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Local and global Lyapunov exponents. (English) Zbl 0718.34080
The authors relate various properties of local and global Lyapunov exponents, which were used in the study of the Hausdorff dimension of the global attractor for the 2D Navier-Stokes equations [cf. P. Constantin and C. Foias, Commun. Pure Appl. Math. 38, 1-27 (1985; Zbl 0582.35092), P. Constantin, C. Foias and R. Temam, Mem. Am. Math. Soc. 314 (1985; Zbl 0567.35070)]. This goal is achieved by posing their problem in the framework of flows of positive operators on a space of continuous functions over a compact set and utilizing the theory developed by G. Choquet and C. Foias [Ann. Inst. Fourier Grenoble 25 (1975), No.3-4, 109-129 (1976; Zbl 0303.47004)]. The key idea is to obtain a flow of positive operators from the nonlinear semigroup of solution operators \(S_ t\), acting on a compact invariant subset X of a Hilbert space H. The main content is as follows:
Section 2. L-exponents associated with positive operators.
Section 3. Semiflows on infinite-dimensional vector spaces and associated L-exponents.
Section 4. Estimates on the dimension of attractors.
Section 5. An estimate on the dimension of the Lorenz global attractor.
Section 6. Evolution equations satisfying a dissipativity condition.

MSC:
34D45 Attractors of solutions to ordinary differential equations
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
35Q30 Navier-Stokes equations
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
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