## Abstract

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.