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Local Lyapunov exponents and a local estimate of Hausdorff dimension. (English) Zbl 0684.58022
Summary: The Lyapunov dimension has already been used to give estimates of the Hausdorff dimension of an attractor associated with a dissipative ODE or PDE. Here we give a slightly different version, utilizing local Lyapunov exponents, in particular we show the existence of a critical path along which the Hausdorff dimension is majorized by the associated Lyapunov dimension. This result is then applied to Lorenz equations to deduce a better estimate of the dimension of the universal attractor. We conclude with an example that shows some of the drawbacks of this estimate.

MSC:
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
28A75 Length, area, volume, other geometric measure theory
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