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Fractal dimension of a random invariant set. (English) Zbl 1134.37364
Summary: In recent years many deterministic parabolic equations have been shown to possess global attractors which, despite being subsets of an infinite-dimensional phase space, are finite-dimensional objects. Debussche showed how to generalize the deterministic theory to show that the random attractors of the corresponding stochastic equations have finite Hausdorff dimension. However, to deduce a parametrization of a ‘finite-dimensional’ set by a finite number of coordinates a bound on the fractal (upper box-counting) dimension is required. There are non-trivial problems in extending Debussche’s techniques to this case, which can be overcome by careful use of the Poincaré recurrence theorem. We prove that under the same conditions as in A. Debussche’s paper [J. Math. Pures Appl. (9) 77, No. 10, 967–988 (1998;Zbl 0919.58044)] and an additional concavity assumption, the fractal dimension enjoys the same bound as the Hausdorff dimension. We apply our theorem to the 2D Navier-Stokes equations with additive noise, and give two results that allow different long-time states to be distinguished by a finite number of observations.

MSC:
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
28A80 Fractals
35B41 Attractors
37C45 Dimension theory of smooth dynamical systems
37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents
37L55 Infinite-dimensional random dynamical systems; stochastic equations
76D05 Navier-Stokes equations for incompressible viscous fluids
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