Lyapunov exponents of the predator prey model Ljapunovovy ...

15and hence E α,βg dµ ≥ αµ (E α,β ) .Note that (−g) ∗ = −g ∗ , (−g) ∗= −g ∗ andE α,β = {x : (−g) ∗ (x) > −β and − α > (−g) ∗(x)} .Ifwe replace g, α, β by −g, −β, −α, respectively in the previous inequality, then we haveE α,β(−g) dµ ≥ −βµ (E α,β ), and hence E α,βg dµ ≤ βµ (E α,β ).Thus we obtain αµ (E α,β ) ≤ βµ (E α,β ), which implies that if β < α then µ (E α,β ) = 0.Therefore g ∗ = g ∗ and 1 n−1n i=1 g f i (x) converges to g ∗ almost everywhere.Now we show that g ∗ is integrable. Letn−11 h n (x) =g f i (x) .ni=0Then lim h n (x) = |g ∗ | andh n dµ ≤ 1 n−1 g f i (x) dµ =ni=0|g| dµ.Fatou’s lemma implies that |g ∗ | dµ = lim inf h n dµ = lim infh n dµ ≤|g| dµ < ∞.It remains to show that g dµ = g ∗ dµ. For n ≥ 1 and k ∈ Z putD n,k = x ∈ X : k n ≤ g∗ (x) < k + 1 .nNote that for each n the collection of D n,k , k ∈ Z, forms a partition of X. For sufficientlysmall ε > 0 we haveD n,k ⊂ B kn −ε.Hence by Corollary 2.21 we haveD n,kg dµ ≥for sufficiently small ε > 0, and henceBy the definition of D n,k kn − ε µ (D n,k )D n,kg dµ ≥ k n µ (D n,k) .D n,kg ∗ dµ ≤ k + 1n µ (D n,k) ≤ 1 n µ (D n,k) +D n,kg dµ.