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Dimension Estimates on Manifolds

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Attractor Dimension Estimates for Dynamical Systems: Theory and Computation

Part of the book series: Emergence, Complexity and Computation ((ECC,volume 38))

Abstract

In this chapter generalizations of the Douady-Oesterlé  theorem (Theorem 5.1, Chap. 5) are obtained for maps and vector fields on Riemannian manifolds. The proof of the generalized Douady-Oesterlé  theorem on manifolds is given in Sect. 8.1. In Sect. 8.2 it is shown that the Lyapunov dimension is an upper bound for the Hausdorff dimension. A tubular Carathéodory structure is used in Sect. 8.3 for the estimation of the Hausdorff dimension of invariant sets.

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Authors and Affiliations

  1. Department of Applied Cybernetics, Faculty of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia

    Nikolay Kuznetsov & Volker Reitmann

  2. Faculty of Information Technology, University of Jyväskylä, Jyväskylä, Finland

    Nikolay Kuznetsov

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  1. Nikolay Kuznetsov
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  2. Volker Reitmann
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Correspondence to Nikolay Kuznetsov .

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Kuznetsov, N., Reitmann, V. (2021). Dimension Estimates on Manifolds. In: Attractor Dimension Estimates for Dynamical Systems: Theory and Computation. Emergence, Complexity and Computation, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-030-50987-3_8

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  • DOI: https://doi.org/10.1007/978-3-030-50987-3_8

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