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 ¯ ¯ = (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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