Exponential Decay of Volume Elements and the Dimension of the Global Attractor

Let Σ₀ be an m-dimensional smooth manifold in θY for some fixed θ ∈ [1, ∞), let u₀ ∈Σ₀, and let u = ϕ(α) be a local parametrization of Σ₀ in a neighborhood of u₀, where a α =(α₁,…,α _m ) runs over a neighborhood of 0 in ℝmand u₀ = ϕ(0). The infinitesimal volume element of S(t)Σ₀ at S(t)u₀ is |υ ₁ (t) ^ … ^ υ _m (t)| where υ _i (ts) evolve according to (2.3) and υ _i (0) = ∂ϕ(α)/∂α _i |_α=o. Using (2.7) and (2.9) we deduce the equation (see [CF1])(8.1)$$\frac{1}{2}\frac{d}{{dt}}|{\upsilon _1}\left( t \right){|^2} + \left( {Tr A\left( t \right)P\left( t \right)} \right)|{\upsilon _1}\left( t \right) \wedge \cdots \wedge {\upsilon _m}\left( t \right){|^2} = 0,$$ where P(t) is the projector on the tangent space to S(t)Σ₀ at S(t)u₀. Thus the volume element will decay exponentially if(8.2)$$\mathop {\lim }\limits_{t \to \infty } \operatorname{int} \frac{1}{t}\int_o^t {Tr} \left( {A\left( s \right)P\left( s \right)} \right)ds > 0.$$