For 1 > 1 ,2 , le t (X^.S^.m^) be a Lebesgue p ro b a b ility space, Tj be a n on -sin gu lar automorphism o f (X^,B|,m{ ) and l e t Aj c 8^ be a sub-o-algebra s a tis fy in g T^A ^ ^ A^ . Suppose also that 4 : Xj X2 Is a measure-class preserving Isomorphism s a tis fy in g *T1 “ T2* * and the cocycle-coboundary equation,
Jm /Al ‘ Tl l " + h#Tl ‘ h • (5.15)
holds, where a ll the functions appearing are 1n L*>( X j,8 j,m j) . Then the follow ing c a lc u la tio n 1s v a lid .
11m sup 1(11^ -ess.sup J ( A j.T ^ ) )
n-H, n
■ 11m sup ^ (m ^-ess.sup(J_ (Ag,T^ )« 4 + h*T!| - h ) )
n-H, ” 2
; using equation (5.15)
■ 11m sup ^ (rfc|-ess.sup Jm
n-H» c
> 11m sup 1 (ly e s s .s u p J (A2,tJ ) )
fH*
Cm; since 4 preserves measure-class.
- 125 -
11m 1nf 1 (ifcj-ess. 1nf (Aj,t!J))
■ 11m 1nf 1 (mg-ess. 1nf Jffl (A g .T ^ )) .
n-*» 2
I f we define
h+(Jm ( Ai*Ti ) ) : ■ 11m SUP ^ - e s s . sup J (Aj.tJ ) )
1 n*H»
and
h’ ( Jm ( A1 * V ) : " l1ra l ^ ^ - e s s . 1nf Jm (A1 ,T ^ )) ,
1 n-*» 1
then the above c a lcu la tio n shows that h+(J m (Aj.Tj) ) and
h"(Jm (Aj.Tj)) are In va ria n ts o f equation (5.15).
The follo w in g theorem enables one to calcu late the In va ria n t h+( Jm(A*T)) fo r Markov s h ifts over f i n i t e l y many state s. A s im ila r re s u lt holds fo r h " (J m( A , T ) ) .
5.25. THEOREM.
Let (X,8,m ,T) be a measure preserving Markov s h if t over f i n i t e l y many sta te s , defined by a stochastic m atrix P ■ P (1 ,J) ,
L<< H a* *4 \ \ \ 1 i* iH * W t y \ f ( h'v { / < v v f j l 1! tfk’ l W t i I 1 i < » * * " M *, 1* t ' > • \ 1 h ' A* i V . * ' * * * *' * \ / l . ' [ \ 1, i
126 -
w ith past 0-algebra A , then
h+(J m(A ,T ))
« max{- 1 log (P (x 0,x 1) .P ( x 1, x 2) . - - - . P ( x n. 1,x 0) } ;
n - 1 , 2 ,. . . , the symbols Xq.Xj, . . . , xn_^
are pairwise d is t in c t , and m([XgX1 • ••xn_1x(j ] 0) >• 0 } .
Proof.
F ir s t note that since T 1s a measure preserving Markov s h if t , we can conclude from equation (5.3) that f o r x ■ (x n) t X ,
< y A ,T ) ( x ) ■ * log P iX g .X j) ♦ lo g p (X j) - log p(xQ) ,
where p 1s the (unique) s t r i c t l y p o s itiv e p ro b a b ility vector s a tis fy in g p.P ■ p (see Def1n1t1ons2.1). When c a lc u la tin g h+(J m(A*T ) ) we therefore o nly need to consider the "adjusted" Inform ation cocycle defined by
Jm(A *T > (x ): " " lo g p( xo*xl ) •
To enable us to work almost e n t ire ly w ith strin gs o f symbols (1n preference to c y lin d e r s e ts ), we shall use the 1-1 correspondence defined by associating each c y lin d e r se t [ X g X j...x n]^ with the s t r in g
- 127 -
Let A - { 0 , 1 , . . . , k-1 > be the alphabet from which the s h if t space X was constructed. For the purposes o f th is proof we shall make the fo llo w in g temporary d e fin itio n s .
(1) A s trin g X g X ^ ...x n o f symbols from the alphabet A 1s calle d admiaaible I f m( [XqX^. . . xn]®) > 0 .
(11) An admissible s trin g o f the form XgX1. . . x nx0 1s called a cycle .
(111) A cycle XqX ^ . - .x^ q 1s c a lle d prime 1f Xq,x, , . . . ,x
are pairwise d is tin c t.
(1v) For an admissible s trin g s - XgX1. . . x n , we define
h (s ): - - J l o g W x Q . x ^ . P t X j . x ^ ... p( xn_ r xn) ) •
t ( s ) : ■ n ,
and we c a ll t ( s ) the len g th o f s . I f the s trin g x rXq
1s admissible then we can construct a cycle s ' from s , by pu tting
s ' : ■ xQX i .•*xnx0 .
For such a p a ir o f s trin g s s and s ' , we define
h '( s ) : - h ( s ') ,
- 128 -
Since there are only f i n i t e l y many prime c y c le s , we can also define
h : ■ max{h(s) ; s Is a prime c y c le ) .
Let s ' be a prime cycle such that h ( s ') ■ h ^ , and define s*
to be the unique s trin g such th at (s * )' ■ s ' , then h '(s* ) - h i s ') « h ^ ^ .
The follo w ing two facts w i l l be useful:
(1) Any admissible s t rin g consisting o f a t le a st k+1 symbols contains a repeated symbol, and hence contains a t le ast one prime c ycle .
(2 ) There 1s a sm allest s t r i c t l y p o s itive e ntry (denoted by Pmin ) In the tra n s itio n m atrix P .
« Le t us define
H: ■ 11m sup ^ (m-ess. sup J_(A,11’ ) ) , fH*
then H e xis ts and 1s f i n i t e , because T 1s a Markov s h if t over o n ly f i n i t e l y many symbols.
For a fixe d but a r b itra r y c ► 0 , choose a corresponding In te ge r K ► k such that
- 129 -
and fo r the same e , choose (and f i x ) an admissible s trin g s ■ XqXj. . .x u o f length t ( s ) ■ u > K , fo r which
|h(s) - H| < J . (5.17)
On the c y lin d e r [XqXj. . .x u3° associated to the s trin g s , the adjusted Inform ation cocycle s a tis fie s
1 - M i ) ,
and the Idea o f the fo llo w in g proof 1s to show that by using the s trin g s* , 1t Is possible to construct a c y lin d e r o f the same length on which
1 j ; ( A , T u) > h (s ) - f .
The a r b itra r y nature o f the choice o f c , and the subsequent choice o f the s trin g s , w i l l then Imply the re s u lt.
Since i ( s ) > K > k , fa c t 1 Im plies that the s trin g s contains a prime c yc le . Let s{ be the prime c ycle that occurs clo se st to the l e f t 1n the s trin g s . I f s| ■ x i xt + i , , , x j+ jx i+ j+ i * where
X1 ’ X1+J+1 • and
- 130 -
then we sh all denote by the s trin g x^x^+-| * * *xi+ j and s " S1 the admissible s trin g
xox i • • * X1 -1 X1 + j+1 X1 + j+2 * * ’ xu *
By removing 1n th is manner the le ft-m o s t prime cycle a t each stage, we can continue th is process and obtain a f in it e set o f s trin g s
S j,s 2, . . . , s t (each constructed from a prime cycle by removing the right-m o st symbol) and a s trin g s ^ : « s - s1 - s2 st , that contains no prime c ycle s . (Note that the above operations on strin g s are to be carrie d out In the order determined by reading the expression from l e f t to r ig h t . ) From fa c t 1 the length o f s ^ must be less than k+1 , and as a consequence o f the above d e fin itio n s we also have
h ( s ) .t ( s ) • h (« w s ) . » ( s w ) + h'( s 1M '( S j ) + . . . + h '( s t ) . t ' ( s t )
(5.18) Using the d e fin itio n o f the c ycle s* , we have
h '( s * ) ( i '( s 1)'*>.. . + t ' ( s t ) ) a h, (s 1) . i ' ( s 1) + ...+ h '( s t ) . t ' ( s t ) , (5.19)
and 1 f r 1s the la rg e s t In te ge r fo r which
r . t '( s * ) i » '( * , ) i ' ( * t ) .
then
- 131 -
Combining th is la s t In e q u a lity w ith In e q u a lity (5.19) gives
h '( s * ) . r . i'( s * ) a h ' ( s 1) . £ ' ( s , ) ♦ . .. + h '( s t ) . f ( s t ) - h '( s * ) . t '( s * ) . (5.20)
For a ll j ( H , we sh all denote by w j(s*) . the s trin g con sisting o f the f i r s t J symbols o f the I n f in it e s trin g s* which 1s obtained by concatenating s* w ith I t s e l f I n f i n i t e l y o fte n.
Noting that t ( s ) ■ u , we now compare the values o f h(s) and h(iru(s* )) as fo llo w s : h (s ).u ■ l U s ^ ) . t ( s w $ ) + h '( s 1) . t ' ( s 1) + ...+ h '( s t ) . t ' ( s t ) ; from equation (5.18) * ( k + l) ( -lo g Pmin) + h '( s 1) . t ' ( s 1) + ...+ h '( s t ) . t ' ( s t ) , (5.21) and K * u(* 2 ))-“ * h '( * * ) . r . t '( t * ) a h*(s1) . £ '( s 1) h '( s t ) . t ' ( s t ) - h '( s * ) .t '( s * ) i by In e q u a lity (5.20) a h (s ).u - ( k + l) . ( - lo g PB in ) - l'(s * )(-1 o g Pmin) (5.22)
; using In e q u a lity (5 .2 1 ). and n o tic in g that h '(* * )* (-1 o g Pm1n) •
- 132 -
In e q u a lity (5.22) im plies that
H (.u( » : ) ) * h (s ) - l . ( - l o g ?n in ) { W + t '( t * ) )
* H - f - f ; from In e q u a litie s (5.16) and (5.17)
>• H - e .
Since the choice o f c > 0 was a r b itra r y ,
h '(s * ) - 11m h(w (s*))