The Perron-Frobenius operators, invariant measures and representations of the Cuntz-Krieger algebras

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1 The Perron-Frobenius opertors, invrint mesures nd representtions of the Cuntz-Krieger lgebrs Ktsunori Kwmur Reserch Institute for Mthemticl Sciences Kyoto University, Kyoto , Jpn For trnsformtion F on mesure spce (X, µ), we show tht the Perron-Frobenius opertor of F cn be written by representtion (L 2 (X, µ), π) of the Cuntz-Krieger lgebr O A ssocited with F when F stisfies some ssumption. Especilly, when O A is the Cuntz lgebr O N nd (L 2 (X, µ), π) in the bove is some irreducible representtion of O N, then there is n F -invrint mesure on X which is bsolutely continuous with respect to µ.. Introduction Invrint mesures (especilly, Hr mesures) ply n importnt role in the representtion theory of Lie groups nd hrmonic nlysis. On the other hnd, invrint mesures of non invertible trnsformtions re studied in [4, 5, 6] by the Perron-Frobenius opertors of dynmicl systems. By using the Perron-Frobenius opertors, the chrcteriztion of given dynmicl system nd the construction of invrint mesure re obtined. We show their roles in representtion theory of opertor lgebrs in this pper. Let L p (X, µ) be the set of ll complex-vlued mesurble functions φ on mesure spce (X, µ) stisfying φ Lp < nd let L p (X, µ; R) be the subset of ll rel-vlued functions in L p (X, µ) for p =, 2,. For nonsingulr trnsformtion F on X(tht is, µ(f (A)) = 0 if µ(a) = 0 for A X), P F is the Perron-Frobenius opertor (or the Frobenius-Perron opertor, the trnsfer opertor) of F if P F is the opertor on L (X, µ) which stisfies (.) (P F ψ)(x) dµ(x) = ψ(x) dµ(x) ( ψ L (X, µ)) A F (A) for ech mesurble subset A of X ([5]). By (.), P F ψ is uniquely determined s n element in L (X, µ) for ech ψ L (X, µ). For ψ L (X, µ) nd θ L (X, µ), we obtin X θ(f (x))ψ(x) dµ(x) = X θ(x)(p F ψ)(x) dµ(x). From this, P F is bounded liner opertor on L (X, µ) nd P F ψ L e-mil:kwmur@kurims.kyoto-u.c.jp.

2 ψ L for ech ψ L (X, µ; R). Further, positive function ρ L (X, µ) stisfies P F ρ = ρ if nd only if ρ is the density of n F -invrint mesure, tht is, the following holds for ny ψ L (X, µ): (.2) ψ(f (x))ρ(x) dµ(x) = ψ(x)ρ(x) dµ(x). X In order to describe both the Perron-Frobenius opertors nd representtions of the Cuntz-Krieger lgebrs simultneously, we introduce brnching function systems on mesure spce (X, µ). A fmily f = {f i } N i= of mps on X is semibrnching function system if there is finite fmily {D i } N i= of mesurble subsets of X such tht f i is mesurble mp from D i to R i f i (D i ), µ(x \ R R N ) = 0, µ(r i R j ) = 0 when i j nd there is the Rdon-Nikodým derivtive Φ fi of µ f i with respect to µ nd Φ fi > 0 lmost everywhere in D i for i =,..., N. A mp F on X is clled the coding mp of semibrnching function system f = {f i } N i= if F f i = id Di for i =,..., N. For semibrnching function system f = {f i } N i= with the coding mp F, define fmily {S(f i )} N i= of opertors on L 2(X, µ) by (.3) (S(f i )φ)(x) χ Ri (x) {Φ F (x)} /2 φ(f (x)) (φ L 2 (X, µ)) where χ Ri is the chrcteristic function of R i. Then S(f i ) is prtil isometry with the initil spce L 2 (D i, µ) nd the finl spce L 2 (R i, µ), nd S(f i )S(f j ) = S(f i f j ) when D j R i. For N 2, let A be n N N mtrix which consists of elements 0 or nd ny column nd row re not 0. A semibrnching function system f = {f i } N i= is n A-brnching function system if µ(d i \ j; ij = R j) = 0 for ech i =,..., N. For n A-brnching function system f = {f i } N i=, (.4) π f (s i ) S(f i ) (i =,..., N), defines representtion (L 2 (X, µ), π f ) of the Cuntz-Krieger lgebr O A. X Theorem.. For n A-brnching function system f = {f i } N i= coding mp F, the following holds: with the (P F ψ)(x) = {(π f (s ) ψ)(x)} {(π f (s N) ψ)(x)} 2 for ny positive function ψ L (X, µ) where ψ(x) ψ(x). Theorem.2. Assume tht F is the coding mp of n A-brnching function system f = {f i } N i= on mesure spce (X, µ) nd b i Φ fi is constnt for i =,..., N nd µ(x) <. Then the following holds: (i) Define subspce V Lin < {χ R,..., χ RN } > of L 2 (X, µ). Then P F V V where R i is the imge of f i. 2

3 (ii) For digonl mtrix B dig(b,..., b N ) M N (R), the following identity of mtrices holds: P F V = BA where P F V is the mtrix representtion over the bsis χ R,..., χ RN of V nd the rhs is the product of mtrices. In Theorem.2 (ii), the eigenvlues of the Perron-Frobenius opertor ssocited with F depend on not only A but lso B. In this sense, eigenvlues of the Perron-Frobenius opertor hve the informtion of representtion of the Cuntz-Krieger lgebr. It is importnt problem to construct the invrint mesure for given dynmicl system. For exmple, Lsot-York theorem shows construction of invrint mesure by using the Perron-Frobenius opertor of dynmicl system ([4]). We show the condition of existence of invrint mesure from the viewpoint of representtion theory of the Cuntz lgebr. f = {f i } N i= is brnching function system if f = {f i} N i= is n A- brnching function system for mtrix A = ( ij ), ij = for ech i, j =,..., N. In this cse, (L 2 (X, µ), π f ) is representtion of the Cuntz lgebr O N. For z = (z i ) N i= SN {y R N : y = }, (H, π) is GP (z) of O N if there is unit cyclic vector Ω H such tht (.5) π(z s + + z N s N )Ω = Ω. We cll Ω by the GP vector of (H, π). In this cse, there is g O(N) U(N) such tht (π α g )(s )Ω = Ω where α is the cnonicl ction of U(N) on O N. This implies tht (H, π α g ) is n irreducible permuttive representtion of O N ([]). Hence GP (z) of O N exists uniquely up to unitry equivlence nd it is irreducible. Further we see tht GP (z) GP (y) if nd only if z = y where mens the unitry equivlence. Theorem.3. Let F be the coding mp of brnching function system f on mesure spce (X, µ). If there is z S N such tht (L 2 (X, µ), π f ) is GP (z) with the GP vector Ω L 2 (X, µ; R), then there is probbilistic F -invrint mesure ν on X which is bsolutely continuous with respect to µ nd it is given s follows: dν(x) {Ω(x)} 2 dµ(x) (x X). In 2, we show the min theorems. It is explined tht (.5) implies the eigeneqution of the Perron-Frobenius opertor. In 3, we show concrete exmples. 2. Proofs of the min theorems For N 2, let M N ({0, }) be the set of ll N N mtrices such tht ech element is 0 or nd ny row nd column is not 0. For A = ( ij ) M N ({0, }), 3

4 O A is the Cuntz-Krieger lgebr by A if O A is C -lgebr which is universlly generted by genertors s,..., s N nd they stisfy s i s i = N j= ijs j s j for i =,..., N nd N i= s is i = I ([3]). Especilly, when ij = for ech i, j =,..., N, O A is the Cuntz lgebr O N ([2]). In this pper, ny representtion is unitl nd -preserving. Proof of Theorem.. L 2 (X, µ) is s follows: By (.3), the djoint opertor S(f i ) of S(f i ) on (2.) (S(f i ) φ)(x) = χ Di (x) {Φ fi (x)} /2 φ(f i (x)) (φ L 2 (X, µ)). For the coding mp F of semibrnching function system f = {f i } N i=, we hve N (2.2) (P F ψ)(x) = χ Di (x) Φ fi (x) ψ(f i (x)) (ψ L (X, µ)). i= By (2.) nd (2.2), we hve (2.3) (P F ψ)(x) = {(S(f ) ψ)(x)} {(S(f N ) ψ)(x)} 2 for ny positive function ψ L (X, µ). By (.4) nd (2.3), the sttement holds. Proof of Theorem.2. Define v i χ Ri for i =,..., N. (i) We see tht χ Di = N k= ikv k. By (2.), (S(f i ) φ)(x) = b i χ Di (x)φ(f i (x)). From this, S(f i ) v j = N k= b /2 i c (j) ik v k for i =,..., N where c (j) ik = δ ij b i ik. Hence S(f i ) V V. By (2.2), the following holds: (2.4) P F = b /2 S(f ) + + b /2 N S(f N). Therefore the sttement is proved. (ii) By the proof of (i), we see tht S(f i ) V = (b /2 i c (i) jk ) s mtrix with respect to v,..., v N. From (2.4), the sttement holds. Corollry 2.. Let X be bounded closed intervl of R nd A M N ({0, }). Assume tht f = {f i } N i= is n A-brnching function system on X nd b i Φ fi is constnt for ech i =,..., N. Then the eigenvlue of BA becomes tht of the Perron-Frobenius opertor of the coding mp F of f where B dig(b,..., b N ). Proof of Theorem.3. Assume tht Ω L 2 (X, µ; R) stisfies π f (z s + + z N s N )Ω = Ω. Define ρ(x) (Ω(x)) 2 for x X. Then ρ L (X, µ). By Theorem. nd π f (s i ) Ω = π f (s i ) π f (z s + + z N s N )Ω = z i Ω, we hve (P F ρ)(x) = N i= {(π(s i) Ω)(x)} 2 = N i= {z iω(x)} 2 = ρ(x). Hence 4

5 P F ρ = ρ. This implies the sttement. Corollry 2.2. Let X be mesurble subset of R. Assume tht piecewise C -clss mp F on X is the coding mp of brnching function system {f i } N i= on the mesure spce (X, dx) where dx is the Lebesgue mesure. If φ 0 L 2 (X, dx; R) stisfies (2.5) F (x) φ 0 (F (x)) = Nφ 0 (x) (.e. x X), then dµ(x) {φ 0 (x)} 2 dx is n invrint mesure on X with respect to F. Proof. By (.3) nd (.4), we see tht (π f (s + + s N )φ)(x) = F (x) φ(f (x)) for ech φ L 2 (X, dx). From this nd (2.5), π f (N /2 s + + N /2 s N )φ 0 = φ 0. By Theorem.3 for z = (N /2,..., N /2 ) S N, the sttement holds. In 6.5 of [5], it is explined tht intertwiners mong dynmicl systems bring new invrint mesures from known ones. We show its unitry version s follows: Proposition 2.3. Let F be the coding mp of brnching function system f = {f i } N i= on mesure spce (X, µ). Assume tht (L 2(X, µ), π f ) is GP (z) for z S N with the GP vector Ω L 2 (X, µ; R). If ζ is mesure spce isomorphism from (X, µ) to other (Y, ν) nd G ζ F ζ, then ρ (S(ζ)Ω) 2 is the density of probbilistic G-invrint mesure on Y which is bsolutely continuous with respect to ν where S(ζ) is unitry opertor from L 2 (X, µ) to L 2 (Y, ν) defined by (S(ζ)φ)(y) {Φ ζ (y)} /2 φ(ζ (y)). Proof. Define brnching function system g = {g i } N i= by g i ζ f i ζ. Then G is the coding mp of g. We see tht S(ζ)π f ( )S(ζ) = π g ( ), π g (z s + + z N s N )Ω = Ω for Ω S(ζ)Ω L 2 (Y, ν; R). Hence (L 2 (Y, ν), π g ) is GP (z) with the GP vector Ω. By Theorem.3, we hve the sttement. 3. Exmples Exmple 3.. Let 0 < < nd X [0, ]. (i) Define mp F on X by F (x) x/ on R [0, ] nd F (x) (x )/( ) on R 2 [, ]. 5

6 f 2 F f 0 0 Then F is the coding mp of brnching function system f {f, f 2 } defined by f i (F Ri ) for i =, 2. Then (π f (s )φ)(x) = /2 χ R (x)φ(x/), (π f (s 2 )φ)(x) = ( ) /2 χ R2 (x)φ( (x )/( )) for φ L 2 (X, dx). (P F ψ)(x) = ψ(x) + ( )ψ( ( )x + ) (ψ L (X, dx)). The Lebesgue mesure dx is the probbilistic invrint mesure of X with respect to F. (ii) Define f (x) x nd f 2 (x) ( )x 2 + on X. The coding mp F of f = {f, f 2 } is given by F (x) = x/ on [0, ], F (x) = (x 2 )/( ) on [, ]. Then function Ω(x) 2x on [0, ] stisfies π f ( s + s 2 )Ω = Ω. Hence the probbilistic F -invrint mesure on X is 2xdx. (L 2 (X, dx), π f ) in both (i) nd (ii) is GP (, ) of O 2. Both invrint mesures re independent in the prmeter. Exmple 3.2. For, b R, 0, define F (x) (x b) 2 / + b 2 on X [ 2 +b, 2 +b]. Define brnching function system f = {f, f 2 } on X by f i (F Ri ), i =, 2 for R [ 2 + b, b] nd R 2 [b, 2 + b]. Then π f (s + s 2 )Ω = Ω for Ω(x) π /2 {4 2 (x b) 2 } /4. Hence ρ(x) π 4 2 (x b) 2 is the density of probbilistic invrint mesure on X with respect to F. When = /4 nd b = /2, we hve F (x) = 4x( x), ρ(x) = π x( x). This ws first obtined by Ulmn nd von Neumnn ([8]). 6

7 F /2 f 2 f 0 /2 0 Exmple 3.3. For 0 < <, define mp F : [0, ] [0, ] s follows: 0 Define R [0, ], R 2 [, ], D( [0,) ], D 2 [0, ]. Then F is the coding mp of the following A = -brnching function system on 0 X = [0, ]: f i : D i R i, f (x) = x for x [0, ] nd f 2 (x) = ( )x/+ for x [0, ]. Define v χ [0,], v 2 χ [,], V Lin < {v, v 2 } >. Then the mtrix representtion of P F with respect to v, v 2 is given s follows: P F V = ( ( )/ 0 Hence its eigenvlue re nd. Their normlized eigenvectors re given s follows: w = χ [0,] χ ( ) [,], w 2 = χ[0,] + χ ( + 2 [,]. ) Especilly w 2 is the density of the invrint mesure on [0, ] with respect to F. Exmple 3.4. We show pplictions of Proposition 2.3. The following (X, F, µ) is trnsformtion F on X R nd probbilistic invrint mesure µ on X: (i) For b R \ [, 0], X [0, ], F (x) ). 2b 2 /(b x) 2b (x D ), 2b 2 { + 3b ( + b) 2 ( + 3b)x + b(b ) 7 } (x D 2 ),

8 where D [0, b/(2b + )) nd D 2 [b/(2b + ), ]. Then dµ(x) = (ii) For 0 k <, X [, ], F (x) b(b + ) dx (x [0, ]). (x + b) 2 2x 2 dx k 2 ( x 2. Then dµ(x) = ) 2 2K ( x 2 )( k 2 ( x 2 )) where K is the positive constnt defined by K (iii) For n integer N 2, X [0, ] nd 0 ( x 2 )( k 2 x 2 ) dx. F (x) ( N x [ N x ] ) 2, we hve dµ(x) = where [ ] is the gretest integer less thn equl x. (iv) For rel number >, X [, ] nd dx 2 x F (x) /x 2 (x [, )), F (x) x 2 / (x [, ]), we hve dµ(x) = dx/x. (v) For X R \ (, ) nd F (x) 2/(2 x ), dµ(x) = dx 2x 2. Exmple 3.5. Let F be mp defined by the following grph: 2/3 /3 0 2/3 /3 Define A 0, B dig (/3, /2, ), R [0, /3], R [/3, 2/3], R 3 [2/3, ], D [0, ], D 2 [/3, ], D 3 [0, /3]. The A-brnching function system f = {f, f 2, f 3 }, f i : D i R i, i =, 2, 3, with the coding mp F is given by f (x) = x/3, f 2 (x) = (x )/2 + /3, f 3 (x) = x +. From this nd Theorem.2 (ii), P F W = BA where W Lin< {χ R, χ R2, χ R3 } >. We see tht 0, /6, re eigenvlues of P F W. Hence they re eigenvlues of P F. Exmple 3.6. For A M N ({0, }) nd n A-brnching function system f = {f i } N i= on mesure spce (X, µ), µ(x) <, ssume tht b i Φ fi is constnt for ech i =,..., N. Then b i = r i /( N j= ijr j ) 8

9 where r i µ(r i ). When A dig ( r r r 2 +r 3, 2 r +r 3, 0 0, we hve B = (b, b 2, b 3 ) = r 3 r +r 2 +r 3 ). For 0 < < b <, consider the cse F on X = [0, ] which grph is given s follows: F b 0 b b 0 b F is the coding mp of n A-brnching function system given s follows: f i : D i R i, i =, 2, 3, f (x) = f 2 (x) = (x ) (x D ), b b+ x + b, (x R ), b b+ (x ) + (x R 3), f 3 (x) = ( b)x + b (x [0, ]), where R [0, ], R 2 [, b], R 3 [b, ], D [, ], D 2 [0, ] (b, ], D 3 [0, ]. From these, we hve B = dig (/( ), (b )/( b + ), b). } } } f 3 f 2 f dx 4 x 2 Exmple 3.7. Let F (x) x 3 3x on X [ 2, 2]. Then π is probbilistic invrint mesure with respect to F. For brnching function system f = {f, f 2, f 3 } defined by f (F [ 2, ] ), f 2 (F [,] ), f 3 (F [,2] ), (L 2 [ 2, 2], π f ) is GP (3 /2, 3 /2, 3 /2 ) of O 3. Exmple 3.8. Let X be the closed bounded region in R 2 which is clipped by 4-curves L {( x 2, x) : x [, ]}, L 2 {(x, ) : x [, ]}, L 3 {(2 x 2, x) : x [, ]}, L 4 {(x, ) : x [, ]}. Define mp F on X by F (x, y) ( 2x 2 + 4xy 2 2y 2 4x +, 2y 2 ) ((x, y) X). Then F (X) = X. There re the following four subregions R,..., R 4 of X: 9

10 0 X 2 R R 2 R 3 R 4 where the new center curve in the right figure is {( x 2, x) : x [, ]}. Then X = R R 4 nd F Ri is bijection from R i to X for ech i =,..., 4. Then dµ(x, y) = dxdy π 2 ( y 2 )(x + y 2 )(2 x y 2 ) is probbilistic invrint mesure on X with respect to F. For brnching function system f = {f i } 4 i= defined by f i (F Ri ), (L 2 (X, dxdy), π f ) is GP (/4, /4, /4, /4) of O 4. Acknowledgement: I would like to thnk Mkoto Mori for his tlk in [7]. References [] O.Brtteli nd P.E.T.Jorgensen, Iterted function Systems nd Permuttion Representtions of the Cuntz lgebr, Memories Amer. Mth. Soc. No.663 (999). [2] J.Cuntz, Simple C -lgebrs generted by isometries, Comm. Mth. Phys. 57 (977), [3] J.Cuntz nd W.Krieger, A clss of C -lgebr nd topologicl Mrkov chins, Invent.Mth., 56 (980), [4] A.Lsot nd J.A.Yorke, Exct dynmicl systems nd the Frobenius-Perron opertor, Trns.Amer.Mth.Soc. 273, (982), [5] A.Lsot nd M.C.Mckey, Chos, Frctls nd Noise, Stochstic Aspects of Dynmics, Second Edition, Springer-Verlg (99). [6] M.Mori, On the convergence of the spectrum of Perron-Frobenius opertors, Tokyo J. Mth. 7 (994), -9. [7] M.Mori, Frctl nd Perron-Frobenius opertor (Jpnese), Representtions of Cuntz lgebrs nd their pplictions in mthemticl physics, Symposium in RIMS Kōkyuroku No.333 (2003). [8] S.M.Ulm nd J.von Neumnn, On combintion of stochstic nd deterministic processes, Bull. Am. Mth. Soc., 53:20, (947). 0