On the fractal dimension of invariant sets; applications to Navier Stokes equations

Abstract. A semigroup St of continuous operators in a Hilbert space H is considered. It is shown that the fractal dimension of a compact strictly invariant set X (X b H, StX = X) admits the same estimate as the Hausdorff dimension, namely, both are bounded from above by the Lyapunov dimension calculated in terms of the global Lyapunov exponents. Applications of the results so obtained to the two–dimensional Navier–Stokes equations are given. 1. Introduction. Let X be a compact set in a Hilbert space H: X b H. We recall the following definitions (see [1]). Definition 1.1. The Hausdorff dimension of X in H is the number dimH X = inf{d|µH(X, d) = 0}, where µH(X, d) = limε→0+ µH(X, d, ε), µH(X, d, ε) = infX⊂U Vd(U). Here th