22 1. Autonomous Linear Differential and Difference Equations be derived from the solution formulas in the generalized eigenspaces. For example, formula (1.5.3) yields for n m 1, (1.5.5) 1 n log ϕ(n, x0) = n + 1 m n log |μ| + 1 n log m−1 i=0 n i μm−1−iN i x0 . One estimates log m−1 i=0 n i μm−1−iN i x0 log m + max i log n i + max i log |μ|m−1−i N i x0 , where the maxima are taken over i = 0, 1,...,m−1. For every i and n one can further estimate 1 n log n i = 1 n log n(n 1) . . . (n i + 1) i! max i∈{0,1,...,m−1} i n (log n log i!) 0. Hence taking the limit for n in (1.5.5), one finds that the Lyapunov ex- ponent equals log |μ| for every initial value x0 in this generalized eigenspace. The same arguments work for complex conjugate pairs of eigenvalues and in the sum of the generalized eigenspaces corresponding to eigenvalues with equal moduli. Finally, for initial values in the sum of generalized eigenspaces for eigenvalues with different moduli, the largest modulus determines the Lyapunov exponent. Similarly, one argues for n −∞. The ‘only if’ part will follow from Theorem 1.5.8. Remark 1.5.7. As in continuous time, we note that a characterization of the Lyapunov spaces by the dynamic behavior of solutions needs the limits for n +∞ and for n −∞. For example, every initial value x0 = x1 +x2 with xi L(λi),λ1 λ2 and x2 = 0, has Lyapunov exponent λ(x0) = λ2. We can sharpen the results of Theorem 1.5.6 in order to characterize the exponential growth rates for positive and negative times and arbitrary initial points. This involves flags of subspaces. Theorem 1.5.8. Consider the linear difference equation xn+1 = Axn with A Gl(d, R) and corresponding Lyapunov spaces Lj := L(λj),λ1 . . . λ . Let V +1 = W0 := {0} and for j = 1,..., define Vj := L . . . Lj and Wj := Lj . . . L1.
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