Dimension Estimates for Invariant Sets of Dynamical Systems

Volker Reitmann
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Volker Reitmann: Fachrichtung Mathematik, Technische Universität Dresden

A chapter in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, 2001, pp 585-615 from Springer

Abstract: Abstract We investigate the relationships between various types of dimension like characteristics (e.g. Hausdorff, box, and Lyapunov dimensions, topological entropy) of dynamical systems and the geometric complexity of invariant and limit sets (e.g. isolated equilibria, global attracting or unstable cycles, products of Cantor-type sets and regular objects). We consider both flows and discrete-time dynamical systems of smoothness C 1 on n-dimensional smooth Riemannian manifolds which possess compact invariant sets. We formulate our results in terms of the singular values of the linearized evolution operator and consider additionally global informations such as the homology group and curvature properties of the manifold, natural Lyapunov functions and Losinskii norms, and the degree of non-injectivity of the generating map.

Keywords: Riemannian Manifold; Lyapunov Exponent; Lyapunov Function; Hausdorff Dimension; Dimension Estimate (search for similar items in EconPapers)
Date: 2001
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-56589-2_25

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DOI: 10.1007/978-3-642-56589-2_25

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