Unwrapping eigenfunctions to discover the geometry of almost-invariant sets in hyperbolic maps
نویسنده
چکیده
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. c © 2007 Elsevier B.V. All rights reserved. PACS: 05.10.-a; 05.45.-a; 02.60.-x; 02.30.Uu
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