Notation
Throughout this text we will use the following notation:
A B = (A \ B) (B \ A) , the symmetric difference of sets
f
−1(E)
= {x | f(x) E} for a map f : X Y and E Y
Ec
the complement of a subset E X,
Ec
= X \ E
IE the characteristic function of a set E, IE(x) := 1
if x E and IE(x) := 0 elsewhere
f
+(x)
= max(f(x), 0), the positive part of f : X R
log+
x
log+
x := log x for x 1 and
log+
x := 0 for x 1
gl(d, R),gl(d, C) the set of real (complex) d × d matrices
Gl(d, R),Gl(d, C) the set of invertible real (complex) d × d matrices
A the transpose of a matrix A gl(d, R)
· a norm on
Rd
or an induced matrix norm
spec(A) the set of eigenvalues μ C of a matrix A
imA, trA the image and the trace of a linear map A, resp.
lim sup, lim inf limit superior, limit inferior
N, N0 the set of natural numbers excluding and including 0
ı ı =

−1
z the complex conjugate of z C
¯
A
¯
A = (aij) for A = (aij) gl(d, C)
Pd−1
the real projective space
Pd−1
=
RPd−1
Gk(d) the kth Grassmannian of
Rd
L(λ) the Lyapunov space associated with a Lyapunov exponent λ
E expectation (relative to a probability measure P )
For points x and nonvoid subsets E of a metric space X with metric d:
N(x, ε) = {y X | d(x, y) ε}, the ε-neighborhood of x
diam E = sup{d(x, y) | x, y E}, the diameter of E
dist(x, E) = inf {d(x, y) | y E} , the distance of x to E
cl E, int E the topological closure and interior of E, resp.
xv
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