zbMATH — the first resource for mathematics

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.

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
Full Text: DOI
[1] Constantin, P., and Foias, C. (1985). Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for 2D Navier-Stokes equations.Comm. Pure Appl. Math. 38, 1-27. · Zbl 0582.35092
[2] Constantin, P., Foias, C., and Temam, R. (1985).Attractors Representing Turbulent Flows, AMS Memoirs, Vol. 53, No. 314. · Zbl 0567.35070
[3] Constantin, P., Foias, C., Nicolaenko, B., and Temam, R. (1988).Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Applied Mathematical Sciences Vol. 70, Springer-Verlag, New York. · Zbl 0683.58002
[4] Choquet, G., and Foias, C. (1975). Solution d’un problème sur les itérés d’un opérateur positif sur ?(k) et propriétés de moyennes associées.Ann. Inst. Fourier Grenoble 25, 109-129.
[5] Douady, A., and Osterle, J. (1980). Dimension de Hausdorff des attracteurs.C.R. Acad. Sci. Paris 290 (Ser. A), 1135-1138. · Zbl 0443.58016
[6] Farmer, J. D. (1982). Chaotic attractors of an infinite dimensional dynamical systems.Physica 4D, 366-393. · Zbl 1194.37052
[7] Kaplan, J., and Yorke, J. (1979). Chaotic behaviour of multidimensional difference equations.Functional Difference Equations and Approximation of Fixed Points, Lecture Notes in Mathematics 730, Springer-Verlag, Berlin.
[8] Kolmogorov, A. N., and Tihomirov, V. M. (1959).?-entropy and?-capacity of sets in functional spaces.Uspehi Mat. Nauk 14, 3-86. · Zbl 0090.33503
[9] Ledrappier, F. (1981). Some relations between dimension and Lyapunov exponents.Comm. Math. Phys. 81, 223-238. · Zbl 0486.58021
[10] Rogers, C. A. (1970).Hausdorff Measure, Cambridge University Press, Cambridge. · Zbl 0204.37601
[11] Ruelle, D. (1979). Ergodic theory of differential dynamical systems.Publ. Mathe. IHES 50, 275-306. · Zbl 0426.58014
[12] Temam, R. (1988).Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences Vol. 68, Springer-Verlag, New York. · Zbl 0662.35001
[13] Walters, P. (1982).An Introduction to Ergodic Theory, Springer-Verlag, New York. · Zbl 0475.28009
[14] Yomdin, G. (1986). Volume growth and entropy. Preprint. · Zbl 0636.26004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.