Abstract
Various properties of Local and Global Lyapunov exponents are related by redefining them as the spectral radii of some positive operators on a space of continuous functions and utilizing the theory developed by Choquet and Foias. These results are then applied to the problem of estimating the Hausdorff dimension of the global attractor and the existence of a critical trajectory, along which the Lyapunov dimension is majorized, is established. Using this new estimate, the existing dimension estimate for the global attractor of the Lorenz system is improved. Along the way a simple relation between topological entropy and the fractal dimension is obtained.
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References
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.
Constantin, P., Foias, C., and Temam, R. (1985).Attractors Representing Turbulent Flows, AMS Memoirs, Vol. 53, No. 314.
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.
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.
Douady, A., and Osterle, J. (1980). Dimension de Hausdorff des attracteurs.C.R. Acad. Sci. Paris 290 (Ser. A), 1135–1138.
Farmer, J. D. (1982). Chaotic attractors of an infinite dimensional dynamical systems.Physica 4D, 366–393.
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.
Kolmogorov, A. N., and Tihomirov, V. M. (1959).ɛ-entropy andɛ-capacity of sets in functional spaces.Uspehi Mat. Nauk 14, 3–86.
Ledrappier, F. (1981). Some relations between dimension and Lyapunov exponents.Comm. Math. Phys. 81, 223–238.
Rogers, C. A. (1970).Hausdorff Measure, Cambridge University Press, Cambridge.
Ruelle, D. (1979). Ergodic theory of differential dynamical systems.Publ. Mathe. IHES 50, 275–306.
Temam, R. (1988).Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences Vol. 68, Springer-Verlag, New York.
Walters, P. (1982).An Introduction to Ergodic Theory, Springer-Verlag, New York.
Yomdin, G. (1986). Volume growth and entropy. Preprint.
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Eden, A., Foias, C. & Temam, R. Local and Global Lyapunov exponents. J Dyn Diff Equat 3, 133–177 (1991). https://doi.org/10.1007/BF01049491
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DOI: https://doi.org/10.1007/BF01049491