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)