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Ergodic Theory : Fractal Geometry

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

The connection between fractal geometry and dynamical system theory is very diverse. There is no unified approach and many of the ideas arose fromsignificant examples. Also the dynamical system theory has been shown to have a strong impact on classical fractal geometry. In this article thereare first presented some examples showing nontrivial results coming from the application of dimension theory. Some of these examples require a deeperknowledge of the theory of smooth dynamical systems then can be provided here. Nevertheless, the flavor of these examples can be understood. Then thereis a brief overview of some of the most developed parts of the application of fractal geometry to dynamical system theory. Of course a rigorousand complete treatment of the theory cannot be given. The cautious reader may wish to check the original papers. Finally, there is an outlook over themost recent developments. This article is by no means meant to be complete. It is...

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Abbreviations

Dynamical system:

A (discrete time) dynamical system describes the time evolution of a point in phase space. More precisely a space X is given and the time evolution is given by a map \( { T\colon X\to X } \). The main interest is to describe the asymptotic behavior of the trajectories (orbits) \( { T^n(x) } \), i. e. the evolution of an initial point \( { x\in X } \) under the iterates of the map T. More generally one is interested in obtaining information on the geometrically complicated invariant sets or measures which describe the asymptotic behavior.

Fractal geometry:

Many objects of interest (invariant sets, invariant measures etc.) exhibit a complicated structure that is far from being smooth or regular. The aim of fractal geometry is to study those objects. One of the main tools is the fractal dimension theory that helps to extract important properties of geometrically “irregular” sets.

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  1. Center for Mathematical Sciences, Lund University, Lund, Sweden

    Jörg Schmeling

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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Schmeling, J. (2009). Ergodic Theory : Fractal Geometry . In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_179

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