By Bruce Kitchens
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Additional resources for Finitary Measures for Subshifts of Finite Type and Sofic Systems
Example text
I s the r i g h t cover f o r (G,A). y = mo<|)~ it To show that i s a semi-group measure f o r (G,A), define the r i g h t t r a n s i t i o n matrix R as f o l l o w s . x I f g e G and a e A are such t h a t ga f 0, w r i t e as a product of generators and put R(9»ga) = u [ x 1 , . . , x n , a ] / y [ x 1 , . . , 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] ^ .
K and y = [ y ^ . . , y ]°°. such that ( x { 1 * . . x j 1 * ) ° jLQ>a z 1 s (yi---yD) (y^--yp) • Now choose = ( x { l } . . x j ^ ) 0 for i = 1 , . . , K and Letting R and R* denote the right transition matrices of y and TT, there exist constants 0 , 0 , , . . yp] ) - C R ' ( ( y r . •y T)n ) . xl")V-^)]V for large enough n. yp) ° + (yr--yp) \ yJ*) must hold for i = 1 , . . , K and, taking (pit)-th roots, we have wu(x ) = w (y). 2) Proposition. Let (S,y) and (T,TT) be sofic systems with semi-group measures, and let
For Since the weight of a periodic poin t given by i -i / ^-> i ) ' , the converse, suppose that m s a t i s f i e s ( * ) . ,a For an S-word [ x , , . . , x ] , l e t 2 [ x i > « " » xIJn ] s [ x , , . . , x n ] = { ( i , j ) e L« : there exists a path i — -> j } . Define an equivalence r e l a t i o n on S-words by puttin g [ x ^ . . , x n ] % [ x ^ , . . ,x'] j)/M(i — ^> j ) is independent of ( i , j ) e s [ x l t . . , x n ] = s [ x ^ , . . , x ' ] . Denote the equivalence class of [ x , , .