1.1. General properties ( Since the Wronskian of c(z,.. no) and s(z,., no) does not depend on n we can eval- uate it at no (1.26) W(c(s,.,7io),s(^.!n0)) = a(no) and consequently equation (1.22) simplifies to (1.27) u(n) = u(n0)c(z, n, n0) + u(n0 + l)s(s, n, n0). Sometimes a lot of things get more transparent if (1.19) is regarded from the viewpoint of dynamical systems. If we introduce u = (u, u+) G -f(Z, C2), then (1.19) is equivalent to (1.28) u(n + 1) = U(z, n + l)u(n), u(n - 1) = U(z, n)_1w(n), where U(z,.) is given by a(n) \ -a( n - 1) c - 6(n) (1.29) t r ' ( S , » ) = X (Z~b{n) " a ( n ) o ( n - l ) V «("•-! ) 0 The matrix U(z, n) is often referred to as transfer matrix. The corresponding (non-autonomous) flow on £(Z, C2) is given by the fundamental matrix $/„ „„• _ f c(2,n,no) s(=,n,n0) l ^' ' o j _ Vc(«,n+l,n„ ) s ( 3 , n + l , n 0 ) C ^(^,n)---t/(2,n 0 + l) n n 0 (1.30) = \ 1 1 n = n0 [U-1(z,n+l)---U-1(z,n0) n n0 More explicitly, equation (1.27) is now equivalent to (1.31) ( "{n\,)=*(z,n,no)( ^ u Using (1.31) we learn that $(2,71, no) satisfies the usual group law (1.32) $(z,n,n 0 ) = $(z,n,7ii)$(5,nj,n 0 ) ? $( ?,n0,no) = 11 and constancy of the Wronskian (1.26) implies (1.33) det*(s,n,no) = ^ . a(n) Let us use $(z,n) = $(3, n,0) and define the upper, lower Lyapunov exponents 1 1 T ^ s ) = limsup r T ln||$(^,n,n 0 )| | = limsup T-T In ||$(z,rc)||, n~±x \n\ n—±x \n\ (1.34) 7 ± (*) = liminf ^ In ||$(*,n,n0)|| = liminf ^ In ||$(z,n)||. Here ii*«iica (1.35) ||$||= sup «6C2\{()} l|W||C2 denotes the operator norm of $. By virtue of (use (1.32)) (1.36) \\$(z,n0)\\-l\mz,n)\\ ||$(.,n,n„)|| ||$(^n 0 )- 1 ||||$( 2 ,n)
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