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August  2007, 17(3): 671-689. doi: 10.3934/dcds.2007.17.671

On Ulam approximation of the isolated spectrum and eigenfunctions of hyperbolic maps

Gary Froyland 1,

1. 

School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia

Received  February 2006 Revised  July 2006 Published  December 2006

Perron-Frobenius operators and their eigendecompositions are increasingly being used as tools of global analysis for higher dimensional systems. The numerical computation of large, isolated eigenvalues and their corresponding eigenfunctions can reveal important persistent structures such as almost-invariant sets, however, often little can be said rigorously about such calculations. We attempt to explain some of the numerically observed behaviour by constructing a hyperbolic map with a Perron-Frobenius operator whose eigendecomposition is representative of numerical calculations for hyperbolic systems. We explicitly construct an eigenfunction associated with an isolated eigenvalue and prove that a special form of Ulam's method well approximates the isolated spectrum and eigenfunctions of this map.
Keywords: almost-invariant set, Ulam's method, isolated spectrum, eigenfunction, hyperbolic map., Perron-Frobenius operator.
Mathematics Subject Classification: Primary: 37M25, 37C30; Secondary: 37D2.
Citation: Gary Froyland. On Ulam approximation of the isolated spectrum and eigenfunctions of hyperbolic maps. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 671-689. doi: 10.3934/dcds.2007.17.671
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