Dimension, entropy and Lyapunov exponents
TLDR
In this article, the authors consider diffeomorphisms of surfaces leaving invariant an ergodic Borel probability measure and define HD (μ) to be the infimum of Hausdorff dimension of sets having full μ-measure.Abstract:
We consider diffeomorphisms of surfaces leaving invariant an ergodic Borel probability measure μ. Define HD (μ) to be the infimum of Hausdorff dimension of sets having full μ-measure. We prove a formula relating HD (μ) to the entropy and Lyapunov exponents of the map. Other classical notions of fractional dimension such as capacity and Renyi dimension are discussed. They are shown to be equal to Hausdorff dimension in the present context.read more
Citations
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Determining Lyapunov exponents from a time series
TL;DR: In this article, the authors present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time series, which provide a qualitative and quantitative characterization of dynamical behavior.
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Measuring the Strangeness of Strange Attractors
TL;DR: In this paper, the correlation exponent v is introduced as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise, and algorithms for extracting v from the time series of a single variable are proposed.
Journal ArticleDOI
Ergodic theory of chaos and strange attractors
Jean-Pierre Eckmann,David Ruelle +1 more
TL;DR: A review of the main mathematical ideas and their concrete implementation in analyzing experiments can be found in this paper, where the main subjects are the theory of dimensions (number of excited degrees of freedom), entropy (production of information), and characteristic exponents (describing sensitivity to initial conditions).
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Percolation, statistical topography, and transport in random media
TL;DR: A review of classical percolation theory is presented, with an emphasis on novel applications to statistical topography, turbulent diffusion, and heterogeneous media as discussed by the authors, where a geometrical approach to studying transport in random media, which captures essential qualitative features of the described phenomena, is advocated.
Journal ArticleDOI
The dimension of chaotic attractors
TL;DR: In this paper, the authors discuss a variety of different definitions of dimension, compute their values for a typical example, and review previous work on the dimension of chaotic attractors, and conclude that dimension of the natural measure is more important than the fractal dimension.
References
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