On the fractal dimension of invariant sets: Applications to Navier-Stokes equations. (Q1433909)

A semigroup \(S_{t}\) of continuous operators in a Hilbert space \(H\) is considered. The aim of the reviewed article is to estimate the fractal dimension of a compact strictly invariant set \(X \Subset H, S_{t}X=X\). It is proved that this fractal dimension admits the same estimation as the Hausdorff one. Namely, both are bounded from above by the Lyapounov dimension calculated in terms of the global Lyapounov exponents. Then, the main estimate proved in the abstract setting is applied to the two-dimensional Navier-Stokes system.