Unwrapping eigenfunctions to discover the geometry of almost-invariant sets in hyperbolic maps |
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Authors: | Gary Froyland |
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Institution: | School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia |
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Abstract: | The numerical approximation of Perron-Frobenius operators allows efficient determination of the physical invariant measure of chaotic dynamical systems as a fixed point of the operator. Eigenfunctions of the Perron-Frobenius operator corresponding to large subunit eigenvalues have been shown to describe “almost-invariant” dynamics in one-dimensional expanding maps. We extend these ideas to hyperbolic maps in higher dimensions. While the eigendistributions of the operator are relatively uninformative, applying a new procedure called “unwrapping” to regularised versions of the eigendistributions clearly reveals the geometric structures associated with almost-invariant dynamics. This unwrapping procedure is applied to a uniformly hyperbolic map of the unit square to discover this map’s dominant underlying dynamical structure, and to the standard map to pinpoint clusters of period 6 orbits. |
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Keywords: | 05 10 -a 05 45 -a 02 60 -x 02 30 Uu |
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