# Algebraic Structure Theory of Sequential Machines [appl by J. Hartmanis, R. Stearns, By J. Hartmanis, R. Stearns,

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This c o n t r a d i c t s our assumptions. Thus, k = n - l . Since we took the shortest loop from I l to i t s e l f , I i to l j f o r j > i + l in the A-graph of f. there are no arrows from From t h i s i t f o l l o w s (use i n d u c t i o n ) t h a t the ordering of the elements of A on the real l i n e must be I n _ l , l n _ 3 . . . 12, l l , I 3 . . . In_4,1n_ 2 up to o r i e n t a t i o n . The common endpoint of In_ 3 and In_ l is 24 mapped o n t o t h e common e n d p o i n t o f In_ 2 and J and f ( I n _ I ) ~ I I .

5 ~(U[J(A(6)) :6>0]) Corollary: show that L A c [fk(w2) : k < 0 } U = ~(L) • , and the proof is complete. has full measure in L and Pesin's results apply to is the union of possibly countably many ergodic components. We show in the next section that with respect to §5 U W3 f and hence f L has only one ergodic component is ergodic. 1 Theorem: P,q E L - B There exists a set then B c- L with ~(B) -- 0 such that if wU(p) n wS(q) ~ ¢ • Before discussing this theorem we will use it to finish the proof of ergodicity of linked twist maps.

1 o n l y i n one some o t h e r e l e m e n t K o f A. even p e r i o d o f p e r i o d i c points of f the A-graph o f f c o n t a i n s a subgraph i s odd and t h e r e f o r e t h e r e are arrows from In_ 1 t o even v e r t i c e s ) , t h e r e i s no e l e m e n t K o f A such t h a t f(K)~ I I. 4) a p e r i o d i c p o i n t o f p e r i o d 2 (the o n l y p o s s i b l e common p o i n t o f In_ 1 and In_ 2 has p e r i o d n). In the second case, [min Orb x, min I I ] f - c o v e r s [max I I , max Orb x] and vice versa.