HK (1) ) H, the H is the graph of a group edomorphism. The other assertios are proved similarly. Corollary 1.7. Let G G be a correspodece ad H G be a closed ormal subgroup. We deote by K (i), i = 1, 2, the closed ormal subgroups of G associated with the correspodece, 2, i (1.4) by (1.9). The sequeces of subgroups (K (i), 1) are odecreasig ad have the followig property. (1) H is ivariat uder Π if ad oly if it cotais 1 K(2) ; (2) H is co-ivariat uder Π if ad oly if it cotais 1 K(1). roof. If a closed ormal subgroup H G is ivariat uder Π the Theorem 1.6 (1) shows that K (2) = η 2 (K (1) K(2) ) η (K (1) H) H, hece η (K (2) ) H ad, by 2 iductio, η (K (2) ) H for every 1. Coversely, if H 1 K(2), the it is obviously ivariat. The proof of the secod assertio is aalogous. If the group G is abelia, the characterizatio of ergodicity of a polymorphism of G is completely aalogous to that of ergodicity of a automorphism of G. Theorem 1.8. Let G G be a correspodece of a compact abelia group G with Markov operator V (cf. (1.5)). The Π is oergodic if ad oly if there exist a otrivial character χ of G ad a iteger 1 with V χ = χ. roof. If χ is a otrivial character of G the the restrictio to of h = χ π 1 is a otrivial character o, ad V χ = E λ (h B (2) ) is either equal to zero or a otrivial character of G (depedig o whether h is costat o {1 G } K (2) or ot). Fourier expasio completes the proof of the theorem. 2. Toral polymorphisms Compact groups do ot have dyamically iterestig polymorphisms uless they have large abelia quotiets. For this reaso we focus our attetio i this sectio o compact abelia groups, ad i particular o fiite-dimesioal tori. Let m 1, ad let f (T m ) be the semigroup of all fiite-to-oe correspodeces of T m. We deote by L the set of all fiite idex subgroups of Z m. For every = ( 1,..., m ) Z m ad x = (x 1,..., x m ) T m we write χ (x) = e 2πi m j=1 jx j (2.1) for the value of the correspodig character χ of T m at x. The aihilator of a subgroup F T m (or F T 2m ) is deoted by F (resp. F ). For Q GL(m, Q) we put Λ Q = Z m QZ m L. (2.2) Fially we itroduce the semigroup with compositio M = {(Q, Λ) : Q GL(m, Q), Λ L, Λ Λ Q } (2.3) (Q, Λ) (Q, Λ ) = (QQ, Λ QΛ ). (2.4)

5 ALGEBRAIC OLYMORHISMS 5 ropositio 2.1. The semigroup f (T m ) is isomorphic to the semigroup M i (2.3), where the isomorphism θ : M f (T m ) is give by θ(q, Λ) = {(Q 1, ) : Λ} (2.5) for every (Q, Λ) M. A correspodece f (T m ) is coected if ad oly if = Q = θ(q, Λ Q ) (2.6) for some Q GL(m, Q) (cf. (2.2)). Fially, if = θ(q, Λ) f (T m ), the = θ(q 1, Q 1 Λ). roof. For every Q GL(m, Q) ad Λ Λ Q, θ(q, Λ) T m T m is obviously a elemet of f (T m ), ad (1.9) guaratees that every f (T m ) is obtaied i this maer. If Λ Λ Q Z m, the = θ(q, Λ) cotais Q = θ(q, Λ Q ) as a fiite idex subgroup ad is therefore ot coected. I order to prove the coverse we set W Q = {(Q 1, ) : Λ Q }. The dual group of Q is of the form (Z m Z m )/W Q. If Q is ot coected, the there exist a elemet (m, ) (Z m Z m ) W Q ad a l > 1 with (lm, l) W Q. Hece (m, ) = (Q 1 k, k) for some k Z m QZ m = Λ Q, ad (m, ) W Q. This cotradictio proves that Q is coected. The last assertio is obvious. Remark 2.2. ropositio 2.1 shows that coected fiite-to-oe correspodeces are i oe-to-oe correspodece with the elemets of GL(m, Q). For every 1 we defie ad K (i) as i Corollary 1.7. Theorem 2.3. Let Q GL(m, R) ad = θ(q, Λ) f (T m ), where Λ Λ Q = Z m QZ m is a fiite idex subgroup (cf. (2.2) ad (2.5)). (1) The followig coditios are equivalet. (a) Π is right odetermiistic, (b) Ξ + = { Λ : Qk Z m for every k 0} = {0}, (c) 1 K(2) is dese i T m. (2) The followig coditios are equivalet. (a) Π is left odetermiistic, (b) Ξ = { Λ : Qk Z m for every k 0} = {0}. (c) 1 K(1) is dese i T m. (3) The followig coditios are equivalet. (a) Π is totally odetermiistic, (b) Ξ + Ξ = { Λ : Qk Z m for every k Z} = {0}. (c) Both 1 K(1) ad 1 K(2) are dese i T m. roof. I order to prove (1) we ote that Ξ + is a group ad that Q 1 Ξ + Ξ+. We set H = (Ξ + ) T m. The the correspodece H = /(H H) is the graph of a cotiuous surjective homomorphism of the group Y = T m /H to itself. The coverse is proved by reversig this argumet. If the group H = 1 K(2) is trivial, the is the graph of a edomorphism. If H is ot dese i T m, the its closure H is otrivial ad is the smallest proper ivariat subgroup of Π (cf. Corollary 1.7). The assertios (2) ad (3) are proved i exactly the same maer.

6 ALGEBRAIC OLYMORHISMS 6 The property of beig left, right or totally odetermiistic ca also be expressed i terms of the Markov group X i (1.8). Theorem 2.4. Uder the hypotheses of Theorem 2.3 the polymorphism Π is right odetermiistic if ad oly if the remote past of the process X is trivial. 2 Similarly, Π is left odetermiistic if ad oly if the remote future of X is trivial. roof. We deote by λ X the Haar measure of the compact abelia group X. For every 1 we defie the subgroups K (1) ad K (2) as i Corollary 1.7. For the proof of the theorem it is eough to otice that the coditioal measure λ X (. x = t) for fixed t T m ad < 0 is the uiform measure o a coset of the group K (2). Accordig to Theorem 1.6 (1), the polymorphism Π is right odetermiistic if ad oly if 1 K(2) is dese i T m, i which case the coditioal measures λ X (. x = t) coverge to λ T m as m. Theorem 2.5. Let Q GL(m, R) ad = θ(q, Λ) f (T m ), where Λ Λ Q = Z m QZ m is a fiite idex subgroup (cf. (2.5)). The Π is oergodic if ad oly if Q has a otrivial root of uity as a eigevalue. Furthermore, if Π is left, right or totally odetermiistic, the it is ergodic. roof. By defiitio, = {(Q 1, ) : Λ}. A direct calculatio shows that, for every Z m, { { χ V (χ ) = Q 1 if Λ, V 0 otherwise, (χ χ Q if Q 1 Λ, ) = (2.7) 0 otherwise, (cf. (2.1)). The existece of a Λ with V kχ = χ for some k 1 is obviously equivalet to Q havig a root of uity as a eigevalue. The last assertio is obvious. We tur to the spectral properties of the Markov operator V associated with a correspodece f (T m ). Theorem 2.6. Let m 1, f (T m ), ad let Sp(V ) D = {z C : z 1} be the spectrum of the liear operator V : L 2 0 (Tm, B m T, λ T m) L2 0 (Tm, B m T, λ Tm) i (1.5), where L 2 0 (Tm, B m T, λ T m) = {f L2 (T m, B m T, λ T m) : f dλ T m = 0} is the orthocomplemet of the costats. (1) Sp(V ) = Sp(V ) = {0} if ad oly if is totally odetermiistic; (2) Sp(V ) = Sp(V ) = S = {z C : z = 1} if ad oly if Ξ+ = Ξ = Λ; (3) Sp(V ) = Sp(V ) S {0} if ad oly if Ξ+ = Ξ Λ; (4) If Ξ Ξ+ the Sp(V ) = D; (5) If Ξ + Ξ the Sp(V ) = D. roof. We choose (Q, Λ) M with θ(q, Λ) = (cf. (2.5)). By defiitio of M, Λ Λ QΛ. If is totally odetermiistic the there exist, for every ozero Λ, a smallest positive iteger k + () ad a largest egative iteger k () such that Q k± () / Λ. If we set O Q () = {Q k ()+1,..., Q k+ () 1 }, 2 The remote past of the process X (T m ) Z is the itersectio A = Z A, where A is the sigma-algebra geerated by the coordiates of the process X with idex less tha or equal to. The remote future of X (T m ) Z is the sigma-algebra A = Z A+, where A + is geerated by the coordiates with idex greater tha or equal to of X.

7 ALGEBRAIC OLYMORHISMS 7 the the restrictio of V to the liear spa O Q() of O Q () i L 2 (T m, B T m, λ T m) is uitarily equivalet to a matrix of the form , which has spectrum {0}. By takig the direct sum of the subspaces O Q (), Z, i L 2 (T m, B T m, λ T m) we see that Sp(V ) = {0}, ad that the same is true for Sp(V ). This proves (1). The assertio (2) is obvious, sice the coditio give there is equivalet to Q beig a elemet of GL(m, Z). I order to prove (3) we set Ξ = Ξ + = Ξ p, S = Ξ, Y = Ξ = T m /S, ad we observe that the restrictio of Q to Ξ is a group automorphism. Hece the restrictios of V ad V to the closed liear spa of {χ : Ξ} i L 2 (T m, B T m, λ T m) are uitary. For / Ξ there exist a smallest oegative iteger k + () ad a largest opositive iteger k () such that Q k± () / Z, ad by combiig the precedig paragraph with the argumet i the proof of (1) we obtai (3). If Ξ Ξ+ there exists a Λ with Qk Λ for every k 0, but Q 1 / Λ. The restrictio W of V to the closed liear spa H of {χ Q k : k 0} has a ozero kerel, sice V χ = 0. Furthermore, if γ C, γ < 1, ad if v γ = k 0 γk χ Q k H, the V v γ = γv γ, i.e., v γ is a eigevector of V with eigevalue γ. This proves that Sp(V ) Sp(W ) = D. The same argumet shows that Sp(V ) Sp(W ) = D if Ξ+ Ξ, ad the remaiig implicatios are immediate cosequeces of what has already bee show. 3. Factors of toral automorphisms ad other examples If A, B are edomorphisms of T m, the (A, B) = {(x, y) T m T m : Ax = By} (3.1) is a fiite-to-oe correspodece, ad every f (T m ) is of this form. Note that (A, B) = (CA, CB) for every C GL(m, Z), ad that (AC, BC ) ad (A, B) are isomorphic if C GL(m, Z). The subgroups K (1) (A,B) ad K(2) (A,B) ad the isomorphism η (A,B) : T m /K (1) (A,B) T m /K (2) (A,B) associated with (A, B) by (1.9) are give by K(1) (A,B) = (A Z m ), K (2) (A,B) = (B Z m ) ad η (A,B) = B 1 A, respectively, where deotes traspose. Ergodicity of (A, B) is thus equivalet to the assumptio that B 1 A GL(m, Q) has o eigevalues which are roots of uity (cf. Theorem 1.8). = B (A ) 1. Ac- I order to characterize coectedess of (A, B) we set Q = cordig to ropositio 2.1, (A, B) is coected if ad oly if i.e., if ad oly if B Z m = Z m B (A ) 1 Z m = Λ Q, η 1 Z m = (A ) 1 Z m (B ) 1 Z m. (3.2) Example 3.1. Let m = 1, k, l N ad = {(u, v) : u, v T, ku = lv}. We deote by s the highest commo factor of k, l ad set k = k s, l = l s. The = θ(q, Λ), where Λ = lz ad Q is multiplicatio by k l (cf. (2.5)). If k, l are coprime (i.e., if s = 1) the = Q is coected by (3.2).

8 ALGEBRAIC OLYMORHISMS 8 If s > 1 the φ is the quotiet map from Z to F = Z/sZ, is discoected, ad K (1) K(2) = {x T : sx = 0 (mod 1)}). Fially, if k/l = 1 the Π is ergodic. If k, l are coprime ad k > 1, l > 1, the Π is totally odetermiistic. Examples 3.2 (Factors of polymorphisms). (1) Cosider the correspodece = {(u, v) : u, v T, 3u = 2v} (cf. Example 3.1 (1)), ad let H = {0, 1/5, 2/5, 3/5, 4/5} T. The is the aihilator of {(3k, 2k) : k Z} Z 2 ad H is the aihilator of {(15k, 10k) : k Z}. Note that ad H are isomorphic. { ( )} (2) Let Q = ( ), ad let = Q = θ(q, Z 2 1/2 ). ut H = 0, T 2 ad 0 set = H. We idetify T 2 /H with T 2 by the map φ ( s t ) = ( 2s t ) ad view as a correspodece of T 2. The is isomorphic to the polymorphism of (A, B) of T 2 with ( ) ( ) A =, B = Sice A, B / GL(2, Z), is ot the graph of a automorphism. Example 3.2 (2) shows that a toral automorphism may have a proper polymorphism as a factor (proper meas that the groups K (1), K(2) i (1.9) are ot both trivial). However, the followig theorem shows that factors of automorphisms always have a otrivial doubly ivariat subgroup. Theorem 3.3. Let (A) f (T m ) be the graph of a toral automorphism A GL(m, Z). For every fiite subgroup H T m there exists a fiite doubly ivariat subgroup H T m cotaiig H. I particular, H is the graph of a automorphism of T m /H. I other words, if a polymorphism is a factor of a toral automorphism the it has a further factor which is agai a automorphism. roof. Sice H is fiite, there exists a q 1 such that H H = {t T m : qt = 0}. As oe checks easily, 1 K(i) H for i = 1, 2. Theorem 2.3 shows that H is ivariat uder H, which proves our claim. Theorem 3.3 allows us to say a little more about the structure of factors of toral automorphisms. Corollary 3.4. Let (A) f (T m ) be the graph of a toral automorphism A GL(m, Z), H T m a fiite subgroup ad H T m a fiite doubly ivariat subgroup cotaiig H. If we idetify both T m /H ad T m /H with T m, the the correspodece = H is the graph of a toral automorphism A (i.e., = (A )), ad the correspodece = H has the graph of the automorphism A GL(m, Z) as a factor with kerel (H/H ) (H/H ). roof. The idetificatios of T m /H ad T m /H with T m yield fiite-to-oe equivariat homomorphisms T m T m /H T m /H where the automorphisms A ad A act o the first ad third torus ad the polymorphism Π o the secod. Remarks 3.5. (1) The automorphism A GL(m, Z) i Corollary 3.4 is obviously cojugate to A i GL(m, Q), but ot ecessarily i GL(m, Z).

9 ALGEBRAIC OLYMORHISMS 9 Coversely, if = (A) is the graph if some A GL(m, Z), ad if A GL(m, Z) is cojugate to A i GL(m, Q), the the graph (A ) is isomorphic to (A) H for some fiite subgroup H T m. (2) There is a miimal choice of the subgroup H T m i Theorem 3.3: the subgroup geerated by Z Ak H (which we kow to be fiite from the proof of Theorem 3.3). (3) Corollary 3.4 shows that a polymorphism Π is a factor of a automorphism A GL(m, Z) of T m if ad oly if there exist a A GL(m, Z) which is cojugate to A i GL(m, Q) ad fiite groups H H T m such that is a skew product over the (the graph of) automorphism A with fibre (H/H ). Note, however, that is coected ad is therefore a otrivial H/H -budle over the base T m o which A acts. (4) I [1] it is show that every polymorphism is a factor of a automorphism with respect to some ivariat partitio (i.e., ivariat sub-sigma-algebra), but Theorem 3.3 shows that this is ot true i the algebraic category. Refereces [1] A.M. Vershik, olymorphisms, Markov processes, ad quasi-similarity, Discrete Coti. Dyam. Sys. 13 (2005), Klaus Schmidt: Mathematics Istitute, Uiversity of Viea, Nordbergstraße 15, A-1090 Viea, Austria ad Erwi Schrödiger Istitute for Mathematical hysics, Boltzmagasse 9, A-1090 Viea, Austria address: klaus.schmidt@uivie.ac.at Aatoly Vershik: Steklov Istitute of Mathematics at St. etersburg, 27 Fotaka, St.etersburg , Russia address: vershik@pdmi.ras.ru