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Dimension Theory of Smooth Dynamical Systems

In: Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems

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  • Jörg Schmeling

    (Fachbereich Mathematik und Informatik, Freie Universität Berlin)

Abstract

One of the basic properties of dynamical systems is that local instability of trajectories gives rise to a global “chaotic” behavior. This local instability can be described as some kind of hyperbolicity. Smooth Ergodic Theory investigates the metric and stochastic properties of measures invariant under differentiate mappings or flows on manifolds. The consideration of invariant measures allows to “tame” the “chaotic” behavior from a probabilistic point of view. This transition from differentiable structures to measurable structures and vice versa makes this field fascinating and paves the way to applications far beyond this field. Due to its generality the methods and results of Smooth Ergodic Theory entered areas as Riemannian Geometry, Number Theory, Statistical Physics, Partial Differential Equations or Numerical Simulations.

Suggested Citation

  • Jörg Schmeling, 2001. "Dimension Theory of Smooth Dynamical Systems," Springer Books, in: Bernold Fiedler (ed.), Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, pages 109-129, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-56589-2_5
    DOI: 10.1007/978-3-642-56589-2_5
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