By Bruce Kitchens
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Additional resources for Finitary Measures for Subshifts of Finite Type and Sofic Systems
Y J < ] whenever [ X p . . , x n , y 1 , . . , y k ] i s an S-word, and l e t t i n g the product equal 0 otherwise, we get a well-defined m u l t i p l i c a t i o n , turning G into a semi-group. I d e n t i f y i n g a e A w i t h [ a ] , the set A generates G. f i n i t e precisely because of ( * ) . We w i l l show that G i s However, l e t us assume the f i n i t e n e s s of G f o r the moment, and show t h a t the sofic pair (G,A) then meets a l l the r e q u i r e ments. C l e a r l y , (G,A) describes S.
X n ] . We must show that this is independent of how g is written as a product of generators. y k of generators to unambiguously write s(h)=s[y1»... ,y k ]. We have .. xa] ^ . . . lx, Choose (iQfJn) € s(g) such t h a t ( I Q ^ J Q 3 ) l € s(ga). xn = j ] C (i,j) = m [ 10 , 10 x l, . . , 1 0 x l. . xn, i ' x ^ . '^,. ••»i0xr*,xn-l,':i0>j0a] 50 BRUCE KITCHENS AND SELIM TUNCEL do not depend on how g i s expressed as a product g = x , . . xn = j ] C e (i,j) = (-j«fj») ro[ioafiQa*i = = * • • • »1Q^X, .
Is to permute (i , j , ) » . . 3) by replacing, in the last part, the integer c by an integer d so large that i f g = x ^ . . x and n > d then in the sequence s [ x 1 ] , s [ x 1 , x 2 ] , . . , s [ x ^ . . , x . 5) Remark. and j such that s = sCx,,.. , x k ] = *-*k+l s " * * **k+£ > j for a l l ( i , j ) e s. ,xn] FINITARY MEASURES 53 when s [ x 1 , . . , x | < , x k + 1 , . . > x k + £ ] = s [ x 1 § . . , x k ] = s and j k+1' *"" ,x k+£ > j for a l l ( i , j ) € s gives a semi-group G' such that the image of every Markov measure satisfying the weight condition on (ZA^>) is a semi-group measure for G'.