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)
Previous Page Next Page