Ergodic Theory of Chaotic Dynamical Systems

THEOREM ([41], [42]). Let f,Λ, m and R be as above. Then: (a) If ∫ Rdm <∞, then f admits an SRB measure μ. (b) If, additionally, gcd{Ri} = 1, then (f, μ) is mixing. (c) If m{R > n} < Cθ for some θ < 1, then ∃ θ̃ < 1 s.t. ∀φ, ψ, Cn(φ, ψ) < Cθ̃ . (d) If m{R > n} = O(n−α) for some α > 1, then Cn(φ, ψ) = O(n −α+1). (e) If R is as in (d) and α > 2, then the CLT holds for all φ. Next, we argue that conceptually m{R > n} is essentially the speed with which arbitrarily small pieces of unstable manifolds grow to a specified size. (This is not the same as Lyapunov exponents, which measure pointwise growth rates.) First we describe the picture. If f has good hyperbolic properties, then we can cover most of phase space with a finite number of sets Γ1, · · · ,Γk with product structures (they look like W u × W s trelises). If f is mixing, then in finite time, fΓi crosses over Γj in the unstable direction for every i, j. These structures give the dynamics the flavor of a finite Markov chain, but one should not carry the analogy too far, for ∪Γi is not all of phase space, nor is it an invariant set. The rest of phase space is made up of small bits of stable and unstable manifolds that twist and turn as described in Section 1.2. Returning to the problem of estimating m{R > n}, suppose that Γ1 is our reference set. Since f is ergodic, it is inevitable that some parts of Γ1 will get into the messy regions of phase space before they return. It is necessary, therefore, to know how long it takes structures of arbitrarily small scales to “straightout out” and grow to the scale of the Γi’s. This is also sufficient, for once a W -leaf reaches a size comparable to the Γi’s, it